The issue behind the student loan question is the question how far state-funded free education should go. If you want a level playing field in careers, everyone who can benefit should get higher education - and that means that almost everybody should be entitled to have a go. At the same time, if people benefit financially, there is a good case for saying that some of that benefit should go back to whoever funded it. Ironically, in the UK, the financial benefit from higher education is rapidly shrinking and, some say, has disappeared, mainly because it has been extended so widely. The proportion of student loans that is actually repaid is astonishingly low. (I can't remember the actual figures.) — Ludwig V

That's right. So the students majoring in unmarketable majors are subsidized by people who skipped school and went into the trades. That doesn't seem fair. It's just that the college grads vote for Democrats and the tradesmen vote for Republicans, so the Democratic administration forgives billions in student loans -- illegally, as the Supreme Court has already ruled -- in an election year.

And not just that. The Democratic party use to be the party of the tradesmen and no longer is. When did the left abandon the workers, and why? I gather the Labour party in the UK has undergone a similar transition, is that right?

So is it possible that a different version of the social justice approach might be more effective? Is it possible that other places may be implementing it in a better way? — Ludwig V

Compassion for criminals is anti-compassion for their victims. New York City is a great lesson in restorative justice gone too far. I think the first duty of civic authorities is to provide for civic order. What's weird is that the voters themselves vote for the faux-compassion that ends up hurting them.

So the real solution to our problems is better voters!

I've watched this debate for a long time - though I don't claim to have understood all of it. But I think those two quotes show that you are talking past each other. — Ludwig V

He'll come around :-)

I didn't like ω at all, when it was first mentioned. I'm still nowhere near understanding it. But the question whether a mathematical symbol like ω is real and a number is simply whether it can be used in calculations. That's why we now accept that 1 and 0 are numbers and calculus and non-Euclidean geometries. ω can be used in calculations. So that's that. See the Wikipedia article on this for more details. — Ludwig V

This paragraph gratified me. If you are struggling to understand my posts then I'm getting through to at least one person. My talk about $\omega$ is something most people haven't seen, but the ideas aren't that hard. For what it's worth there's a Wiki page on ordinal numbers. The page itself isn't all that enlightening, but it does at least show that the ordinal numbers really are a thing in math, I'm not just making it all up.

You have a great insight that what makes a mathematical concept real is, in the end, its utility. Sometimes not even to anything practical, but just to math itself. We want to solve the equation x + 5 = 0 so we invent negative numbers. That kind of thing. In that sense, the ordinals exist.

But another way to think about it is that it's just an interesting new move in a game. As if you were learning chess and they told you how the knight moves. You don't say, "Wait, knights slay dragons and rescue damsels, they don't move like that." Rather, you just accept the rules of the game. You can think of ordinals like that. Just accept them, work with them, and at some point they become real to you. Just like the moves of the chess pieces any other formal game.

But I have tried to give a very concrete, down to earth example of how this works.

Suppose that we have the sequence 1/2, 3/4, 7/8, ... It converges to 1.

Now we can certainly form the set {1/2, 3/4, 7/8, ..., 1}. It's just some points in the closed unit interval.

But it gives us a model, or an example, of a set that contains an entire infinite sequence that "never ends" blah blah blah, and also contains its limit.

If you believe in the set {1/2, 3/4, 7/8, ..., 1}, then you should have no trouble at all believing in the set

{1, 2, 3, ..., $\omega$}. That's also just a set that contains an entire infinite sequence, along with its limit. We typically don't encounter this concept in the math curriculum that most people see, but it's perfectly standard once you go a little further. Also a lot of people have seen the extended real numbers with $\pm \infty$ and nobody complains about that, or do they?

It's true that the distances are different inside the two sets. But in terms of order, the two sets are exactly the same: an infinite series, along with its limit.

Anyway, this framework is very handy for understanding supertask type problems. That's why I'm mentioning it.

So if you don't like $\omega$, that' s no problem. Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.

Does that help?

But it is also perfectly true that a recitation of the natural numbers cannot end. — Ludwig V

That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.

And if we can't imagine that, we can certainly imagine {1/2, 3/4, 7/8, ..., 1}. There's nothing mysterious about that. An entire infinite sequence is in there, along with an extra point. It's a legitimate set.

If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work. They are containers for infinite collections of things.

By the way, $\omega$ is the "point at infinity" after the natural numbers. And $\omega + 1$ is the name for the set {1, 2, 3, 4, ..., $\omega$}.

$\omega + 1$ is the natural setting for all supertask puzzles. We have the state at each natural number, and we inquire about the final state at $\omega$.

That's why I like $\omega + 1$ as a mental model for these kinds of problems.

As I said earlier, it is remarkable that we can prove it. Yet we cannot distinguish between a sequence of actions that has not yet ended from one that is endless by following the steps of the sequence. So we are already in strange territory. — Ludwig V

This business about actions is what confuses people. They set up scenarios that violate the laws of physics, like the lamp that switches in arbitrarily small intervals of time, and then they try to use physical reasoning about them. Then they get confused.

In the way I'm describing this, you may think that the difference is between the abstract world (domain) of mathematics and another world, which might be called physical, though I don't think that is right. — Ludwig V

Well yes, you are correct to feel that it's not quite right. Because there is nothing physical about the lamp or the staircase. So it's a category error to try to use everyday reasoning about the physical world. That's why people get confused.

I'm very puzzled about what is going on here, but I'm pretty sure that it is more about how one thinks about the world than any multiverse. — Ludwig V

I think it all comes down the fact that calculus classes care about computation and not theory. That, and the fact that we don't know the ultimate nature of the world, and there's are good reasons to think it's not anything like the mathematical real numbers.

So on the one hand, the continuity of the world is an open question. And two, calculus classes are not designed to teach people how to think about limits in the more general ways that mathematicians sometimes do. Put those together with quasi-physical entities like physics-defying lamps, and you have a recipe for confusion.

There is no such thing as "going by pure logic", toward understanding the nature of reality. [/quore]

Agreed. But that does not justify using some means OTHER than logic to understand reality, and calling it logic! That's @Michael's fallacy. Saying something's a logical contradiction when it merely makes no sense to him. You agreed with me earlier that this is a fallacy. But you defend it when YOU do it.

To be clear: I have no objection to using extra-logical means of understanding reality. But then don't turn around and all it logic.

