I agree with you that a uniform probability measure on N is impossible. I think the resolution to this paradox lies elsewhere. — keystone

Perhaps, but you are kind of all over the map in what follows.

While I initially used the rolling of the die to visually express my idea, your critique concerning the non-existence of perpetual motion machines suggests that my approach failed. Let me therefore explain in broader terms: — keystone

I didn't critique perpetual motion. I just asked, since you seem to have gravity but not friction, what are the rules of your physics?

Objects

I see two possible resolutions to the paradox: — keystone

The business with the processes and objects doesn't seem to bear on the problem at hand.

1) The game never starts because infinite objects don't exist.
2) The game never ends because infinite processes never terminate. — keystone

There are no infinite processes. You stick your hand into God's fishbowl and pull out a ticket and read the number. I don't understand why you're attacking the premises of your own problem. Conceptually, we pick an arbitrary natural number. That's very straightforward. You're just confusing yourself by going into all these different directions.

I'm not suggesting that labeling the undecided state as (Win or Lose) is enigmatic. However, the notion of a superposition of multiple states isn't generally embraced by mathematicians and philosophers. If it were, why wouldn't we resolve the Liar's Paradox by accepting (True or False) as its core solution, or use (Alive or Dead) to solve the Unexpected Hanging Paradox, as I have previously proposed? — keystone

I don't think pop quantum theory is helpful here. I was only pointing out the superpositions are not all that mysterious. They're just linear combinations in the state space.

My argument is that limits correspond to processes, not objects. I know textbook problems are often handpicked where shortcuts can be used to determine the limit (e.g. L'Hopital's Rule). In such a case, you can exibit your work (the object) and you're set. Seems like an object, right? However, the vast majority of limits don't allow for shortcuts and involve the unending work of narrowing epsilon further and further (let's put a pin on this idea of shrinking intervals). There's no complete object you can exhibit and say that that's the limit. The best you can do is work through the unending process. That's why I believe that fundamentally limits correspond to processes. — keystone

I don't think discussing the foundations of calculus is all that helpful either. I really think you have a lot of things in your mind and you're just tossing them out. There is an interesting problem that you originally posed, but this is going nowhere.

So do I believe in pi and all of it's usefulness? Yes, BUT I believe it corresponds to a process. Just as I believe 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... describes a process not an object. — keystone

That's fine. It's perfectly ok to identify computable real numbers with the algorithms that generate them.

However, noncomputable real numbers exist, and they do not have algorithms.

Please consider my version of the Stern-Brocot Tree: — keystone

I fail to see the relevance. This is just not helpful. Not to me, anyway. You seem to want to discuss the nature of the real numbers, but that's very far afield from the original question.

This is paradox screaming at us telling us that we're missing something. And at the heart of the issue is our belief that calculus is a study of objects (real numbers as if they were vertices on the tree), not processes (reals as if they described an endless journey down the tree corresponding to ever shrinking intervals). — keystone

If so, fine. But far afield again. Real numbers are not vertices on the tree, all the vertices are rational.

I'm really enjoying our discussion and finding it incredibly beneficial. Thank you for your patience and the knowledge you share. I feel very lucky to have you sticking around. — keystone

I'm glad to help. I wonder if we could have a more focussed conversation. The bit with the Stern-Brocot tree threw me for a loop. I have no idea where you were going with that. Wasn't there a thread about that on his board a while back? Here it is.

https://thephilosophyforum.com/discussion/14273/real-numbers-and-the-stern-brocot-tree

Is your concern with the nature of the real numbers? That's really got nothing to do with the original post, which is trying to find a logical basis for Adam's strategy of always switching.

You know, there's a thing called the counting measure. The Wiki article's not very good, gets too technical. The idea is that the measure of a set is its number of elements, or infinity if it's infinite.

In this case, the serpent's choice partitions the natural numbers into a left-hand segment, with finite counting measure; and the right hand segment, with infinite counting measure. In that sense, the right-hand segment is always larger, and that "explains" the strategy.

The problem is that counting measure is not a probability measure because the total measure's not 1. But I think that's a sensible way to resolve the problem.

I've given much thought to your critiques of my proposed resolution, and largely, I find myself in agreement with you. While I believe I'm onto something profound, my arguments have been somewhat muddled, and I've mistakenly mixed up the concepts of the null set with infinite sets. I aim to refine my approach moving forward. — keystone

After reading this, I feel I must have totally misunderstood your post. I thought you were trying to put a sensible probability measure on $\mathbb N$ that formalizes the obvious intuitive correctness of Adam always switching. But now it seems you are more interested in the physical sense of rolling an infinite die. I thought that got dispensed with early on. Of course this problem is only an abstract thought experiment, there are no infinite-sided dice. But physical infinite-sided dice seems to be what you are interested in. I am confused.

The more faces a die has, the more it needs to bounce around to ensure fairness. — keystone

There is no die bouncing around. This is not a physical experiment. There is no physical infinite-sided die. That's why I suggested God's fishbowl, containing a countable infinity of identical movie theater tickets.

If you are concerned with bouncing, then we are not even having the same conversation, and never were.

Infinite faces - The game never concludes since the dice continue bouncing indefinitely. — keystone

No friction? A spherical ball bearing, machined to be as perfect a sphere as engineers can create in this world, set rolling on a perfectly smooth steel floor, will eventually come to a halt with exactly one point at the highest height. So you are proposing a physical experiment, but with alternate physics.

What are the rules for your alternate physics? Why is there no friction?

Previously, I incorrectly conflated the null set with infinite sets. It was largely because I incorrectly conflated 0 faces with infinite faces because their histories both summarize to (W or L). However, I failed to appreciate that their histories are fundamentally different—one doesn’t begin, while the other never ends. — keystone

Can't parse this.

Thus, my answer to the paradox is that the narrative isn’t fairly told because when Adam opens his eyes, he should see the dice still in motion. In such an undecided state, it doesn’t matter whether he chooses to switch rolls with the serpent or not. — keystone

I am certain we have never been having the same conversation. I thought you were interested in putting some sort of measure on the set of naturals that makes Adam's switching strategy rational. But apparently not.

As long as the roll can't be completed, there is no paradox. This raises a more significant question: what, if any, endless processes can be completed? If supertasks are unachievable, does this imply that infinite objects are also impossible? — keystone

What do you mean by object? Physical objects? Those are impossible, unless you believe in the eternal inflation theory of cosmology.

In math, we have infinite objects all the time. Even finitists, who deny the axiom of infinity, still allow for the endless collection of the succession of natural numbers. They just don't allow it to be called a set.

SETUP PART 1 may seem superfluous but inclusion of this undecided state is extremely important to my approach to resolving paradoxes. Take a look at my recent post about the Unexpected Hanging Paradox. I believe the universe uses this same approach to avoid paradoxes/singularities, but in physics speak this (W or L) state would be called a superposition. — keystone

A superposition is just a linear combination of states, in principle no more mysterious than the fact that the point (1,1) in the plane is the linear combination (1,0) + (0,1). You are throwing spaghetti at the wall now.

I do not reject the value of limits and their importance at making calculus rigorous, however I interpret them to describe a journey not a destination. In other words, when I consider the limit of 1/x at x = 0, I do not see a need to say that the there is a destination at x=0 corresponding to number called infinity but rather I see an unending journey to increasingly and unboundedly larger function values as we approach x = 0. While you may agree that there is no destination in this case, we would end up disagreeing on a lot of other limits where the limit is a real. — keystone

The formal definition of a limit, the epsilon-delta definition, is perfect rigorous and leaves no room for metaphysical ambiguity.

With my view, reals retain all of their value in calculus, they just aren't numbers in the sense that rationals are numbers. In summary, I think that limits, the reals, and calculus represent significant achievements, but they require a fresh philosophical interpretation. — keystone

You don't believe in the real numbers? Ok. I don't think it would be productive for me to respond to this paragraph here. But if you start a thread entitled, "Do the real numbers exist?" I could talk about that all day. Yes they have mathematical existence. And unless there is a great leap forward in physics someday, they do not have physical existence.

Now suppose we play the same game, but with rationals.
— fishfry

I think the dice would keep bouncing around and so Adam's status would remain undecided (W or L). — keystone

Why? Just use the theater ticket metaphor, and label each ticket with a rational number, using your favorite bijection between the naturals and the rationals.

I am baffled that you are hung up on the idea of a physical infinite-sided die; and that in your conception, such a die would bounce forever, violating the known laws of physics that include friction.

I am also chagrined that I misunderstood your OP so completely that I jumped in at all. We have not been having the same conversation since the beginning.

It's impossible to place uncountably many numbers on countably many sides. — keystone

Just as there is a conceptual countably infinite-sided die, there is a conceptual real-number sided die.

But unlike the naturals or the rationals, there is indeed a uniform probability measure on the unit interval of real numbers. As I indicated a while back, Adam should switch if and only if he rolls a real number between 0 and 1/2. And there is a rigorous mathematical basis for that conclusion.

It seems like you're venturing into the realm of the Dartboard Paradox. If every point has a probability of zero of being hit, how could any point on the dartboard possibly be hit? — keystone

Every point on the dartboard has probability zero. The total probability is 1, assuming the dartboard has area 1. That's consistent with Kolmogorov's axioms.

Additionally, how does this reconcile with Kolmogorov's axiom that the sum of the probabilities of all possible events must equal 1? — keystone

Kolmogorov requires only countable additivity. That's the point. Nobody knows how to logically account for the fact that uncountably many zero-area points can sum up to a positive area. We just accept it, and we have many formalisms to express it. The length of the unit interval of the real numbers is 1, even though each point has length 0. It's a philosophical mystery, but a mathematical fact.

That being said, I do see the value in Measure Theory and the concept of probabilities on continua. Those aspects make sense to me. — keystone

I'm glad if I said something you found useful.

Why is there gravity but no friction in your alternative physics? Why wouldn't the die just float up into the air?

I'll close with this xkcd, which I just ran into yesterday.

https://xkcd.com/704/

On further reflection the infinite sided die shouldn't need a choice axiom in its construction (e.g a sphere can be painted by working clockwise and outwards from a chosen pole — sime

Is it possible you're misunderstanding what the axiom of choice says? It's surely not contradicted or made irrelevant by painting a sphere.

But then what of the idea of rolling said die an actually infinite number of times? — sime

In the problem posed by the OP, there are exactly two rolls: one by the serpent, and one by Adam. Not sure what rolling infinitely many times has to do with this.

That surely is equivalent to choice, — sime

Why?

In the Peano axioms I can invoke the successor function infinitely many times, but not only isn't there a need for any choice axioms, there isn't even a concept of sets.

I took the idea to mean that the faces of an infinite die isn't a well-ordered set — sime

The natural numbers are well ordered in their usual order.

Any countably infinite set can be well ordered simply by bijecting it to $\mathbb N$ and using the order induced by the bijection.

, unless the Axiom of Countable Choice is assumed. — sime

No choice is needed to well-order the natural numbers or any countably infinite set.

If this axiom isn't assumed, then the sides of the die can only be ordered in terms of their order of appearance in a sequence of die rolls, which implies that unrolled sides are indistinguishable. — sime

I don't follow this. The real numbers in their usual order are not well ordered, but they are certainly distinguishable.

You don't need to well order a set to distinguish its elements. The elements of a set are all distinct from each other by the the axiom of extensionality. A set is completely characterized by its elements. There are no duplicates.

Are you saying the remaining five faces on a standard 6-sided die can't be distinguished if we've only rolled it once?

fishfry: I previously posted a message here but have decided to retract it and spend more time reflecting on the comments before continuing our discussion. I'll get back to you in the next couple of days with a more considered response. Apologies if you were already in the process of replying! — keystone

I just came here late at night to reply before bed but I'll stand by for further developments.

For what it's worth, the fact that we can't put a uniform probability measure on the natural numbers doesn't mean they have to be "all the same number." They're all different numbers. And I can't understand the idea you're getting at.

You know, here's a variation that challenges the strategy of Adam always taking the serpent's number.

As you know, the rational numbers are countably infinite. Now suppose we play the same game, but with rationals. We can even use the same die. All we need to do is repaint the faces, replacing natural numbers with rationals.

But now, whatever number Adam picks, there are infinitely many numbers smaller, and infinitely many larger. There is now no sensible strategy at all.

Suppose we play the game in the unit interval of reals, which has total measure 1.

If we were playing with a real-number sided die, there's an obvious strategy. I gave this example earlier. If Adam's number is less than 1/2, he should switch; else not. Because the measure, or probability, of an interval of real numbers is its length.

But if we now play the game with the rational numbers in the unit interval, that no longer works. The unit interval of rationals has measure 0 (because all countably infinite sets have measure 0).

If Adam picks, say, 1/googolplex, a tiny number, the serpent's number has probability 0 of being to the left, and probability 0 of being to the right.

But if we conceptually filled in the rationals with the rest of the reals, then we can assign sensible probabilities and Adam should switch.

I do believe this is a pretty good paradox or at least a highly counterintuitive situation.

Note please that we can analyze this situation, and be completely confused by it, without the need to deny infinite sets. That seems to complicate the issue. Once we are thinking about choosing a random natural, we are already contemplating making a selection from an infinite collection, whether or not we call it a set.

Ok I'll leave all this to you.

That's fair. I didn't actually think you were making an argument, it just didn't know where you were coming from. To me it looked like you had misunderstood the intention of my quote. — Apustimelogist

I was trying to be helpful, but I only ended up hijacking and essentially terminating the thread. Not my intention at all. Perhaps someone has something interesting to say about information, randomness, entropy, and murky Youtube videos. I've seen better work from Veritasium.

In practise the math always refers to something. — Metaphysician Undercover

In application, math is used to refer to something. But math itself does not refer. This was the great philosophical insight that followed the discovery of the logical consistency of non-Euclidean geometry.

It's actually an ongoing process, the evolution of thought. Look at Russel's paradox for example. — Metaphysician Undercover

In the evolution of thought, people are going to decide math is wrong because it doesn't actually refer to anything? I thought that was a feature.

I see what's going on with the Lounge. All the political posts, Trump and Ukraine and Gaza, got moved over there.

What kind of philosophy considers only lofty, abstract issues, and turns its face from matters of life and death in the world today? Of course those topics generate heat. From what I can see, the Lounge is now the best part of this site.

Have we hopelessly hijacked this thread? What happened to information and randomness?

I believe this probability chart captures the all of the essentials of the infinite dice game and yet I do not see how it violates Kolmogorov's treatment. Can you explain? — keystone

I didn't understand the picture.

I wouldn't venture to disagree with Kolmogorov on this matter. — keystone

But you are. You are trying to develop a probability measure on $\mathbb N$, which contradicts the Kolmogorov axioms. So embrace your infinite-measured probability distribution on $\mathbb N$ and see if you can make it work.

Although it's not an easy undertaking, particularly for me, I find it relatively easier to contest the notion of infinite sets than to formulate rules for a system of probability spaces with actually infinite measure. — keystone

Correct. It's difficult to make your idea work.

But you are really conflating two entirely different topics. One, your puzzler is interesting. But two, you're promoting some sort of finitistic argument that's entirely beside the point.

I'm not saying I've convinced you, — keystone

You convinced me.

but no one—especially a mathematician—has ever responded like this to my mathematical/philosophical thoughts. — keystone

That's the second time in two days that I've been called a mathematician here. If only. It was once a failed dream. I studied math in school many years ago and keep up a bit online.

It makes me feel a bit less out of my mind. Thank you. — keystone

You're welcome.

You're likely familiar with the Principle of Explosion, where a single contradiction can undermine an entire logical system. I have a different take on what 'explosion' actually means, perhaps because I hold consistency paramount. Let's consider my system of arithmetic, which starts with universally accepted statements such as:

Statement 1: 1+0=1
Statement 2: 1+1=2
Statement 3: 1+2=3
Statement 4: 1+3=4

These statements are not in question. Now introduce the following into the system:
Statement 5: 1+2=2

To maintain logical consistency in this updated system, our only choice is to accept that 0=1=2=3=4=... Realizing this, the system remains consistent but becomes trivial and loses all distinction. This situation resembles a singularity, where distinctions that exist in more sensible systems dissolve.

Moreover, dividing by zero (a classic error leading to mathematical singularities) can yield absurdities like Statement 5. You likely have come across arithmetic tricks using division by zero to demonstrate fallacies like 1=2. (EXAMPLE)

Returning to the infinite-sided dice game, consider the successor function S(). Some statements would be:
Statement 1: S(0)=1
Statement 2: S(1)=2
Statement 3: S(2)=3
Statement 4: S(3)=4

I'm not suggesting these statements are incorrect or trivial. However, if we theoretically extend this pattern infinitely, insisting on a complete sequence of natural numbers, then we must accept 0=1=2=3=4=5=... In an infinite set, natural numbers lose their distinctiveness. Even if different sides of a die show different numbers of dots, in an infinite scenario, every roll results in a tie because all numbers effectively become one. The real twist in the story is that Adam lost everything for nothing—the game invariably ends in a tie but Adam never lets it end, a truly cunning maneuver by the serpent. — keystone

This entire passage is insane. I only say that to clarify that when you said that I have made you feel less insane, that is not my intention. My purpose in posting was to note that there is no uniform probability measure on the naturals. I offered no opinions on the sanity of any member of this forum. Would you like me to? Oh my.

Does it still seem like the information content of my idea is nil? — keystone

That last bit about your version of the principe of explosion, and your idea of infinite sets containing only one element, has made things considerably worse.

Let me give it another go. In calculus, we handle singularities by using limits to approach (but never actually reach) a singularity. We can apply a similar principle here. If I roll a 42, my probability of winning can be illustrated as follows:

Number of faces, Probability of Winning
42, 1
100, 0.42
1000, 0.042
10000, 0.0042

As the number of faces approaches infinity, my probability of winning approaches zero. However, it never actually reaches zero because we never consider a truly infinite-sided die—it simply doesn't exist. — keystone

You reject the modern theory of limits? This is a real problem. The Adam puzzler is genuinely interesting. You are confusing the issue by rejecting infinite sets. If there are no infinite sets there's no game in the first place. What do you mean an infinite sided die doesn't exist? A googolplex-sided die doesn't exist either, but we can still put a uniform probability measure on it. Nothing in math "exists" in a physical sense.

In my initial example with four arithmetic statements, they seemed meaningful, right? Each one features a unique set of type characters, creating the impression of distinct statements. — keystone

Yes, you're defining the names of the Peano natural numbers.

However, as I explained, in the context of an inconsistent system, they lose significance. We might as well condense them into a single statement: 0=1=2=3=.... The situation is similar with the concept of an infinite die. Each face of the die appears different, suggesting a variety of numbers, but upon closer examination, we realize the distinctions are superficial. The dots essentially hold no value. We might as well be dealing with a die that has only one face. — keystone

Nutty.

I subscribe to the concept of completed infinite sets, — keystone

So you're NOT promoting a finitistic agenda.

but with a twist: I believe they encompass just one unique element. — keystone

1 is a finite number.

As a related example, when natural numbers are defined as nested sets of empty sets, I don't perceive an infinite collection of distinct objects; instead, I see a single entity: the void - emptiness. — keystone

The set {{{{}}}} is a singleton. It has cardinality 1. It's finite. You are making no sense.

"The void?" Did Nietzsche say that when you look into the empty set, the empty set always looks back? Maybe he should have. You're just waving your hands and not saying anything meaningful.

Conventionally, we begin with natural numbers and develop our systems upward from there. I contend that this approach is fundamentally backwards, though that's a conversation for another time. — keystone

What?

Ew. Actual infinitesimals are no better than actual infinities. — keystone

Actually I was trying my best to be charitable. With your talk about Newtonian infinitesimals and 17th century calculus, I thought nonstandard probability theory was exactly what you were getting at. If not, then I can't conceive of any referent for your explanations.

Considering the significance I attribute to the tie outcome in resolving the paradox, it's surprising how carelessly I addressed it in my previous two messages to you. — keystone

Ties seem incidental to the problem. I'd just ignore them.

I assume you understand my general stance, — keystone

I'm absolutely baffled by it. I mean that. I can not find any referent to your ideas. I don't know what you are talking about. Singleton sets are not infinite, for example. They have cardinality 1. Every set may only be nested downward a finite number of times, so there is nothing to be said about them.

3) The probability of each event is 1. — keystone

Fine. Then your total probability space has measure $\infty$. Nothing inherently wrong with that, you just have to try to make it work.

Number of faces, Probability of Winning
42, 1
100, 0.42
1000, 0.042
10000, 0.0042 — keystone

All true for any finite number of faces.

Earlier you tried to make some sort of limit argument, but that doesn't work. If you assign probability 0 to each natural number, countable additivity requires that the total probability is 0.

Wow, if the last time you went to a theater they were still using those raffle-like ticket stubs, you've missed out on quite a few great theater experiences. — keystone

Sticky floors and noisy patrons? I watch at home these days. I hadn't gone to a live theater in many years, then I went once for a film I absolutely had to see, and then I remembered why I don't go to theaters.