— Metaphysician Undercover

"Pure logic" would be form with no content, symbols which do not represent anything. All logic must proceed from premises, and the premises provide the content. And premises are often judged for truth or falsity. But as explained in the passage which ↪wonderer1 referenced, in the case of an "appeal to consequences", there is no fallacy if the premises are judged as good or bad, instead of true or false. That's why I said that this type of logic is very commonly employed in moral philosophy, religion, and metaphysical judgements of means, methods, and pragmatics in general. So for example, one can make a logically valid argument, with an appeal to consequences, which concludes that the scientific method is good. No fallacy there, just valid logic and good premises. — Metaphysician Undercover

Call it anything you like, but not logic! Logic means something else. That term is already taken. You are using extra-logic. Morality, right or wrong, productive/nonproductive. All well and good, but not logic. If logic is to mean anything, it has to mean something.

Therefore it is not the case that the reasoning is "extra-logical", it employs logic just like any other reasoning. What is the case is that the premises are a different sort of premises, instead of looking for truth and falsity in the premises we look for good and bad. So this type of judgement, the judgement of good or bad, produces the content which the logic gets applied to. — Metaphysician Undercover

Let's agree to disagree on that point.

No, that is not the case, because there are two very distinct senses of "determined". One is the sense employed by determinism, to say that all the future is determined by the past. The other is the the sense in which a person determines something, through a free will choice. In this second sense, a choice may determine the future in a way which is not determined by the past. And, since it is a choice it cannot be said to be random. Therefore it is not true that if the world is not random then it's determined (in the sense of determinism), because we still have to account for freely willed acts which are neither determined in the sense of determinism, nor random.[/qouote]

You can't have determinism and free will. Frankly if the world is random and we have some kind of influence on it through our will, or spirit, I find that much more hopeful than a universe in which I'm just a pinball clanging around a well-oiled machine.

Determinism is the nihilistic outlook, not randomness. In randomness there is hope for freedom. Say that's a pretty catchy saying. The church of Kolmogorov. In randomness lies the hope of freedom.
— Metaphysician Undercover

As I said above, it is not a matter of transcending logic, the conclusions are logical, but the premises are judged as to good or bad rather than true or false. So from premises of what is judged as good (rejecting repugnant principles), God may follow as a logical conclusion. — Metaphysician Undercover

God's going to hurl thunderbolts at you for so blithely enlisting him on your side to make such a specious argument. If I'm choosing good versus bad I'm not using logic, I'm using feelings. Logic says kill the one rather than the million. But if the one's you or yours, you kill the million. It's been done. Feelings trump logic. But your feelings are not logic!!

No I was not arguing that. In that case I was arguing that the idea ought not be accepted (ought to be rejected) unless it is justified. In the case of being repugnant, that in itself is, as I explained, justification for rejection. You appear unwilling to recognize what wonderer1's article said about the fallacy called "appeal to consequences". It is only a fallacy if we are looking for truth and falsity. If we are talking principles of "ought", it is valid logic. Therefore the argument that the assumption of randomness ought to be rejected because it is philosophically repugnant, cannot be said to be invalid by this fallacy, and so it may be considered as valid justification. — Metaphysician Undercover

Yes but the contrary proposition of determinism is even more repugnant, as I've noted. Shouldn't we (logically!) choose the lesser of two repugnancies?

But Michael did not show that supertasks are philosophically repugnant. — Metaphysician Undercover

And you have not shown randomness philosophically repugnant. By the time I thought about it a little, I realized that randomness is our only hope for salvation. It's the only way we're not automatons. Clockwork oranges. So you haven't made your point here. I am a proud randomite.

He showed that they are inconsistent with empirical science, — Metaphysician Undercover

There's no empirical science in these silly omega sequence paradoxes like the effing lamp and the effing staircase. That's the massive category error everyone makes. They posit these physics-defying scenarios then claim they're talking about the physical world.

and his prejudice for what is known as "physical reality" (reality as understood by the empirical study of physics) influenced him to assert that supertasks are impossible. — Metaphysician Undercover

I believe I made the same claim, but qualified it to "presently known physics."

As I explained in the other thread, in philosophy we learn that the senses are apt to mislead us, so all empirical science must be subjected to the skeptic's doubt. So it is actually repugnant to accept the representation of physical reality given to us by the empirical sciences, over the reasoned reality which demonstrates the supertask. And this is why that type of paradox is philosophically significant. It inspires us to seek the true reasons for the incompatibility between what reason shows us, and what empirical evidence shows us. We ought not simply take for granted that empirical science delivers truth. — Metaphysician Undercover

This is way past the lamp. The lamp is not a physical thing. These puzzles have no bearing on physical reality. That's a cognitive error everyone makes about them.

Also, there's more bad reasoning than "reason" in the discussions about these problems.

As explained above, I am not taking a standpoint of determinism. There are two very distinct senses of "determine", one consistent with determinism, one opposed to determinism (as the person who has a very strong will is said to be determined). I allow for the reality of both. — Metaphysician Undercover

You say randomness and determinism are compatible, and your justification is to use an alternate and unrelated meaning of the word determined?

But as of now, in this very post, I've convinced myself that I'm a randomist. But then again I've always suspected I'm a Boltzmann brain, and that's how randomists come into existence.

So...you're thinking of a limit in a vauge way ("symbolic"), and vaugely asserting the series "reaches" infinity, and then rationalize this with a mathematical system that defines infinity as a number. — Relativist

No. My thinking about limits is extremely precise and perhaps a bit more general than what you're accustomed to. I have never said that a series (or sequence if that's what you mean here) reaches infinity. I would not say that, and I did not say that.

What I said was that there is a mathematical view that sheds light on the subject, and makes it clear in where the limit of a sequence lives. The sequence 1/2, 3/4, 7/8, ... has the limit 1. Of course it never "reaches" 1. But you would have no objection to my putting {1/2, 3/4, ..., 1} into a set together. After all, I am allowed to take unions of sets: and {1/2, 3/4, ...} U {1} = {1/2, 3/4, ..., 1}. So it's a legit set.

Now 1 is in no way a "point at infinity," after all it's just the plain old number 1. And no member of the sequence ever "reaches" it. But it does live there as the limit; as the result of a well-defined limiting process.

I have suggested this mathematical model as a thought aid to these kinds of paradoxes. If you find it helpful all to the good, but if not, that's ok too. I find it helpful.

For what it's worth, in math, the natural numbers have an upward limit, called $\omega$, that plays the same role for the sequence 1, 2, 3, ... that the number 1 is for the sequence 1/2, 3/4, ...

It's the limit. It's more general notion of limit, one that allows us to reason about a "point at infinity." Which is exactly what these puzzles are about. That's why it's a handy framework for thinking about these kinds of puzzles.

You have a sequence that's defined (on/off, on a step, whatever) at each member of a convergent sequence; and you want to speculate on the definition at the limit. $\omega$ is exactly what you need; or rather, a set called $\omega + 1$, which is like the set {1/2, 3/4, ,,,, 1}. It's a set that contains an entire infinite sequence and its limit. It's exactly what we need to analyze these problems.

If it helps, here's the Wiki page on ordinals, at least so that you know they're a real thing. You can "keep counting past the natural numbers," and you get some very cool mathematical structures. Ordinals find application in proof theory and mathematical logic.

Although it's true that there are such mathematical systems, it doesn't apply to the supertask. Time is being divided into increasingly smaller segments approaching, but never reaching, the 1 minute mark. — Relativist

I'm going to defer talking about supertasks today, had enough for a while.

There is a mathematical (and logical) difference between the line segments defined by these two formulae:
A. All x, such that 0<=x < 1
B. All x, such that 0<=x <= 1 — Relativist

Please reread what I wrote. This is not on topic if you understand what I'm saying.

Your blurred analysis — Relativist

I'm doing my best to fit you with a sharper pair of mathematical eyeglasses to unblur your vision ... but you keep making a spectacle of yourself!!

conflates these, but it is their difference that matters in the analysis. The task maps exactly to formula A, but not to formula B (except in a vague, approximate way). Mathematics is about precise answers. — Relativist

You might consider using words like "reach" and "approach" with precision. They are not part of the mathematical definition of a limit. They're casual everyday synonyms that you are allowing to confuse you.

Then rather than recite the natural numbers I recite the digits 0 - 9, or the colours of the rainbow, on repeat ad infinitum.

It makes no sense to claim that my endless recitation can end, or that when it does end it doesn't end on one of the items being recited – let alone that it can end in finite time. — Michael

The natural numbers do not end, yet they have a successor in the ordinal numbers, namely $\omega$. This is an established mathematical fact.

I regard this as a helpful point of view when analyzing these kind of puzzles. I've explained it as best I can.

"It makes no sense" is not a logical argument. It's only a description of your subjective mental state. Once, violating the parallel postulate or the earth going around the sun or splitting the atom made no sense. You are not making an argument.

So I treat supertasks as a reductio ad absurdum against the premise that time is infinitely divisible. — Michael

If you only demonstrated the reductio. All you have is "it makes no sense," and that is not an argument.

Quite so. That's why these puzzles are not simply mathematical and why I can't just walk away from them. — Ludwig V

I think a lot of people feel that way.

The problem with Margaret Thatcher is that she thought that a dumb quip is a substitute for serious thinking. But then, she was a politician. She also believed that there is no such thing as society. — Ludwig V

I thought it was on point. People in the US like "forgiving student debt." But every nickel is just passed on to the taxpayers. Government doesn't have any money that it doesn't take from someone else. Or borrow and print, that's a nice game that has to end at some point too.

I agree that equality of outcome is not a reliable index of equality of opportunity and that people often talk, lazily, as if they were. But if equality of opportunity does not result in changes to outcomes, then it is meaningless. The only question is, how much change is it reasonable to expect? If 50% of the population is female and only eight of UK's top 100 companies are headed by women (Guardian Oct. 2021), don't you think it is reasonable to ask why? I agree that it doesn't follow that unfair discrimination is at work, but it must be at least a possibility. No? — Ludwig V

I agree. We need a balance between trying to homogenize society, and old-fashioned notions of merit.

Perhaps it's a matter of pendulum swinging and patience.

There are always issues with the NHS in the UK. But that's not about universal health care or not. It's about what can be afforded, what priority it has. Difficult decisions, indeed, but anyone with sense knows they must be made. That's why we have the national institute of clinical excellence. It is not perfect, but it is an attempt to make rational decisions; other systems do not even attempt to do that.
Of course, when my life, or my child's life, is at stake, I will put the system under as much pressure as I can to try everything. And to repeat, it's not about charity or robbing the rich. It's about insurance. — Ludwig V

Health care policy's hard, I agree. I've only heard anecdotal evidence about NHS.

I have no reason to give a flying fig about New York politics. — Vera Mont

It's a beautiful living experiment in what's known as restorative justice.

Crime is rampant and the DA is busy prosecuting the victims. People don't feel safe. It's going to sink Mayor Adams's once-promising political career.

I would think that many people interested in politics do follow New York City politics. But if you don't, that's cool. Not sure you are qualified to comment on the social justice approach to crime, though. It's failing in New York City in a very obvious way.

I can explain it very easily. There is two different senses of "limit" being used here. One is a logical "limit" as employed in mathematics, to describe the point where the sequence "converges". And "unlimited" is being used to refer to a real physical boundary which would be place on the process, preventing it from proceeding any further. There is no such "limit" to a process such as that described by the op. The appearance of paradox is the result of equivocation. — Metaphysician Undercover

Mathematicians would just refer to it as an "upper bound."

But you talk about a "real physical boundary." Here you imagine that the staircase is physical. It's not. The conditions of the problem violate known laws of physics.

It's only a conceptual thought experiment. And why shouldn't math apply to that?

But anyway, it's an upper bound. If it's a least upper bound, it's a limit.

I do think that there are members of the court who have an agenda. It is not that they are on Trump's side but that they see Trump as useful to their side. An expedient for attaining their conservative goals. — Fooloso4

Maybe. Dobbs certainly. But that's been a disaster for the GOP. It cost them the 2022 red wave and many local and special elections. It put abortion back on the table as a political issue. Centrist voters that could trend GOP on the economy, crime, and immigration, now have to vote Dem for abortion.

The percentage of the public for whom abortion is their top issue, represents a certain percent of the vote lost to the GOP and won by the Dems in every election at every level of politics forever, till Congress hashes out a law everyone can live with. And good luck with that.

In the long run, the conservative justices have done more harm to the conservative cause than if they'd left Roe in place. The Dems are going to win a lot of races they'd otherwise have lost, as long as abortion is on the ballot. And overturning Roe put abortion back on every ballot.

If this is part of some secret plan by the conservative Supes, I wonder what it is.

They do, they are just playing dumb. — Lionino

LOL. I'm just trying to take the subtle approach.