You definitely need to see the next Avatar movie in the theater in 3D. — keystone

I didn't see the last one in any dimension. Not my cup of tea. Just checked the IMDB page. Maybe I'll watch it sometime.

Are you certain? By definition, a roll of tickets that has no end can't be completed (for that would mark the end of the roll) — keystone

I don't see why not. Hilbert's hotel is completed. That's the entire point of the exercise. Conceptualizing infinite sets. I visualize God's ticket roll just like Hilbert's hotel. Hilbert's movie theater.

If you reject infinite sets that's perfectly legitimate, but it's an entirely different discussion.

Your OP contemplates randomly choosing an element of $\mathbb N$. If you reject the mathematical existence of infinite sets, you have no $\mathbb N$ and you have no game.

attempting to do so is akin to trying to create a married bachelor. Nevertheless, I agree that God could do it, though it would mean losing the distinction between numbers in the first example and words in the second. — keystone

I just don't follow your idea that all the natural numbers are actually one single number, yet the set containing it is infinite, yet you don't believe in infinite sets.

Anyway I may have lost a paragraph or two in my trying to respond to several comments at once, so let me know if I missed anything.

Whatever your point or vision is about the infinite singleton, I can't figure it out, nor how it would bear on the problem in the first place.

That completely makes sense. However, not every mathematician is as reasonable as you are. If you look at what TonesoffTheDeepEnd is writing here, you'll see great effort to support some kind of formal identity theory. That is not a "casual shorthand for the condition expressed by the axiom of extensionality". — Metaphysician Undercover

I'm not a mathematician. I studied math in school, long ago.

Thanks for calling me reasonable.

I can't defend the views of other posters, and I can't engage with what someone else might have said.

Regarding placement of threads: Some of the moderation of this forum is quite irrational. — TonesInDeepFreeze

For sure. The mods have an aversion to math-related topics that I don't understand.

t's in the Lounge.

It was deemed not Philosophical enough, or just math. Or lousy math. — ssu

Thanks.

Having to do with the appearance that a Philosophy Forum site would discuss eloquently Philsophy, I guess. — ssu

I'm sure that might happen someday .... /s

That's very observant of you fishfry. Generally speaking, one's language use is a reflection of the conventions and habits which the society has immersed that person in. I like to seek, determine, and then exaggerate within my own usage, the various ambiguities, misleading implications, false representations, and overall misgivings of deceptive habits and conventions which permeate our communications, thereby laying them bare, exposed for the world to see, so that perhaps, at some point in time, the general population will start to realize that something needs to be done about this situation. — Metaphysician Undercover

Your thesis is that someday, Internet archeologists are going to discover this thread and go, "My God, math is wrong!"

Wayfarer knows me as the obfuscator. — Metaphysician Undercover

Wish I'd figured that out long ago.

Why is your opinion of particular relevance? — wonderer1

Quite right. Carry on.

there is an isomorphism between the sort of information processing that occurs in modern AIs and a substantial amount of the information processing that occurs in our brains. — wonderer1

I am quite late to this thread and have not read any of it, so my comment is based only on this one post. But this is an important point, one I take great exception to.

What you claim is an isomorphism, I claim is an equivocation ("calling two different things by the same name"), an informal fallacy resulting from the use of a particular word/expression in multiple senses within an argument.

The information processing in a digital computer is nothing at all like the "information processing" in a brain.

In the computer, information is a bitstring, a sequence of 0's and 1's. The bitstrings are processed in a finite state machine. If you conceptually allow arbitrary amounts of memory you have a Turing machine. We know exactly what Turing machines can compute and what are their limits, things they can not compute.

Brains -- I can't believe I even have to explain this. Brains don't work this way. They don't have an internal clock that inputs the next bit and flips a pile of yes/no switches and takes another step along a logic path. Neurons are not bits, and connections between neurons are mediated by the neurotransmitters in the synapses between the neurons. It's a very analog process in fact.

I know the idea you expressed, "Computers process information, brains process information, therefore computers = brains" is a very popular belief among highly intelligent and competent people who in my opinion should know better.

You have no mapping from a Turing machine to the brain. You have no isomorphism. You have a bad metaphor that leads people to false conclusions.

Can you see that you are making a metaphor and that you don't actually have an isomorphism, which is first and foremost a mapping? What's your mapping from the concept of a Turing machine or finite state machine, to the brain?

Neural nets are a clever way to organize a computation. They are not, in my opinion, the way to AGI. Neural nets are only about their input corpus and training. You can't get creativity from that.

My apologies, for misrepresenting what we argued about. I thought you argued that the axiom of extensionality indicated identity. — Metaphysician Undercover

Apology accepted. I do see how my view may have seemed that way to you. For example I am certain that I'd have maintained that if A and B are sets, and we can write A = B, then A and B are the same set.

That is certainly true in set theory. But I think it's really more true in the metatheory or the way we talk about set theory, than set theory itself.

In set theory, "same" is a shorthand for "satisfies the premises of the axiom of extensionality." You are trying to overload the word with metaphysical baggage that it simply does not have in math. The axiom of extentionality is syntax. You are imbuing it with semantics that you are making up or bringing over from other meanings of the word you may know. You need to take things on their own terms when studying any technical field.

What I mean is, formal set theory says:

"If such-and-so, then we can write A = B."

In casual, everyday talk about set theory, we say, "A and B are the same set if they have exactly the same elements."

So you see there is a gap between those. Set theory is a purely syntactic exercise. If, given the definition of "=" as in the axiom of extensionality, we can derive a formal proof from the axioms that A = B, then we can write A = B from now on.

But all this talk of "sameness" is really a very loose casual adaptation of the axiom of extensionality. And in so doing, we seem to add semantics to it. As if we are making a metaphysical claim that A = B.

But in actuality we are not doing that!! Rather, we're simply claiming that the symbol "=" is to henceforth be defined as this other condition.

So the set theory is syntactic; and it's a mistake to confuse our everyday casual talk about set theory, with some kind of ontological claim.

tl;dr: When a set theorist says two sets are "the same," there is a formal derivation from first principles that A and B satisfy the premises of the axiom of extensionality. It's a purely syntactic exercise.

They are NOT implying any kind of metaphysical baggage for the word "same." If pressed, they'd retreat to the formal syntax.

Make sense? You are using "same" with metaphysical meaning. Set theorists use "same" as a casual shorthand for the condition expressed by the axiom of extensionality. It's a synonym by definition. The set theorist's "same" is a casual synonym; your "same" is some kind of ontological commitment. So all this is just confusion about two different meanings of the same word.


Also, meta: This thread, "Infinity," is active, and I keep getting mentions for it and replying. But this thread does not show up in my front-page feed! Anyone seeing this or know what's going on?

You are adding "identity" — Metaphysician Undercover

I'm pretty sure I never said that, but if I did, please supply a reference to my quote.

when the law of extensionality is really a definition of "equal". — Metaphysician Undercover

Now you're getting it.

Nice to see you, fishfry on the forum again! It's been a while. — ssu

Thank you. This forum drives me to extended vacations sometimes.

It's from a channel now called Verisatium, although under an earlier name. The original video wasn't cited until the top of page 2, you can review it here https://www.youtube.com/watch?v=sMb00lz-IfE . (I've viewed quite a few Verasatium presentations and overall found them pretty good, but I'm very dubious about some of the claims in this one.) I can definitely see the relevance of the Kolmogorov complexity idea, the video would have been better if it had been informed by it. — Wayfarer

Veritasium, right. Thanks for the correction. I like Veritasium but didn't watch this video. He has some off days as we all do. If he mentioned compressibility but didn't credit Kolmogorov, I can see why that would lead people to think he was expressing a fringe idea, or whatever the criticism of the video was.

ps -- Just watched half the vid. Hopelessly muddled, definitely a misfire for Veritasium. I gave up when he started trying to claim that quantum mechanics is the cause of increasing information. This video was totally confusing. I can see why it sent the thread down a rabbit hole.

the unintelligible is adequately hidden within what is proposed as intelligible, and it will appear like you are saying something intelligent. — Metaphysician Undercover

You just described your own posting style.

Ha, ha, very funny. — Metaphysician Undercover

I wasn't being funny.

Are you suggesting that the gambling event can occur but that we can't discuss it in mathematical or probabilistic terms? That's hard to accept. — keystone

That's why I gave the example of throwing all the numbers into a bag and picking one. You remember those big rolls of tickets that movie theaters used to use, do they still have cardboard movie tickets? I haven't been to a movie theater in a while.

Of course God, being God, has an infinite roll of tickets. He has an angel separate them all at the perforations and throw them into a big fish bowl. Then Adam reaches in and pulls out a ticket.

I can't think of any reason this is conceptually impossible. It's perfectly clear.

So what is it that we can't do? We can't do probabilistic reasoning about the ticket draw until we make all of our assumptions extremely clear. What's a probability, are you allowed to add and compare them, and so forth. Mathematically you can't do that without violating Kolmogorov's axioms.

There are other variations. You can drop countable additivity and replace it with finite additivity, which is strictly weaker but obviously not so problematic.

The issue isn't whether some alternate concepts and rules of probability might save the situation.

The issue is whether you can be super-duper crystal clear about the concepts and rules you are using. That's a standard you haven't met. The googolplex example is very murky, you are counting one point for each number and then comparing a finite total to an infinite one. But then your probability space has a total measure of infinity. That's not in and of itself illegal. It's just not Kolmogorov. So you have to tell me exactly what are the rules of your system of probability spaces of infinite measure.

Even if we set aside mathematical reasoning, can you truly say that you have no opinion on whether Adam should exchange numbers with the serpent? — keystone

I go back and forth on whether that's a meaningful question. I think I just convinced myself you're right. Once the serpent chooses, it's a heck of a lot more likely the next number will be in the unbounded segment, even if we can't formalize what we mean. You just convinced me. My intuition agrees with yours. You're right, it's a paradox.

I agree with your point and also agree with Kolmogorov's axioms. However, I think the flaw in your argument lies in presenting a false dichotomy by suggesting that there are only two possible scenarios in the game.

1) The probability of each event is 0.
2) The probability of each event is some tiny positive number. — keystone

I hope it's clear that this is Kolmogorov's idea and not mind. But surely you're not suggesting that there could be negative probabilities. That seems like it would take us far afield. Also, probabilities don't have to be tiny. Kolmogorov's first axiom on Wiki is that probabilities are nonnegative real numbers. It's then a consequence of the total measure being 1, and the finite additivity (implied by countable additivity) that together show that each probability must be between 0 and 1 inclusive.