"Repugnant", is a commonly used word in philosophy. The argument I gave is logical, but what is concluded is that the assumption, "there is ontological randomness" is philosophically repugnant, because it would be counter-productive to the desire to know. Therefore it's more like a moral argument. The desire to know is good. The assumption of ontological randomness hinders the desire to know. Therefore that assumption is bad and one ought not accept it. — Metaphysician Undercover

I can agree with your reasoning that one "ought" not to accept it, but the reason is extra-logical. That is, if we are going by pure logic, you have not argued against it. It's like solipsism. Can't refute but pointless to believe it.

But consider: If the world is not random, then it's determined. And is that not equally repugnant? Nothing matters because we have no choice.

What do you say to that? It's repugnant either way. Either there's no meaning or ... there's no meaning. Is there a way out?

Since the argument concerns an attitude, the philosophical attitude, or desire to know, you're right to say that it is an argument concerning "feelings". But that's what morality consists of, and having the right attitude toward knowledge of the universe is a very important aspect of morality. This is where "God" enters the context, "God" is assumed to account for the intelligibility of things which appear to us to be unintelligible, thereby encouraging us to maintain faith in the universe's ability to be understood. Notice how faith is not certainty, and the assumption that the universe is intelligible is believed as probable, through faith — Metaphysician Undercover

God transcends logic, fair enough. But again, that's not a logical argument.

Not only is it pointless to believe it, but I would say it is actually negative. Choosing the direction that leads nowhere is actually bad when there are good places to be going to. — Metaphysician Undercover

Determinism is worse.

I agree that it is very important to leave as undecided, anything which is logically possible, until it is demonstrated as impossible. Notice what I argue against is the assumption of real randomness, that is completely different from the possibility of real randomness. — Metaphysician Undercover

In that case we are entirely in agreement. I never pretend to know the ultimate nature of the world. It may be random, it may be determined, it may be a combination of both, or it may be something entirely else such that the random/determined dichotomy is rendered meaningless.

That we ought to leave logical possibilities undecided was the point I argued Michael on the infinite staircase thread. Michael argued that sort of supertask is impossible, but I told him the impossibility needed to be demonstrated, and his assumption of impossibility was based in prejudice. — Metaphysician Undercover

But yes!! Here you are arguing that just because an idea is repugnant is no logical reason to reject it! So you should apply the same reasoning to randomness.

I believe that paradoxes such as Zeno's demonstrate an incompatibility between empirical knowledge, and what is logically possible. — Metaphysician Undercover

I think it's highly unlikely that the world will turn out to be a mathematical continuum like the real numbers. The real numbers are far too strange.

Most people will accept the conventions of empirical knowledge, and argue that the logically possible which is inconsistent with empirical knowledge is really impossible, based on that prejudice. But I've learned through philosophy to be skeptical of what the senses show us, therefore empirical knowledge in general, and to put more faith and trust in reason. So, to deal with the logical possibility presented in that thread, we must develop a greater intellectual understanding of the fundamental principles, space and time, rather than appeal to empirical knowledge. Likewise, here, to show that the logical possibility of ontological randomness is really impossible, requires a greater understanding of the universe in general. — Metaphysician Undercover

I agree with you there. I agree with most of what you wrote. Still I do want to understand why you see that @Michael is wrong to say that supertasks are logically impossible, when they are merely repugnant; yet you seem to reject that same reasoning when applied to randomness.

Also, don't you think determinism is at least as repugnant as randomness?

Imagine the nerve of somebody demanding fair treatment for all kinds of people, even the designated victims! Appalling, innit? — Vera Mont

Indeed it is. I quite share your sensibilities, or at the very least I have great sympathy for them.

But the larger point is that you have heard about people these days who prefer equity to equality, equality of outcome over equality of opportunity. You in fact seem to happen to be one of those folks.

But earlier, you claimed there were no such people.

So I take it that you have conceded my point. I'm not arguing the point of view pro or con; only that the point of view exists. That in fact you exemplify and represent it. So what you initially said, that you did not believe there were many of these people, was not quite true. Have I got that right? I don't want to presume, I may have misunderstood you.

Secondly, and again purely for conversation, on the issue of criminal justice. Do you follow New York City politics and current events? Do you support Alvin Bragg? Can you see how some people might think that compassion to criminals, no matter how well intentioned, can end up becoming a pronounced lack of compassion for their victims? Some of the folks pushed onto subway tracks by individuals previously treated gently by the criminal justice system might see it that way. Can you at least see that?

Indulge me in an analogy.

I see the Matrix (pictures): — keystone

This entire idea was completely lost on me.

Both perspectives accurately correspond to the simulation. So I agree that sets are fundamental, and I could even be convinced that digital rain is more fundamental than the Matrix. — keystone

Digital rain is more fundamental than the Matrix. That's very poetic.

But Let's not go there. I'm specifically talking about the (continuous version of the) Matrix where I believe continua are more fundamental than points. But I don't even want to debate this further, I'd rather show you what could be done with a Top-down approach and let you decide. — keystone

You know, it might be better if you would write a draft then edit it. This seems like stream of consciousness. It has a groovy vibe to it but it doesn't say anything.

I bring up the Matrix because, I want you to know that I recognize the unique purity and precision of the digital rain, but there are times, especially in discussions on geometry, when it's more effective to visually interpret the geometry from within the Matrix. Please allow yourself to enter the Matrix, try to understand my visuals, just for a little while. End of Matrix analogy. — keystone

As it happens I hate that stupid movie. It's a kung-fu flick with silly pretensions to pseudo-intellectuality. Also someone did the calculation and it turns out that humans make lousy batteries. Very inefficient.

Where is the line between your indulging yourself, and your trying to communicate a clear idea to me?

Okay, I lost you because I made a mistake. Let me try again:

Set: { (0,0) , (0,0.5) , (0.5,0.5) , (0.5,1) , (1,1) } where x1 and y1 in element (x1,y1) is a rational number

Metric: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 | — keystone

No idea what you are trying to do, what you are doing, why you are doing it, and why you are telling me about it.

Upon further consideration, I've decided to significantly restrict my focus to a smaller enclosing set. I am now interested only in what I want to call 'continuous sets' which are those sets where, when sorted primarily by the x-coordinate and secondarily by the y-coordinate, the y-coordinate of one element matches the x-coordinate of the subsequent element. For example, we'd have something like: — keystone

Like a triangular section of the plane? Why?

You're right, |x-y| doesn't qualify as a metric. Let me try again. Forget about Universal Set. Instead, I aim to define a Continuous Exact Set. A set is defined as an exact set if all elements satisfy |x-y|=0. I propose that within my enclosing set, the only Exact Set is the trivial set, containing just one element. Once again, this isn't a groundbreaking revelation; I am simply emphasizing that rational numbers by themselves are insufficient for modeling a continuum. — keystone

I just don't know what you're doing. You seem to be having fun, and I don't mind because this like taking a rest after the mortal combat of the staircase thread.

Zeno greatly inspires me, yet from my viewpoint, his paradoxes serve merely as an aside. I assure you, the core thesis I'm proposing is much more significant than his paradoxes. But to save me from creating a new picture, please allow me to reuse the Achilles image below as I try again to explain the visuals.


The story: Achilles travels on a continuous and direct path from 0 to 1.
The bottom-up view: During Achilles' journey he travels through infinite points, each point corresponding to a real number within the interval [0,1].
The top-down view: In this case, where there's only markings on the ground at 0, 0.5, and 1, I have to make some compromises. I'll pick the set defined above and describe his journey as follows:

(0,0) -> (0,0.5) -> (0.5,0.5) -> (0.5,1) -> (1,1) — keystone

Wasted on me, hope you got something from it.

In words what I'm saying is that he starts at 0, then he occupies the space between 0 and 0.5 for some time, then he is at 0.5, then he occupies the space between 0.5 and 1 for some time, and finally he arrives at 1. — keystone

No idea, eyes glazed long ago.

Inconsistent systems allow for proving any statement, granting them infinite power. While debating the consistency of ZFC is beyond my current scope and ability, my goal is to develop a form of mathematics that not only achieves maximal power but also maintains consistency. Furthermore, I aim to show that this mathematical framework is entirely adequate for satisfying all our practical and theoretical needs. — keystone

Quite a tall order.

I haven't studied his original work, so I can't say with certainty, but I don't believe I'm referring to Euclid's formulation. — keystone

Well Euclid considered points fundamental, along with lines and planes. But modern set-theory based math takes sets as fundamental. In fact there is nothing other than sets. You start with the empty set and the rules of set theory and build up everything else.

The word point is only used casually, to mean an element of some set, or a tuple in Euclidean space, or a function in a function space.

I'm familiar with these methods. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them. — keystone

I'm just throwing things out that seem related to what you're saying.

I'm getting there, and your feedback has been instrumental in enhancing my understanding of this 'digital rain'. Up until now, my approach has primarily been visual. — keystone

I'm very glad I can help. What is the digital rain? Do you remember the Church of the Cathode Ray from the movie Videodrome?

Aside: Please note that I will have a house guest for several days, which may cause my responses to be slower than usual. — keystone

No problem, take your time. I hope you and your guest have a lovely visit.

That's not quite what I'm saying. The process described by the op has no limit. — Metaphysician Undercover

Oh I had no idea we were still talking about the OP. This thread's gone way beyond that.

I thought you were making a more general point, that the limit lives in a different kind of conceptual space than the sequence itself, or that the limit was imposed on the sequence by observers.

If I misunderstood then nevermind. I've long forgotten the staircase problem. I don't think I ever actually understood it.

That should be clear to you. It starts with a first step which takes a duration of time to complete. Then the process carries on with further steps, each step taking half as much time as the prior. The continuity of time is assumed to be infinitely divisible, so the stepping process can continue indefinitely without a limit. Clearly there is no limit to that described process — Metaphysician Undercover

Well 1/2 + 1/4 + 1/8 + ... is a well known convergent sequence. It converges to 1. And surely we've all experience one second going by. So that's the paradox, right?

I think what's confusing you into thinking that there is a limit, is that if the first increment of time is known, then mathematicians can apply a formula to determine the lowest total amount of time which the process can never surpass. Notice that this so-called "limit" does not actually limit the process in any way. The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass. — Metaphysician Undercover

It has not been productive in the past for us to discuss mathematics, and your misunderstanding of limits is not my job to fix. I gave at the office. Nothing personal but you know we have been down this road. I sort of get what you're saying but mostly not. "The process carries on, unlimited, even though there's a limit." I haven't the keystrokes to untangle the myriad conceptual difficulties with that statement, and the beliefs and mindset behind it; even if I had the inclination. I hope you'll forgive me, and understand.

Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician. — Metaphysician Undercover

LOL. And the meaning of Moby Dick is only because of what we all determined the symbols to mean. Man and His Symbols, Jung. Yes we are symbolic beasts.

But within the sphere of math, the definition of a limit is as objective as can be. We lay down a definition, you know the business with epsilon and L, and we confirm that the sum converges; just as in the sphere of the English language, Moby Dick is a story about a bunch of guys who go whaling and it mostly doesn't end well.

To understand this, imagine the very same process, with an unspecified duration of time for the first step. The first step takes an amount of time, and each following step takes half as much time as before. In this case, can you see that the mathematician cannot determine "the limit"? All we can say is that the total cannot be more than double the first duration. But that's not a limit to the process at all. It's just a descriptive statement about the process. It is a fact which is implied by an interpretation of the described process. As an implied fact, it is a logical conclusion made by the interpreter, it is "not inherent to the sequence", but implied by it. — Metaphysician Undercover

I'm sorry, I can't really talk about the staircase problem specifically, I never paid much attention to it at the beginning. I mostly got interested in this thread when other issues were introduced. But mathematicians are very good at determining limits, and the one in question is perfectly well known to everyone who ever took a year of calculus. You might take a look at the Wiki page on limits.

That it is not inherent, but implied, can be understood from the fact that principles other than those stated in the description of the process, must be applied to determine the so-called "limit". — Metaphysician Undercover

You don't need any esoteric "principles other than those stated in the description of the process" to determine the sum of a geometric series as a particular limit.

We must subvert our tendency to compete — Benj96

I just noticed this. What means would you use to bring this about?

This appears to be the case. — Vera Mont

Just for light conversation ... when I say that a lot of people these days are advocating for equality of outcome rather than equality of opportunity ... you do not know what I am referring to? The DEI movement, social justice, wokitude, and the like? Disciplinary standards relaxed in schools, admission criteria relaxed in universities, the criminal justice system biased in favor of criminals, massive social change for the purpose of balancing out racial categories?

This news has not yet reached your province?