3) The probability of each event is 1. — keystone

Your next paragraph is a bit out there. But by the end I sort of think I might know what you are getting at.

Of course, this is only feasible if there exists just one natural number, meaning that when you deal with the set of all natural numbers, you are essentially dealing with a singularity where every natural number is identical. — keystone

Ok this is concerning. First, I don't know what a singularity is in math or set theory. It's a term from cosmology. There are singularities in math, but they are not to be found among the natural numbers. So you are tossing in a buzzword that should cause you to try to be much more precise about the idea you're getting at.

While this notion may seem preposterous, — keystone

That's the problem. It's not even preposterous. It's not a well-formed idea. All the natural numbers are the same number and they form a singularity? I'm sorry, there is not an idea there at all.

And besides, by the axiom of extensionality, a set can't have two elements that are the same. So you are conceptualizing a set {x} where x is some particular natural number or all the natural numbers, and you are calling it a singularity.

I hope I am being clear that the informational content of this idea is nil. I have no idea what you are trying to say.

similar issues emerged with calculus, which were resolved using limits. — keystone

I think you are making an analogy where there isn't one.

For instance, finding the tangent by dividing by zero results in a singularity, yet one can sensibly approach a zero denominator. — keystone

Yes, the limit of the difference quotient was 0/0 and Newton got the right answers but could not find the right formalism. Can't blame him, it took almost another 200 years. I take your point but you are a long way from your {x} "singularity," can you see that?

In a similar vein, I argue that dealing with the set of all natural numbers also results in a singularity — keystone

I'm not feelin' it. There are singularities in analysis, such as the blowup of 1/x at x = 0. In complex analysis they classify singularities in terms of how bad they are.

But singularities in the natural numbers? I must insist on your clarifying this point. It means nothing as far as I can understand.

, but probabilities can be sensibly managed by approaching an infinite set. — keystone

I don't know what that means.

In other words, infinite sets as completed objects do not truly exist. — keystone

Uh oh. You contradicted the game. You can't make a random choice from a bag that never contains all the numbers. God's fishbowl contains ALL the tickets. A completed set of tickets, a completed set of natural numbers.

If you reject completed infinite sets, you can't play the game in the first place. Right?

Although my proposed resolution has significant implications, I believe none of these are insurmountable. — keystone

I don't believe you expressed a coherent idea. I don't mean to say that as a reflection on you. I only intend to convey my own state of mind. I do not understand your idea. All the natural numbers are the same and you call them a singularity and suddenly you don't believe in infinite sets. You have lost me.

What do you think? — keystone

It occurs to me that perhaps you're getting at infinitesimal probability theory. After all what we'd really like to do is assign the probability $\frac{1}{\infty}$ to each natural number in such a way that the infinitesimals add up to 1, if we could only figure out how to make that rigorous.

There are researchers working in that area, but it's a bit arcane and I don't know much about it. I found a link.

https://www.journals.uchicago.edu/doi/full/10.1093/bjps/axw013

From that wiki page:

"or in words:

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B."

I see the phrase "A is equal to B", but where does it indicate that A is the same as B? — Metaphysician Undercover

It's better for me if I defer continuing this discussion at this time.

But for what it's worth, the symbol string "same" has no meaning in ZF. I do not know what it means, and I do not need to know what it means in order to do set theory.

The symbol "=" is being defined by the axiom of extensionality. You're adding things that aren't in the game. It's as if I'm trying to teach you chess and you say, "Where are the zebras?"

I just don't understand what the intention of your initial comment was. From my perspective it doesn't follow from the rest of the thread I was following. — Apustimelogist

Oh I see. I can explain what I had in mind.

The OP was inviting discussion on the intuitively confusing idea that a random sequence of symbols contains the most "information." Many people think that if you're transmitting information, it's NOT random. It's noise. So the compressibility argument seems to be saying that there's more information in the noise than in the signal. So it's a seeming paradox.

Then the thread as I understood it, though I was not reading closely, became a discussion of how sensible a definition of randomness that was. I gather this had been exposited by 3Blue1Brown Youtube channel, and some people were even questioning the video. I'm repeating all this from a very shallow glance at the sense of the thread over the last few days.

Hold that thought and let me change the subject for a moment. In another thread, a paradoxical puzzle was presented that seemed to violate our intuitions about how probability works when randomly choosing a natural number.

I happened to know the puzzle could be explained through an understanding of the axioms of probability devised by the Russian mathematician Andrey Kolmogorov in 1933. So I wrote that up over in that thread.

Then I remembered that it was the very same Andrey Kolmogorov who had come up with the idea that you could measure the randomness of a string of symbols by its incompressibility. And that this was a measure of information. Kolmogorov is both the probability guy and the incompressibility guy.

I said to myself, "In that other thread [meaning this one], people are discussing the compressibility idea, and whether it makes sense. I thought pointing the thread to the originator of the idea might help add context.

So I popped in over here to toss in a little factual nugget that was not intended to change anyone's mind about anything. I was not referencing anything in the thread. I only wanted to let people know where the compressibility idea came from, so they could place it in context in the study of information.

Just intending to add a factoid, not making any argument at all.

I can see how I could have contextualized my remark better.

That would suggest you are implying information is randomness; the original point of my post presupposes this is not necessarily the case. — Apustimelogist

I'm not Kolmogorov. I only identified him as the originator of the idea that randomness is measure by the degree of incompressibility. I said in my post, "There are other definitions of randomness in various other fields."

I'm having trouble following your posts. I gave Kolmogorov's definition then said there are other definitions. I have no disagreement with anything you wrote, but I don't see how it bears on what I wrote.

As I said, the point of K's idea is that when we give a finite-length description of the digits of pi, as we can easily do, we are showing that pi only encodes a finite amount of information. That's Kolmogorov's idea.

Suppose I want to transmit the sequence of the decimal digits of pi to my friend via telegraph, back in the day when telegrams were expensive. (Are telegrams still expensive? Do they still exist? I have no idea). I could sent each of the infinitely many digits. Or, I could just sent one of the closed-form, finite-length expressions for pi. Much cheaper. That's compressibility. It shows that the digits aren't random at all, but strictly deterministic.

Now, I can see how one would object to saying that shows pi doesn't contain much information. But it doesn't! I can express all of its infinitely many digits with a short, finite-length string of symbols. That takes very little information.

I am not planting a flag and saying this is the only definition of information. I'm just trying to motivate Kolmogorov's idea. And failing badly, apparently.

Yes, but then it is another issue how that might relate to what people call information. — Apustimelogist

There are other definitions of the word, as I indicated in my post. Back in the day you could dial 411 and a nice lady would come on the phone and say, "Information." Alas there are no more dials.

Incidentally, I argued extensively with fishfry, that to read the axiom of extensionality as indicating identity rather than as indicating equality is a misinterpretation. — Metaphysician Undercover

After much back and forth, you finally revealed to me that you do not understand the basic material implication $P \implies Q$ of propositional logic; and that you are incapable of understanding the statement of the axiom of extensionality on its Wiki page. I'm afraid I can't dialog with you further till you remedy these very basic misunderstandings.

Yeah, I guess that is fair. I haven't watched the video so I can't comment too much. Maybe it again comes down to this whole use of the word 'information' being ambiguous again. — Apustimelogist

The definition of randomness as incompressibility is due to Kolmogorov.

https://en.wikipedia.org/wiki/Kolmogorov_complexity

There are other definitions of randomness in various other fields.

The rationale of Kolmogorov's idea is that, for example, there is absolutely nothing random about the digits of pi. They're the deterministic output of any number of closed-form expressions that can be programmed on a computer. The same goes for every other familiar irrational constant like sqrt(2), $e$, the golden ratio, and so forth. These are all computable numbers. Their infinitely-long digit sequence can be deterministically cranked out by a computer program or mathematical expression having relatively few characters.

Whether this satisfies or offends anyone's individual philosophical sensibilities is a matter of personal preference.

Is this truly a paradox? If not, why not? — keystone

This interesting puzzler has a clear and unambiguous mathematical resolution.

First, the standard axioms of probability were laid down in 1933 by Russian mathematician Andrey Kolmorogov.

Kolmogorov is also the originator of the definition of randomness as incompressibility, currently being discussed in https://thephilosophyforum.com/discussion/15105/information-and-randomness

Kolmogorov's axioms of probability state (paraphrasing the Wiki exposition):

1) The probability of an event is a nonnegative real number.

2) The sum of the probabilities of all possible events is 1. This make sense, right? On a standard 6-sided die, each face has probability 1/6, and the probabilities of all the faces add up to 1.

3) If you have countably many mutually independent events, the probability that one of them occurs is the sum of the probabilities of the events.

This rule is designed to accommodate the following situation. Suppose we randomly pick a real number from the unit interval [0,1]. Endpoints don't matter for this discussion. We want to be able to say that the probability of choosing a number in [0, 1/2] is 1/2; and the probability of choosing a number in [1/2, 3/4] is 1/4; and the probability of [3/4, 7/8] is 1/8, and so forth. In other words, the probability of choosing a number in an interval is the length of the interval.

On the one hand, the chance of choosing in [0,1] must be 1. And we know from the theory of infinite series that 1/2 + 1/4 + 1/8 + ... = 1. Countable additivity lets us compute the sum of the whole by adding up the countably infinite collection of individual probabilities.

A probability distribution is uniform if each event has an equal probability. For example a standard dice roll is uniform, because each face has probability 1.

A familiar example of a non-uniform probability distribution is he bell curve, or Gaussian distribution. The total area under the curve is 1. But events near the middle have a much higher probability than events near the left and right tails.

Now we are in a position to prove the crux of the matter.

There is no uniform probability distribution on a countably infinite set of events.

Why is this? Say we have a countably infinite set of events {E1,E2,E3,…}.

Suppose we have a uniform probability distribution on this set of events. What must be the probability of each event? We have two cases:

- The probability of each event is 0. But then by countable additivity, the probability that any event at all happens is 0. Contradicts the axiom that says the total probability of the entire set of events must be 1.

- The probability of each event is some tiny positive number. But then the total probability is infinity, no matter how small each probability is. Another contradiction.

Conclusion: It is not possible to place a uniform probability distribution on a countably infinite set. In particular, there is no uniform probability distribution on ℕ
.

With this knowledge in hand, we can analyze this intuitively appealing but sadly meaningless statement:

With trepidation, Adam opened his eyes to see a googolplex dots on his die. It dawned on him then: with his roll being finite, only a finite number of outcomes were lower, yet an infinite number were higher, making the serpent's probability of beating him exactly 100%. — keystone

We can see what's wrong with this. First, you haven't got any kind of probability distribution at all. You're thinking that there are infinitely many events, with each event getting one vote. But that adds up to infinity. It's not a probability space. Remember the probabilities of each face on a six-sided die add up to 1. The same principle must apply to an infinite-sided die. So the logic of this paragraph makes no sense.

Secondly, you are trying to compare one set of nonexistent probabilities to another. Since there is no uniform probability distribution on ℕ, you are trying to reason about a nonexistent mathematical entity.

So that's the answer to the puzzler. There is no uniform probability distribution on the natural numbers that allows the quoted paragraph to have any meaning at all.

Even God can not place a uniform probability distribution on the natural numbers.

Now you COULD conceptually throw all the natural numbers in a bag and reach in and select one. But you could NOT then try to use mathematical reasoning on that situation. That's the flaw in the paradox. You can't reason mathematically about a uniform probability distribution on the natural numbers, because there isn't any such thing. You can't add the individual probabilities, because adding probabilities only applies to well-defined probability spaces, and you haven't got one. Even God does not have a uniform probability distribution on the natural numbers.


For additional insight, I'll give two variations on the game.

* An example of a non-uniform probability distribution on ℕ

For simplicity, let the natural numbers be 1, 2, 3, ... In other words I'm not considering 0 just to make the argument a little simpler; else I'd be off-by-one throughout.

Let the probability of the number n be $\frac{1}{2^n}$.
.

For example the probability of 1 is 1/2. The probability of 2 is 1/4. The probability of 3 is 1/8, and so forth.

The probabilities add up to 1, and countable additivity is true, so we have a probability distribution. It just doesn't happen to be uniform, since each event has a different probability.

Now, let's play the game. The serpent randomly obtains a number, let's say it's 5.

What should you do? The probability that you get 1, 2, 3, or 4 is 1/2 + 1/4 + 1/8 + 1/16 = 15/16.

The probability that you also get 5, tieing the game, is 1/32.

And the probability that you get a larger number is the tail of the distribution, which adds up to 1/32.

So you should switch.

It's clear that the second player is at a big disadvantage half the time. If the snake plays first, Adam should always switch.

The only time this fails is if the snake picks 1. That happens half the time, since 1/2 is the probability of randomly picking 1.

Then when you pick, half of the time you'll tie, by picking 1. The other half you win, by picking a larger number.

But if the snake picks any other number besides 1, which happens half the time, you should switch.

* Example two: A die with the cardinality of the reals that has every real number between 0 and 1 on some face, and no two faces have the same number.

In this case, say you randomly pick a number between 0 and 1/2. Chances are, the snake has a large one. You should switch.

If your number is greater than 1/2, you should keep your number.

If you get exactly 1/2, it doesn't matter what you do.

In conclusion:

* There are variations on the game that make sense and that have a strategy that confers an advantage on Adam.

* In the game as stated in the OP, it's meaningless to try to reason probabilistically. And you certainly can't try to count "finite versus infinite," since the sum of the probabilities of the events is infinite to start with. Probability spaces must have probability 1. So says Kolmogorov.

Let me know if this was sufficiently clear, and please let me know if there are questions.

Now, putting trivial matters aside, do you understand that Adam's probability of winning becomes exactly 0% once he sees his roll? If you disagree, what do you calculate his probability to be? — keystone

I hope you can see now that there is no sensible way at all to apply any numerical probability to the events in this game. Any number you pick violates the axioms of probability.

mispost

And your criticism is belied by the fact that the poster himself explicitly said that my answer was clear and helpful, and his followup questions do show that he basically understood my answer. — TonesInDeepFreeze

Yes but he understood the opposite of the correct answer!

So, I really don't know what your trip is. — TonesInDeepFreeze

I've been feeling the same way about you.

I'll quit while I'm behind here.

And my rigorous, mathematical and standard use and explanations are not refuted (or whatever your disagreement is supposed to be) by your own informal usage. — TonesInDeepFreeze

As I'm sure I've agreed several times. If you don't want to call quantification over a proper class a domain of discourse, I'm fine with that. We frequently quantify over proper classes, however you call it.

The original question was informal. The original question was in invitation to explain a seeming contradiction. That merits a response that is rigorous and definitive, in order to appreciate that mathematics indeed does not tolerate such a contradiction, not just by informal hand waving, so that when we look at the matter with exactness, we do show that the seeming contradiction actually is avoided. — TonesInDeepFreeze

I thought your responses to the recent OP @Sunner were too detailed and technical to be of use at the level the question was being asked. And, frankly -- not really wanting to get back into this -- wrong. The Russell class DOES define a perfectly good collection. That was the question. @Sunner had the impression (rightly or wrongly) that you said it didn't. I pointed out that it did. Perhaps OP misinterpreted what you said. In that case you were right, and I added clarity. So everyone can be happy, yes?

Not sure what contradiction "mathematics indeed does not tolerate." The referent of this paragraph is unclear.

I can't comment on that quote without a link to it. — TonesInDeepFreeze

I hotlinked it. Here's the link.

https://math.stackexchange.com/questions/2724236/how-do-i-quantify-over-the-proper-class-of-all-the-cardinal-numbers

I guess you mean "question of semantics" in the sense of how we use words. — TonesInDeepFreeze

Yes.

"Then what are your rigorous, mathematical definitions of 'domain of discourse' — TonesInDeepFreeze

As I stated up front earlier, my usage is informal.

@TonesInDeepFreeze

Note the quantifier ranges over the universe, but it happens that the formula is a conditional in which being x being a group is the antecedent.

So that is a relativization of P to groups. — TonesInDeepFreeze

Well yes, we agree on that. But there is no set of all groups! The class of groups is a proper class. So you seem to be conceding my point.

In any event, much of the rest of your post is pretty technical and I'm not sure how it bears on the question. I did find this MathSE thread:

How do I quantify over the proper class of all the cardinal numbers? where there's a lot of learned back and forth about quantifying over proper classes.

The consensus seems to be that when we quantify over all sets or all cardinals or whatever, we are really restricting to the particular predicate that defines the object in question. Which doesn't help us, since the extension of a predicate is a class and not necessarily a set.

And then there is a comment from Asaf Karagila, a professional set theorist and prolific SE contributor. He says:

No, quantifiers quantify over everything. All sets. Bounded quantifiers are shorthands to make it clearer that we are only interested in a particular set. But you can bound them to a class just as easily.

As far as I'm concerned, if Asaf says we can quantify over classes, we can quantify over classes. That is good enough for me.

As far as whether a domain of discourse must necessarily be a set, this seems like a matter of which definition we choose. When we make general statements about sets or groups or cardinals, we are quantifying over proper classes. Whether you call that a domain of discourse or not seems like a question of semantics.

It is only by models that the domain of discourse is definite, and so it is only by models that it is definite what the quantifier ranges over. — TonesInDeepFreeze

This post crossed paths with mine. As far as I'm concerned if I have Asaf on my side I'm happy. And when we say "every set has a powerset," we are quantifying over the proper class of sets. And again, as far as what a domain of discourse is, that seems like a matter of semantics. If you say it has to be a set, then so be it. When we say every set has a powerset, we are quantifying over a proper class; but if you don't want to call that a domain of discourse, that's ok by me.

The official, formal definition of a 'model' is that the domain of discourse is a set. — TonesInDeepFreeze

Agreed. We're not talking about models here, though. We're talking about domains of discourse.

I'll admit that for me, domain of discourse is an informal phrase meaning, "the collection over which we are quantifying," rather than a formal or technical definition. So there may be subtleties I'm missing.

But model theory is not relevant to this conversation as I understand it.

My main point in posting was to address this concern of @Sunner:

What I mean is just that the sentence «there is no set of which every set is a member» clearly says something about every set. But how does this not define some kind of collection or set? — Sunner

I wanted to assure @Sunner that indeed, there is a collection defined by the phrase, "the collection of all sets." The collection is just not a set.

The unrestricted complement of a set always exists. It just may not be a set.
— fishfry

The context of my remark was set theory. In that context, there is no operation of absolute complement but only relative complement. — TonesInDeepFreeze

Fair enough. But in general, there is an operation of unrestricted complement. I pointed that out and gave an example.

Even within set theory, there are unrestricted complements. The complement of the set {1,2,3}, within set theory, is the collection of all sets that are not {1,2,3}. That complement is a well-defined collection, but it's not a set. Of course the relativized complement of {1,2,3} in the powerset of the integers is a set. But the unelativized complement is NOT a set, even in the context of set theory. It's a proper class.

In mathematical logic, a domain of discourse is a set. You may look it up anywhere. — TonesInDeepFreeze

I semi-agree. In prepping my post I looked up Domain of Discourse on Wikipedia, and they did indeed say a domain of discourse is a set. I assumed they were mistaken, and were simply using "set" in its everyday, casual meaning, without regard for the issues of set-hood versus proper classes.

So I agree that if I looked it up, I'd find at least one source, namely Wiki, that claims a domain is a set. I just think they're wrong, and gave many examples to show why.

Well, we are quantifying over the collection of all sets.
— fishfry

Not formally. Formally, any model of set theory has as a set, not a proper class, as its domain of discourse. For any model, the universal quantifier ranges over the members of the domain of discourse, which is a set. — TonesInDeepFreeze

Oh my goodness. I do see your point, but I can't agree with it.

You are saying that when I make a statement such as, "Every set has a powerset," I am really saying:

1. I assume ZF is consistent.

2. By Gödel's completeness theorem, if ZF is consistent it has a model, which is a set.

3. The powerset axiom is implicitly quantifying over that set.

I simply can not believe that this is the implicit chain of logic behind every universal statement about sets. Indeed, the assumption of consistency is NOT part of set theory. Set theory can not prove its own consistency. The claim that every set has a powerset is true whether or not set theory has a model. All that is required is the axiom of powersets.

Indeed, "Every set has a powerset" is NOT a semantic claim; it's a syntactic one. It follows from the axiom of powersets. There are models lacking the axiom of powersets where the claim is false.

Perhaps we're arguing about syntactic versus semantic domains. "Every set has a powerset" is a purely formal statement in the language of set theory. It does not talk about models at all. And it does quantify over the universe of sets, which we know is not a set.

Likewise it can not possibly be the case that when we say, "The binary operation of every group is associative," we are implicitly quantifying over a "set" of all groups. There is no such set, and I can not believe there's a group theorist living who would agree with your point of view. Of course I have not asked them. But nobody carries around this implicit belief that universal statements about groups quantify over a mythical set of all groups, which provably does not exist. There is no set of all groups and we are not quantifying over it when we make general statements of groups.