No. What Trump says and does and what the Supreme Court says and does are not the same. — Fooloso4

I only mentioned it because this is a bit of Trumpy thread. A lot of people think the court's on Trump's side and not being judicially impartial. And opinions about that correlate with people's opinions on Trump. So this is really a Trump thread. Or at least a Trumpy thread. That was my thought process anyway. But I'm not actually participating in the thread, so I haven't got any strong feelings, I had just noted that there's a zillion-page long Trump thread, and I assumed that was there to soak up the gusher of opinion on the guy.

Gotta say, the man was on reality tv for ten years, he knows what the American people love, or love to hate. The historians will have a field day, if we all live that long.

You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact. — Relativist

You can think of it that way. Or you can think of it "reaching" its limit at a symbolic point at infinity. Just as we augment the real numbers with plus and minus infinity in calculus, to get the extended real numbers, we can do something analogous with the natural numbers, and augment them with a symbolic point at infinity $\omega$, so that the augmented natural numbers look like this:

1, 2, 3, 4, ..., $\omega$

Now a sequence is just a function that for each of 1, 2, 3, ..., we assign the value of the sequence, the n-th term. And we can simply assign the limit as the value of the function at $\omega$. It's perfectly legitimate. We can define a function with ANY set as its domains. So a sequence is a function on $\omega$, and a sequence augmented with its limit (or any other value!) is just a function on $\omega + 1$.

This is a key point. I've stated it a number of times recently and I'm not sure I'm getting through. The natural numbers augmented with a point at infinity is a perfectly good domain for a function; and we can use such a function to identify each of the points of a sequence, along with the limit.

I also think you are misinterpreting the meaning of limit. — Relativist

On a forum our words must speak for themselves. But in this instance I can assure you that nothing could possibly be farther from the truth.

This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L. — Relativist

Wiki is not necessarily a good source for mathematical accuracy or subtlety of expression, and in this case they have led you astray.

I hope very much that you will give some thought to what I wrote about defining the limit of a sequence as the value of some function on the naturals augmented with a symbolic point at infinity; or more concisely, as a function on $\omega + 1$. A sequence is just a function on $\omega$, which is a synonym for $\mathbb N$. You can "attach" the limit to the sequence by extending the same function to one on $\omega + 1$ .

I hope this is clear. I find it an extremely clarifying mental model of what's going on with a sequence and its mysterious limit. "Where does the limit live?" I get that it's kind of confusing. The limit lives at the point at infinity stuck to the right end of the natural numbers.

1, 2, 3, 4, ... $\omega$

That's how people need to learn how to count in order to better understand supertasks and limits.

You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?
— fishfry
No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does? — Relativist

Sure, after 1 second. It's perfect obvious from daily existence. When I got up to make a snack I did first walk halfway to the kitchen then halfway again. So now I'm arguing for supertasks. But I could just as well argue against supertasks. So whatever you said, I could probably convince myself to agree with it.

I think the word "reach" is being abused in this conversation. It comes out of badly taught calculus classes, and Wikipedia.

There is no limiting process in the premises of the op, nor in what is described by ↪Relativist . The "limiting process" is a separate process which a person will utilize to determine the limit which the described activity approaches. Therefore it is the person calculating the limit who reaches the limit (determines it through the calculation), not the described activity which reaches the limited. — Metaphysician Undercover

Wow that's deep. Deep and wrong at the same time. That's interesting.

If I am understanding you: You say that if we have a sequence; that if that sequence happens to have a limit, then the limit is not inherent to the sequence, but is rather imposed by the observer.

I suppose the analogy is color, which is in the eye-brain system of the observer, not in the object or even in the light.

But actually, the limit can be considered part of the sequence. Just as a sequence is a function defined on the natural numbers; a sequence along with its limit can be defined as a function on the natural numbers augmented with a point at infinity, which I've been calling $\omega$.

It's really no different than taking the set {1/2, 3/4, 7/8, ...} and augmenting it with the number 1, to yield the new set {1/2, 3/4, 7/8, ..., 1}. Surely you can see that 1 is a perfectly sensible number on the number line. In many ways it's the ONLY sensible number. All other numbers are derived from it. That and 0. Give me 0 and 1 and I'll build all the numbers anyone needs.

So if that's what you're saying, I find that a very interesting thought. But there is no reason to imbue limits with mysticism. They're very straightforward. They're just the value of a sequence at the augmented point at infinity; which, if you don't like calling it that, is just adding the number 1 to the 1/2, 3/4, ... sequence.

This isn't the sense of "counting" I'm using. The sense I'm using is "the act of reciting numbers in ascending order". I say "1" then I say "2" then I say "3", etc. — Michael

Yes, I agree with you that math and physics use different definitions.

I apologize for getting crabby last night. As I went to bed I was thinking, Why am I snarling at someone about supertasks, I don't even care about supertasks.

You're right, I was not the one you were originally addressing. I jumped in because I was annoyed by your total lack of logic in claiming that supertasks are metaphysically impossible or logical contradictions. I agree with you that supertasks don't exist physically today, but I allow for the possibility of new physics in the future, just as there's always been new physics in the past. I don't think you've supported your metaphysical or logical arguments. That's why I jumped in.

Also it's perfectly clear that I can walk a mile, and I first walked the first half mile, etc., so if someone (not me, really!) wanted to argue that supertasks exist on that basis, I'd say maybe they have a point.

P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum.

P2. 30 + 15 + 7.5 + ... = 60

C1. The sequence of operations1 described in P1 ends at 60 seconds without ending on some final natural number.

But given that ad infinitum means "without end", claiming that the sequence of operations described in P1 ends is a contradiction, and claiming that it ends without ending on some final operation is a cop out, and even a contradiction. What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?

So C1 is a contradiction. Therefore, as a proof by contradiction:

C2. P1 or P2 is false.

C3. P2 is necessarily true.

C4. Therefore, P1 is necessarily false.

And note that C4 doesn't entail that it is metaphysically impossible to recite the natural numbers ad infinitum; it only entails that it is metaphysically impossible to reduce the time between each recitation ad infinitum. — Michael

I think "reciting natural numbers" is a red herring, because it's perfectly clear that there are only finitely many atoms in the observable universe, and that we can't physically count all the natural numbers.

But let me riddle you this. Suppose that eternal inflation is true; so that the world had a beginning but no end, and bubble universes are forever coming into existence.

And suppose that in the first bubble universe, somebody says "1". And in the second bubble universe, somebody says, "2". Dot dot dot. And bubble universe are eternally created, with no upper bound on their number.