Rather, we are quantifying over a proper class. From where I sit, you are being a bit unreasonable in your claim that there's a set of all groups that we're implicitly quantifying over. That's just not true.

There are set theories in which classes are formalized
— fishfry

Right. But even with those theories, the domain of discourse for a model for the language of the theory is a set.

Even a class theory such as NBG has only models that have a domain of discourse that is a set. — TonesInDeepFreeze

Nice to know. Not relevant to our discussion here IMO.

Moreover, if we tried to allow a proper class to be a domain of discourse, we'd get a contradiction:

For example, suppose we are doing model theory in a class theory in which there are proper classes. Okay, so far. Now suppose U is a proper class and, for simplicity, we have a language with just one nonlogical symbol. And let R be the relation on U that, per the model, is assigned to the nonlogical symbol.

Then we have the structure <U R>. But then, unpacking the ordered tuple by the definition of tuples (such as Kuratowski), we get that U is a member of a class, which contradicts that U is a proper class. — TonesInDeepFreeze

I haven't sufficient technical knowledge, but for sake of discussion I'll concede your point that if we work in NBG or Morse-Kelley set theory, domains are sets. But that's beyond the scope of the discussion. In everyday math, domains of discourse frequently are proper classes. "Every set has a powerset" quantifies over the proper class of sets; NOT, as you seem to claim, over a set model of sets whose existence depends on assuming the consistency of ZF. That's just not right.

There also is the notion of proper classes as models, or more specifically, inner models. However, I think (I am rusty on this) that when we state this formally, it actually reduces to the syntactical method of relativization, so that when we say L is an inner model of set theory, we mean something different from the plain notion of a model. If I recall correctly, roughly speaking, relative to a theory T, saying 'sentence P is true in "class model" M' reduces to: In the language for T, we define a unary predicate symbol 'M', and P relativized to M is provable in T. So, for example, when considering the consistency of the axiom of choice relative to ZF, we find that the axiom of choice is true in the constructible universe L ("L is a model of AC"), which, in one way of doing this, reduces to: In the language of set theory, define a unary predicate symbol 'L', then we show that AC relativized to L is a theorem of ZF. So if we have a model D of ZF, then the submodel that is D intersected with L (the intersection of a set with a proper class is a set) is
a model of ZFC. That entails the consistency of ZFC relative to the consistency of ZF. But I am rusty here, so I may be corrected. — TonesInDeepFreeze

I was never well-oiled enough to even aspire to being rusty in these subjects. I'm sure we're far beyond any considerations relevant to @Sunner. And in that impressively buzzword-compliant paragraph, there is no mention of the domain of discourse. So again, none of this is relevant. You're perfectly correct that ZFC is consistent if ZF is, but what has that got to do with the conversation?

As a layman, it is interesting to hear that no domain of discourse is truly universal. — Sunner

On the contrary, domains of discourse are often truly universal. They're just not always sets.

Let's take as an example the simplest and most important thing we can say about sets.

Two sets are equal if and only if they have exactly the same elements.

That is, the sets $\{1,2, 3\}$ and $\{2, 3, 1\}$ are exactly the same set. Sets are characterized by their membership, without regard to order.

What domain are we quantifying over when we make this statement? We are saying, "For all sets X, and for all sets Y, if X and Y have the same elements, then X = Y."

Well, we are quantifying over the collection of all sets. Twice, for that matter. And as Russell showed us, the collection of all sets is not a set. It is a perfectly well-defined collection. It just isn't a set.

So in fact we can, and commonly do, quantify over domains that are not sets.

What are these collections that are "too big" to be sets? They're called proper classes.

First, a class is any collection defined by a property, or predicate. So, "Is a set" is a property that's true or false about any given individual. The collection of all things for which the property is true, is a class.

From the Wikipedia page on Classes:

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. — Wikipedia

Some classes are sets. Others are "too big" to be sets, such as the class of all sets. Those are the proper sets.

Yet, we can still use a proper set as a domain of discourse. We do that every time we state a property of sets. "For every two sets, their union exists." Quantifying over the proper class of all sets. "Every set has a power set." Quantifying over a the proper class of all sets.

It's a perfectly everyday occurence in math to use a proper class as the domain of discourse. It's so commonplace that we don't even notice ourselves doing it.

In ZF (standard set theory), there are no official classes, so the usage is informal. There are set theories in which classes are formalized, but those set theories are not usually encountered except by specialists.

Of interest to this thread is the Russell class, $R = \{x : x \notin x \}$.

Russell showed that $R$ can not be a set. But it's a perfectly well defined proper class. It's the collection of all the things that satisfy the property $x \notin x$.

To sum up: Sometimes domains of discourse are sets, as when we say, "All positive integers other than 1 have a unique factorization into prime powers." Here, the domain of discourse is the positive integers, because we explicitly stated that. The statement becomes false if we change the domain to the real numbers, for example.

Other times, the domain of discourse is a proper class that is too big to be a set. For example, when we say, "Every set is completely characterized by its elements," we are quantifying over the universe of sets; which as Russell showed, is not a set. But it's still a proper class, and may be spoken of and used as a domain of discourse.

A domain of discourse is a set. — TonesInDeepFreeze

I'm afraid I can't agree. Many obvious counterexamples come readily to mind. We literally could not do math without quantifying over domains that are not sets.

* Every set is in bijective correspondence with itself, via the identity map. Quantifying over the proper class of all sets.

* The identity element of every group generates a 1-element subgroup. Quantifying over the proper class of all groups

* Every vector space has a basis. Quantifying over the proper class of all vector spaces.

* Every topological space contains at least two open sets. Quantifying over the proper class of all topological spaces.

Quantifying over proper classes is so common that we don't even notice it.

And there is no set of all sets that are not in that domain of discourse. — TonesInDeepFreeze

Correct. But there is a class of all such sets. That class is not a set. It's a proper class, characterized by the property of being "not in that domain of discourse" that was being discussed.


A final example involving complements of sets, relative and otherwise, that shows how you can annoy your teacher.

You are asked, "What is the complement of the set of even numbers?" You answer, "The odd numbers, of course." Being a good student, you implicitly assumed that the domain of discourse is the set of integers.

But the literally correct answer is: "Everything that is not an even number." That includes the odd numbers as well as Captain Ahab, the Andromeda galaxy, and the Mormon Tabernacle Choir. The complement of the even numbers, without any further domain restriction, is the proper class of everything in the universe, abstract entitied included, that are not even numbers.

The unrestricted complement of a set always exists. It just may not be a set. If the teacher wanted the complement of the even numbers relativized to the set of integers, they should have said so!

Wow I'm really not following this "logic". The argument that a random individual is unlikely to cause harm is generally an excellent argument against separating that individual from society. Infection, unvaxxed status, serial killerhood are all reasons for separation of that person from society. — hypericin

The problem with this line of argument is that the vaxxed are every bit as contagious as the unvaxxed. The number of breakthrough infections is significant. How significant we don't know, because The CDC stopped tracking breakthrough cases in May. Wonder why.

Now it's true that if you are vaxxed and you get covid anyway, your symptoms will be reduced. That's a good reason to be vaccinated. But if you do get a breakthrough case, you're just as contagious. And there are a lot of breakthrough cases. Israel, which has an 80% vaccination rate, reports that the Pfizer shot is only 39% effective.

So your numbers just don't add up. The average person you meet is highly unlikely to be contagious at that moment in the first place. Of those that are, they're more likely to be vaccinated for the simple reason that most people are vaccinated, and the vaccines simply aren't that effective.

So there's no scientific argument at all to decree that unvaccinated individuals should "forfeit the right to move freely in society," which was @Wayfarer's original quote that I objected to.

And even if you could make such an argument, the downsides of such Othering of a group -- dirty, disease-ridden, danger to society -- would outweigh any good. I defy you to name any time in history that such an Othering came out well. And I'm sure you know all the bad instances I could name.

The vaccinated and infected are rare. If they are identified as such, they should be restricted.
Drunk drivers are rare. If they are identified, they should be restricted. — hypericin

Ok. Now that is entirely different than what @Wayfarer said. You agree that the vaccinated yet infected should be isolated. That's not even what we're talking about, and you are in effect conceding my point.

Vaccination should be a requirement for entry to high risk areas such as transportation, supermarket, bars, restaurants, movie theaters, etc. — hypericin

I'm not even talking about that. @Wayfarer said that the unvaccinated should "forfeit the right to move freely in society." I'm pointing out that this is one, not scientifically supported because of the high vaccine failure rates, the fact that the vaxxed are just as contagious as the unvaxxed, and the Othering that would inevitably produce results that nobody would want to see.

I'm not talking about any other issues, whether the unvaxxed should be allowed into bars and whatnot. I'm talking about "forfeit[ing] the right to move freely in society." That's a tremendous overreach and poorly thought out position. I wouldn't be surprised if @Wayfarer would be willing to say, "You know, I just typed that in, but I didn't really think about it, and it's wrong on many levels and totally unworkable." I don't know. You're taking up an argument on someone else's behalf but you yourself don't seem to remember what the argument was.

The rest of your post is slippery slope — hypericin

Asking someone to drill down a level of detail is not a slippery slope. If you propose to restrict the free movement of the unvaccinated, how do you determine who they are? You have to interrogate everyone. Perhaps you discount an additional, say, one million police/citizen encounters per day. Maybe you haven't read the papers about public opinion of police encounters. It's not a slippery slope argument to challenge a highly impractical suggestion by asking the proposer to supply the details of how their idea would be implemented.

hysteria — hypericin

How so? The proposal is to restrict the free movement of 75 million or so Americans. I think I understated the likely consequences of such a nonsensical and dangerous idea.

and race baiting. — hypericin

How so? I pointed out that only 31% of blacks are vaccinated, so that if you start restricting their movement or rights in large numbers, you would create social problems that hardly need to be mentioned to be perfectly obvious to anyone who follows the news. You act like you don't follow the news.

In any event, the New York Times made much the same point when they reported four days ago, Why Only 28 Percent of Young Black New Yorkers Are Vaccinated.

Perhaps you could read that article in its entirety and explain whether you think the New York Times is race baiting too.

In any event, my comments were regarding New York City's plan to require vaccinations for entry into public spaces. The New York Post reported on the details of the plan today. They said that the venues themselves would be fined, not the individuals. Seems that unlike you, New York City actually put some thought into the consequences of their policy, and enacted enforcement mechanisms intended to avoid the obvious racial consequences instead of exacerbate them.