Therefore: Under these assumptions, there is no number that doesn't get spoken. And therefore, all the numbers are eventually counted.

You see we don't have to "reach the end," since we can't do that. All we have to do is show that there is no number that never gets counted. Therefore they all do. It's a standard inductive argument. You show something's true for all natural numbers because there can't be a smallest number where it's not true.

I remind you that while eternal inflation is speculative but is taken seriously by a lot of smart people.

Therefore I claim that it is metaphysically possible to physically count the natural numbers; and that no logical contradiction is entailed. I'll grant you that I haven't yet shown how to do it in finite time, and so I have not refuted your point. I'm giving more of a plausibility argument that someday, there might actually be a finite-time supertask. We just don't know. You personally can not know. That's my real point, bottom line.

You cannot know what future physics will allow or conceptualize. That's my whole argument. That's why I say that supertasks violate contemporary physics, Planck scale and all that. But based on the shocking paradigm shifts of the past, there will be shocking paradigm shifts in the future; and physically actualized infinitary processes are as good a candidate as any for what comes next.

I wrote a response to @NoAxioms above in which I laid out my thoughts, it might be of interest ... https://thephilosophyforum.com/discussion/comment/900398

Thanks again for your good cheer in not firing back!

What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something. — noAxioms

Well physics is of course exempt from math and logic. The world does whatever it's doing. We humans came out of caves and invented math and logic. The world is always primary. Remember that Einstein's world was revolutionary -- overthrowing 230 years of Newtonian physics. The world told us what new math to use. The world is not constrained by math, nor logic, nor by any historically contingent work of fallible man.

Math and even logic have always been drawn from looking at the world around us. So just as an aside to the main discussion, but responding to this one sentence that caught my eye ... physics IS exempt from math and logic. Meaning that historically, and metaphysically, physics is always ahead of math and logic and drives the development of math and logic.

But to the main question, the physical/mathematical distinction is important. I can never count all the integers in the physical world (as far as we know -- to be clarified momentarily); but in math I can invoke the axiom of infinity, declare the natural numbers to be the smallest inductive set guaranteed by the axiom, and count its contents by placing it into order-bijection with itself. That is: The identity map on the natural numbers is an order-preserving bijection that shows that the natural numbers are countable.

The former is a physical activity taking place in the world and subject to limitations of space, time, and energy. The latter is a purely abstract mental activity. How meat puppets such as ourselves come to have the ability to have such lofty abstract thoughts is a mystery. And if we are physical beings; and if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation? That's another good question.

Perhaps our very thoughts of infinity are nature's way of manifesting infinity in the world.

So bottom line it's clear to me that we can't count the integers physically, but we can easily count them mathematically. And the reason I say that we can't physically do infinitely many things in finite time "as far as we know," is because the history of physics shows that every few centuries or so, we get very radically new notions of how the world works. Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.

You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so? — noAxioms

Not a flaw, of course, any more than general relativity revealed a flaw in Newtonian gravity. Rather, I expect radical refinements, paradigm shifts in Kuhn's terminology, in the way we understand the world. Infinitary physics is not part of contemporary physics. But there is no reason that it won't be at some time in the future. Therefore, I say that supertasks are incompatible with physics ... as far as I know.

I utterly reject the notion that supertasks are a logical contradiction or metaphysical impossibility. They're only a historically contingent impossibility. We split the atom, you know. That was regarded as a metaphysical impossibility once too.

I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time. — noAxioms

I'm not being specific like that. I'm only saying this:

There have been radical paradigm shifts in physics in the past;

There will certainly be radical paradigm shifts in the future; and

The next shift just may well incorporate some notion of physically instantiated infinities or infinitary processes; in which case actual supertasks may be on the table.


I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist; then when shown to be logically consistent, considered only a mathematician's plaything, of no use to more practical-minded folk; and then shown to be the most suitable framework for Einstein's radical new geometry of spacetime.

Mathematical curiosities often become physicists' tools a century or more later. I think it's perfectly possible that physically instantiated infinities may become part of mainstream physics at some point in the future.

I will close with two contemporary examples of where speculative physics is starting to think about infinity.

One, eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending. In this eternal multiverse are many bubble universes; either infinitely many, or at least a very large finite number. Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.

And two, the many-world interpretation of quantum physics. Most people have heard of the Copenhagen interpretation, in which observing a thing causes the thing to be in one state or another; whereas before the measurement, it was neither in one state nor the other, but rather a superposition of the two states.

In Everett's many-world's interpretation, an observation causes the thing to be in both states in different universes. The universe splits in two, one in which the thing is in one state, and another universe it's in the other state. In some other universe I didn't write this. I know it sounds like bullshit, but Sean Carroll, a very smart guy and a prominent Youtube physicist (he's a real physicist too) is a big believer. He's recently moved away from mainstream physics, and more into developing a new philosophy of physics that incorporates many-worlds. How many worlds are there? Again this is a little vague, infinitely many or a large finite number.

These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists. Would anyone really bet that they personally can predict the next 200 years of physics?

By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask. — noAxioms

Well I can walk a mile, and I first walked the first half mile, and so forth, so it's a matter of everyday observation that supertasks exist. That would be an argument for supertasks. Zeno really is a puzzler. I don't think the riddle's really been solved.

Well that's for reading, there's been a lot of back and forth lately and I hope I was able to at least express what I think about all this.

Appeal to consequences — wonderer1

Thanks.

I agree — Benj96

I'll quit when I'm ahead here then :-)

A healthy society can have universal healthcare — Benj96

Many issues with long wait times at NIH in Great Britain. And in Canada, they offer assisted suicide for depression. I'd like to see some datapoints where universal health care has worked. Not an expert on health care policy, just repeating anecdotal evidence re Britain and Canada. Not necessarily defending the expensive US system, but it's a complicated issue. Just giving people free stuff is not a panacea. Who pays for the free stuff? As Margaret Thatcher once noted, "The problem with socialism is that you eventually run out of other people's money."

Which has no bearing on what I'm arguing. — Michael

You are not arguing, you're repeating your lack of argument. I'll let you have the last word, you are incapable of rational discussion.

I'm not talking about infinite sets and transfinite ordinals. I'm talking about an infinite succession of acts. If you can't understand what supertasks actually are then this discussion can't continue. — Michael

A discussion can't continue when you keep making unsubstantiated, evidence-free claims.

I would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. Yes it's speculative, but nobody is saying it's "metaphysically impossible" or "logically incoherent."