See how that works? People have an idea, but then they have to think through the consequences and modify and implement their policies accordingly. That's what you call "hysteria" and "race baiting." I call it basic thoughtfulness and common sense.

This doesn't address the larger harm the unvaccinated, and the scumbag public figures that encourage them, do to society. — hypericin

Yes, you're just the person I'd pick to Other 75 million Americans and select them for special treatment. What could possibly go wrong?

But as I pointed out, your statistical logic is flat out wrong. The vaccines aren't even 50% effective. There are huge numbers of breakthrough cases, so many that the CDC won't even report them. And while the vaccines keep you from getting as sick as you would without them, you are just as contagious. So there is no scientific argument to be made that the unvaxxed are any more dangerous to society than the vaxxed.

What's dangerous is people letting their fear cloud their common sense. And don't you think the lockdowns themselves are harming society? Children born during pandemic have lower IQs, for one thing. The increase in alcoholism, domestic violence, and substance abuse are noteworthy. Every action has consequences both good and bad. There's no thoughtfulness and balance in the simplistic "lock everyone down and shoot the unvaxxed" kind of thinking.

What, am I being hysterical again? UP AGAINST THE WALL: California Congressional Candidate Says Anti-Vaxxers Should Be Shot. I'm not hysterical, I'm just someone who follows the news from a variety of sources.

If everyone was vaccinated, and diligently performed basic social distancing and hygiene during local outbreaks, we might be done with the pandemic, at least in the US. — hypericin

Might. And might not. The adverse reactions to the vaccine are off the chart, and nobody knows the long term consequences because there haven't been any studies. You're making a claim that can't be backed up by science. You're letting your lizard brain flood you with fear. Take a step back and try to think. Who's violating hygiene and social distancing? Who's arguing against it?

I'm arguing against the thoughtless and mindless claim that the unvaxxed should "forfeit the right to move freely in society." That's what I'm arguing against and that's ALL I'm arguing against. Are you sure you're not the one who's hysterical?

Instead, hospitals and morgues are filling up again, and actual freedom, the freedom to enjoy life without risk of death or mutilation, has slipped away. — hypericin

Well that's just terrible, I agree. I'm disagreeing that it's practical to selectively restrict the freedom of movement of 75 million Americans without a lot of unexpected and highly negative consequences. Why don't you stick to the actual topic of what I said, and try to think the issue through?

Really, from that perspective the restriction of freedom of movement is too mild. Vaccination should be mandatory, full stop. — hypericin

The vaccines don't even work all that well. The adverse reactions, including death, are off the charts. Nobody knows if they're safe long term because the studies haven't been done. They have no FDA approval. I think you are so panicked by the media hysteria you'd send Jews to the ovens if someone told you they carried disease, which is exactly what the German media told people. You're just that kind of person. I hope you'll step back and get a grip on your own hysteria.

I haven't considered any government enforced denial of freedom of movement, so any disagreement I might raise isn't to that effect. — Cheshire

Ok, and I appreciate your saying that. Because other than that one point, I haven't said or advocated anything. Except to push back on my hysterical and propaganda-addled friend @hypericin.

My issue is with the pronouncement that the possibility of a vaccinated person spreading a virus and the possibility of an unvaccinated person spreading the virus are treated as equal. Or the first makes the latter not matter. It seems to me a strong argument could acknowledge that one is taking place regularly and the other is somewhere between rare and not impossible. You disagree above, but maybe I missed something. — Cheshire

The numbers don't bear you out. As I posted, the Israelis, who are 80% vaccintaed, report that the Pfizer shot is only 39% effective. The numbers for the other shots are in that range. And now everyone is supposed to get a booster shot. So in terms of effectiveness, the vaccines are essentially a bust. Yes they do make you less sick than you'd be otherwise; but you are just as contagious.

And since most people are vaxxed, the chances that the next person you run into is contagious and vaxxed or contagious and unvaxxed are more or less the same. So there's no statistical argument to be made about treating one class differently than the other on the basis of contagion.

The Wall Street Journal is a Murdoch paper, is it not? — Wayfarer

Yes, but are you denying their factual claim that only 31% of black are vaccinated? As I posted above, the New York Times reported that only 28% of young blacks are vaccinated. If the WSJ prints a fact, then it's a verifiable fact no matter how you feel about their editorial stances. Right? If you don't like the WSJ's 31% number, then just take the NYT's 28% if you prefer that.

More likely crying crocodile tears over the poor benighted black population to feed meat to their civil-libertarian right-wing audience than out of any genuine concern for the former. — Wayfarer

Well I don't much care either, I just want to see the hilarity ensue. A bit of sarcasm, don't get excited.

But as I also linked above, the New York Post reported that the enforcement actions in New York City will be against the venues and not against individuals. Meaning that they took my point to heart and realized that the optics of arresting or ticketing or shooting (as one California congressional candidate wants to do) unvaccinated black people would not look too good in heavily black NYC.

You see once again that I am trying to get people to be thoughtful about what they're saying; and as support for my position, New York City itself was thoughtful about this point. Whether they are genuinely concerned about black people or whether they just don't want the bad optics; they're only fining venues and not individuals.

Murdoch media worldwide are probably alone responsible for tens hundreds of thousands of infections by spreading their anti-vaccination nonsense along with all the many other lies and propaganda they peddle around the world every day. — Wayfarer

LOL. Don't hold back, tell us how you really feel.

You're right, I actually quoted too much from the WSJ article. All I cared about was the 31% number. If I'd known it would trigger you I wouldn't have bothered.

But it's a fascinating point. The stanard mainstream belief is that the unvaxxed are MAGA-hat wearing racist deplorables. But it turns out that the real unvaxxed are blacks and Latinos. And Ph.D.s. That's right, the single group with the greatest degree of vax avoidance is people who hold PhDs. The article didn't say why, but my guess is that people who have done actual scientific research can recognize the sham, politicized pseudoscience for what it is.

I would never cite or refer to any articles published by any Murdoch outlet in support of any point whatever. — Wayfarer

I gave you a New York Times link reporting much the same information. You seem to have forgotten to argue your own point, you got so triggered by the WSJ.

I acknowledge that all forms of lockdown and restriction of movement are an infringement on civll liberties, but in light of the severity of this illness, I believe that imposing a lockdown is a lesser of two evils. I mean, giving up some freedom of movement and even income, is generally preferable to getting a life-threatening illness, in my opinion. — Wayfarer

Now that is an entirely different matter that your original suggestion that the unvaxxed should forfeit the right of free movement.

I might (or might not) argue against a general lockdown, but I'm not discussing lockdowns here. Lockdowns affect everyone equally. To implement a lockdown you don't have to Other 75 million people (in the US), subject everyone to demands to show their papers, and add a few million or so daily police/citizen interactions. Those are the issues I'm concerned about.

Lockdowns, regardless of their merits, apply to everyone equally, and therefore don't have the problems I'm concerned about regarding your earlier idea.

Australia generally has succeeded in controlling the infection, although the Delta variant outbreak that started in Sydney June 16 has well and truly escaped the net. — Wayfarer

I looked this up. Australia has some 25M people versus 300M in the US. And the US has only 1.27 times the area. So Australia has a much smaller population and much much sparser population density. You have no idea what it's like to get 300 million crabby Americans to do anything.

And besides, having just discovered that the US government has been lying to us for 20 years about the progress of the war in Afghanistan, which we are even as we speak losing in a majorly humiliating fashion, I doubt that American are inclined to believe anything the government says. I remember the anti-government sentiment of the 1970's after our loss in Vietnam, and I expect the same to happen now. So you can't lock down the US. Can't be done even if was the black plague.

And from what I hear, Australia has surrendered its civil liberties in ways that. to this American, are truly frightening. I always thought of Austrlians as liberty lovers, Crocodile Dundee kinds of folks. Guess that was only a movie.

I'm not saying lockdowns wouldn't be effective. Only that American is an unruly country full of unruly people not currently inclined to believe anything the government says. It's just a practical matter.

There is a lot of commentary that the mistake the NSW Govt made was in not locking down faster and harder - there was a super-spreader event on June 26th that transmitted the virus from Sydney’s East to the vast Western Suburbs, which is when it really began to escape, as there are many more large households and a high degree of geographic mobility. That’s where it remains - yesterday’s case numbers were 344, two deaths, and also cases appearing in regional centres. — Wayfarer

I don't disagree that locking everyone down works in a pandemic. China was apparently successful doing that. But they're a majory authoritarian regime. And like I say, Australia has a much smaller population.

But mainly, why are we talking about lockdowns? I'm not talking about lockdowns. Lockdowns are imposed on everyone equally. To lock down a country you just patrol the streets and shoot or fine or chastize everyone who's out without a good excuse. Your original idea, to restrict the free movement of only the unvaxxed, involved interrogating everyone, necessitating millions of cop/citizen interactions every day, many of which, if you read the US papers, don't go particularly well, especially when there are minority groups involved. Black people are not interested in being accosted by the police in the US and frankly I can't say I blame them.

And by the way, where are you getting all these extra cops? As a result of the anti-cop sentiment in the US, cops are quitting in droves. You can't find enough cops to enforce a selective lockdown that involves asking everyone for their papers.

So a lockdown for all, whether it's a good or bad thing, is completely different than a lockdown for some.

As I think I said earlier, community attitudes to vaccination have dramatically shifted in the last month, due to the insidious nature of this variant, and the fact that there’s a lot of younger people in ICU, with two otherwise healthy and comparatively young people dying. I think everyone now realises that getting a severe case of COVID-19 is a life-changing event even if it doesn’t kill you. So vaccination rates have ticked up enormously, supply problems are being overcome, the Moderna vaccine has now been approved and the country is on track to be around 80-90% vaccinated by year’s end. — Wayfarer

Not disagreeing. Only pushing back hard on the idea that the unvaxxed should have their freedom restricted; especially because there are a lot of breakthrough infections, and that the vaxxed are just as contagious as the unvaxxed. So the statistical argument for restricting only the unvaxxed is false. Let alone the problems of asking for everyone's papers in a country like the US that is in the midst of both an anti-cop hysteria and a crime wave.

As to whether lockdowns have to be enforced, I still don’t see any other option. — Wayfarer

Why did you so radically change the subject? You can enforce a lockdown easily, just shoot/arrest/fine/shame anyone you see on the street.

A selective lockdown, on the other hand, entails interrogating EVERYONE and asking for their papers. Which entails a lot of cop/citizen interactions; which, in the US, often go south in terrible ways. So that's a bad idea.

AND it's statistically unsound, since your chances of meeting a contagious vaxxed or a contagious unvaxxed person are about the same, and they are equally contagious. So you haven't got a case, and you have a very poorly thought out position.

Have you backed off your earlier proposal, or just changed the subject to universal lockdowns?

The laissez faire approach of some of the US GOP governors simply results in higher rates of infections and more deaths. — Wayfarer

Statistics are mixed. Some red states are doing better. But I am not discussing laissez faire approaches. I'm pointing out that restricting the free movement of ONLY the unvaxxed would one, be a complete policing disaster; two, would in fact fall heavily on minorities, as I've documented; and three, is flat out wrong anyway since the vaxxed are just as contagious and there are a lot of breaktrhough infections, which in the US the CDC will not even report.

So your idea is a non-starter. Is that why you changed the subject?

Some US states with comparable populations to NSW are having thousands of cases and hundreds of deaths every day, which NSW might easily be matching, had not the lockdowns been enforced. — Wayfarer

Well we're not talking about lockdowns at all. You proposed selective lockdowns against a population that can't be distinguished from the vaxxed and therefore needs to be challenged for their papers; would mostly consist of harassing minorities; would be an unmitigated policing disaster; and wouldn't help anyway, since the vaxxed are potentially just as contagious.

And that's the only point I was making. A selective lockdown against the unvaxxed wouldn't work and wouldn't help.

Square circle as a genuinely contradictory object would look like a square and like a circle from the same perspective (and at the same time and under all other same circumstances). Such an object cannot exist — litewave

In the taxicab metric the unit circle is a square. There's a picture of a square circle on that page. A circle is the set of points equidistant from a given point. If you choose your distance function appropriately, a circle can be a square.

Note that this is very different than an unmarried bachelor. A bachelor by definition is a male who is not married. so that a married bachelor is indeed a contradiction.

But a circle and a square are NOT defined as each other's opposites, nor are they mutually exclusive at all. People should stop using square circles as an example of a contradiction, because in fact there are square circles.

Note that if you define the distance between two points to be the sum of the horizontal and vertical distance between them, then the distance of each red point from the blue point is the same in each case, and that these are therefore square circles. (This is the Wiki image).

A square circle would be a regular polygon with four sides, the perimeter of which is equidistant from a given point on the same plane.

Draw me one of those. — Banno

Done. It all depends on how you define distance. Standard Euclidean distance (square root of the sum of the squares of the respective differences of the coordinates) is only one way of defining distance. Even in physics, Euclidean distance is only a special case of a more general way of defining distance.

Whether or not they are as contagious once infected, they are infected at lesser rates. As continual testing of everyone is impractical, they therefore present less danger to the public than the unvaccinated. — hypericin

I have a small point and a large point to make. I'll start with the small. First, to recap. @Wayfarer posted:

Perhaps their freedom of movement may also be curtailed, though less so. Perhaps 'social distancing', the wearing of masks, and other hygeine measures, will henceforth remain as part of civil society. — Wayfarer

I pointed out that when the vaccinated acqire a breakththrough infection, they are just as infectious as the unvaxxed. Therefore their movement should be restricted too. And that's when you pointed out that they're infected at lesser rates.

This point is easily refuted. The fact that the average vaxxed person is statistically unlikely to infect you means nothing. After all, the average person is not a serial killer, but we endeavor to take serial killers out of society to protect the public. The argument that a random individual is unlikely to cause harm is no argument against separating that indvidual from society.

Likewise drunk drivers, which you mentioned.

The unvaccinated are making this choice to (in their mind) improve their well being, at the expense of the public well being. It is therefore rational public policy to restrict their freedom of movement, to both protect the public well being, and to discourage this selfish choice. — hypericin

In terms of protecting the public well being, you need to restrict the movement of the vaccinated as well, since they are just as contagious as the vaccinated, even if perhaps fewer in number.

So in the end, your point is purely punitive and unrelated to public health.

The situation is rather similar to driving. Everyone on the road presents some danger. But drunk drivers, as a result of their selfish decision to be drunk drivers, present a greater danger. Therefore their freedom of movement is restricted, to protect the public and to discourage drunk driving. — hypericin

But by your logic the contrary conclusion is forced on us. The average driver is statistically rare, even if all too common. Since contagious vaxxed people and drunk drivers alike are statistically rare, they should both be free to travel. After all, your likelihood of encountering either one is relatively low.

So your statistical argument is wrong, and all you have left is your feelings that the unvaxxed should be punished for their "selfishness," as you put it. How about people who don't get their flu shots? People who don't contribute enough to charity? Those with unpopular political opinions? If punishment is your only argument, you yourself wouldn't want to live in the world you wish for.

Now to the larger point. @Wayfarer suggests,"Perhaps their freedom of movement may also be curtailed ..."

Ok. Let's think that through. I can think of two extremes. One is what is done by the a grocery store near me. They have a sign out front that non-vaccinated people must wear masks. They don't check, and rely on the honor system. Then again I live in a relatively small, laid-back town with a relatively low infection rate.

The other alternative is full on police-enforced compliance. You're walking down the street, and the police may ask to see your papers. If you can't produce a vax card, you're arrested on the spot.

Those are the extremes. Perhaps you and @Wayfarer would like to say, specifically, how you think the restriction of free movement in the US (or your country, whatever it may be) should be implemented.

I well remember a few years back when the US state of Arizona wanted to implement a "show your papers" law to challenge brown-skinned people on their immigration status. Decent people across the country were rightfully outraged. Most people think of "show me your papers" as something said in a German accent in a late-night black and white movie from the 1940's. In the US, at least, we don't "show our papers" to the authorities without the police having probable cause or a damn good reason.

So perhaps you think this is a good reason, and that American citizens should be required to show their papers on demand. Can you see how this would quickly go south? Did you get your flu shot? Have any unapproved political opinions? Maybe you tweeted that "All lives matter," or that you believe in rationality and hard work. Those ideas are racist, according to the Smithsonian.

Can you look at history and give me an example of when "show your papers" ever came out well for a society and didn't quickly get abused?

How about when you're driving? Surely if freedom of movement is to be constrained, we need highway checkpoints. That's not so farfetched; there are already interior immigration checkpoints as far as 75 miles inside the US border, where travelers staying entirely within the US may be stopped, interrogated, and searched. Of course if they happen to find a joint or some other contraband, that's your bad luck. What, the Fourth Amendment to the Constitution forbids such an abomination? Sadly, courts have repeatedly allowed these interior checkpoints. It would be easy to set up a lot more of them to check people's vaccination status.

Do you think that's a good idea? Is that the country you want to live in?

Let me point out one more "inconvenient truth," as Al Gore once put it. Who in fact are the unvaxxed in the US? In the popular imagination they're white, MAGA hat-wearing deplorables with unapproved ideas.

In fact, the unvaxxed are blacks and Latinos. Don't believe me?

https://www.kff.org/coronavirus-covid-19/issue-brief/latest-data-on-covid-19-vaccinations-race-ethnicity/

https://thenewamerican.com/leftists-vaccine-passports-are-racist-under-the-lefts-own-thinking/

https://www.nytimes.com/2021/01/31/nyregion/nyc-covid-vaccine-race.html

So what are you going to do? Start pulling over or checkpointing black drivers, accosting blacks and Latinos on the streets and demanding their papers, refusing access to great numbers of blacks and Latinos to restaurants and movie theaters? Can't wait to see how that works out.

We have a real-life datapoint coming up. In New York City, restaurants and other indoor venues will soon require proof of vaccination for entry.

https://www.nytimes.com/2021/08/03/nyregion/nyc-vaccine-mandate.html

But it turns out that only a fraction, one third or so, of NYC blacks are vaccinated.

The policy takes effect this Monday, August 16, and enforcement begins in September. It will be administered by the health department and not the police. So can you imagine what it's going to be like when two thirds of the black people in New York City are banned from restaurants?

The WSJ has the summary. Most of the article is paywalled but the free part says plenty.

https://www.wsj.com/articles/bill-de-blasio-new-york-city-covid-vaccine-mandate-coercion-11628022693

The modern progressive speaks the language of high-minded purpose but always ends with coercion. Witness New York Mayor Bill de Blasio, the uber progressive, who announced Tuesday that New Yorkers will soon need proof of vaccination to do everything from dining out to working out at a gym. He’s proud that New York is the first U.S. city to impose such a mandate.

“It’s time for people to see vaccination as literally necessary to living a good and full and healthy life,” he said at his press conference. You gotta love Mr. de Blasio telling you what is necessary for a good and full life. According to the data, roughly 55% of the city’s residents are fully vaccinated, ranging from 46% in the Bronx to 67% for Manhattan.

His response is to exclude the unvaccinated from many of the functions of daily life. He doesn’t seem to care that this burden will fall heaviest on the city’s black population, which is only 31% fully vaccinated (versus 71% for Asian Americans, 42% for Hispanics and 46% for whites). — WSJ

@Wayfarer and @hypericin, is this what you want? 69% of black people in NYC excluded from public life? And if not, then what DO you mean when you talk about restricting people's movement?

Feedback appreciated. You disagree with my facts? My reasoning? Or are you you all in on "show me your papers" to every non-white face in New York City? You want to bring back stop-and-frisk but for vax cards instead of guns and knives?? And if you did implement nationwide walking and driving checkpoints, how long do you think it would be before the inevitable scope expansion and mission creep set in? Check for your vax card, check your wants and warrants. Behind on your child support? Carrying any unapproved contraband? Tweet any unapproved thoughts recently?

You serious? Anyone thinking this thing through? Or do you all want to live under the Chinese social credit system and can't wait till it's implemented here? I'm afraid that's exactly what some people want.

Some second hand books that I have leafed through, seemingly of the period when I was at school, averred that calculus was all about areas and speeds, though that had never had anything to do with the lessons I had "had". Is this a class thing? — Fine Doubter

Velocities and areas are applications of calculus, as in Newton's fluxions and fluents, corresponding to today's derivatives and definite integrals. In pure math, one only uses velocity and area as illustrations to help students understand, but they're irrelevant to the mathematical content.

Find Doubter, nice pun on "Find Outer."

reforms of the calculus in the 19th C? In terms of limits, as explained copiously hereabouts by fishfry — bongo fury

Thanks for remembering :-)

Well if you want a perfectly unhackable root, this idea might suffice. — Shawn

Do you know that in normal operations, the OS needs to constantly make changes to the kernel in privileged mode? How would you determine what's a legitimate change versus a malicious one? If you disallow all kernel changes the computer won't boot and won't run.