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

Here's a definition for you: "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time".

The key parts are "sequence of operations" and "occur sequentially". — Michael

Please stop embarrassing yourself.

If I write the natural numbers in ascending order, one after the other, then it is metaphysically impossible for this to complete (let alone complete in finite time). This has nothing to do with what's physically possible and everything to do with logical coherency. — Michael

It's physically impossible. I have no idea why you keep claiming it's "metaphysically" impossible or logically incoherent. What's logically incoherent about infinite sets and transfinite ordinals? You just keep repeating the same unsupportable claims. You can count the natural numbers by placing them into bijective correspondence with themselves. This is the standard meaning of counting in mathematics.

And it doesn't address the issue. — Michael

I asked you to consider a hypothetical world and you pretended I was talking about mathematical sets.

If I write the natural numbers in ascending order, one after the other, then this can never complete. — Michael

Yes, the observable universe is finite. We're agreed on that. How many times are you going to try to convince me of something I've already agreed with many times?

To claim that it can complete if we just write them fast enough, but also that when it does complete it did not complete with me writing some final natural number, is just nonsense, — Michael

I have not claimed otherwise.

and so supertasks are nonsense. — Michael

According to current physics. That's as far as we can go.

That we can sum an infinite series just does not prove supertasks. — Michael

Nor does it disprove their metaphysical possibility. We just don't know at present.

No, I'm responding to you to explain that your reference to mathematical sets and mathematical limits does not address the issue with supertasks. — Michael

I gave you a mathematical model that puts your unsupported claims into context.

I've provided arguments, and examples such as Thomson's lamp that shows why. — Michael

Thompson's lamp shows nothing of the sort. I've explained that to you repeatedly as well.

Would you prefer the term "act"? It is metaphysically impossible for an infinite succession of acts to complete. — Michael

Metaphysically impossible? Repeating a claim ad infinitum is neither evidence nor proof.

Have you even looked up supertasks? I don't know how you can confuse them with mathematical sets. — Michael

I'm not the one advocating for supertasks, yet you keep arguing with me that they are impossible.

If there are an infinite number of whole numbers, and an infinite number of decimals in between any two whole numbers, and an infinite number of decimals in between any two decimals, does that mean that there are infinite infinities? — an-salad

There are indeed infinitely many infinities, but not by the argument you gave.

And an infinite number of those infinities? And an infinite number of those infinities? And…(infinitely times. And that infinitely times. And that infinitely times. And…) … — an-salad

Yes. There are many online resources.


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

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

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

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

The task consists of a sequence of actions occurring at intervals of time that decrease by half at each step: 1/2 minute, 1/4, 1/8,.... It is logically impossible for this sequence of actions to reach the 1 minute mark (the point in time at which the descent is considered completed), it just gets increasingly close to it. — Relativist

Zeno again?

Say (in some hypothetical world, say current math or future physics) that we have a "sequence of actions" as you say, occurring at times 1/2, 3/4, 7/8, ... seconds.

It's perfectly clear that 1 second can elapse. What on earth is the problem?

You are falling into the trap of thinking a limit "approaches" but does not "reach" its limit. It does reach its limit via the limiting process, in the same sense that 1/2, 3/4, 7/8, ... has the limit 1, and 1 is a perfectly good real number, and we all have had literally billions of experiences of one second of time passing.

I can't imagine what you are thinking here, to claim that one second of time can't pass.

I have repeatedly noted in this thread that we can symbolically adjoin a "point at infinity" to any countably infinite sequence, and that's where the limit lives. We can note that 1/2, 3/4, 7/8, ... has the limit 1, which lives in the ordered set {1/2, 3/4, 7/8, ..., 1}.

We can also do the same thing in the integers as 1, 2, 3, 4, ..., $\omega$, where $\omega$ can be thought of as a formal symbol that's greater than every natural number. It also has technical importance as the first transfinite ordinal.

Either way, sequences do "reach" their limit via the limiting process, though the sequence itself does not necessarily attain the limit. It's just semantics.

You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?

Based on this picture, what I want to say is that Achilles can occupy any position on the continuous line, but, for this specific example where the ruler only has a few tick marks on it, I'm limited to describing his location using one of five specific intervals:
(0,0)
(0,0.5)
(0.5,0.5)
(0.5,1)
(1,1) — keystone

Sorry what? We're doing Zeno now? I must pass on that.

I believe what I want to do is define a 2D metric space on set S={(0,0),(0,0.5),(0.5,0.5),(0.5,1),(1,1)} where each element is an ordered pair (x1,x2).

While I will eventually explore higher dimensional spaces, for now, let's say that my sandbox is limited to sets of ordered pairs of rational numbers. — keystone

I do not know what you are talking about now.

You're right. Scratch the Universal Metric. If my metric is |x2-x1| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. This is yet another trivial conclusion since we know that rational numbers alone cannot model a continuum. — keystone

Lost me again. In a metric space the distance between two points is 0 if and only if they are the same point.

Elements of sets are sometimes called points, but it's possible to do set theory without elements!
— fishfry
Is it sets all the way down or do you eventually get to points? Anyway, you don't have to answer that question. I'm willing to agree that it doesn't matter which is more fundamental. What matters is what approach yields the most powerful math. Let's move on. — keystone

It's sets all the way down. In set theory everything is a set.

Points are just elements of a set. Sometimes a "point" in a function space can be a function. Sometimes a point is just a tuple of coordinates in Euclidean space. Points aren't fundamental. Perhaps you're thinking of Euclid's original formulation of geometry.

You are trying to invent something more powerful than contemporary math?

I don't get the top-down idea. 'Splain me please.
— fishfry
I was hoping to get closure on the open topics first, but if you don't have any problems with this post then I think we're there. [/quoote]

I don't understand what you are doing. Seems like random flailing.

— keystone

By the way, if you ever feel like my time is running out then please let me know and I'll plow through. But at the current pace I'm extracting a lot of value from our conversation. — keystone

I'm fine.

By the way I wanted to mention that there are often two ways of describing a mathematical object, internal and external. For example we can define the real numbers internally, by building them up from the empty set to get the naturals, integers, rationals, and finally reals.

Or, we can define the reals as the unique Dedekind-complete totally ordered set. That characterizes the reals without bothering to construct them. Perhaps you're getting at this.

You also talked about cuts, and perhaps you're interested in Dedekind cuts, which are used to construct the reals out of sets of rationals.

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

You seem to want to make points out of cuts in a line, but I don't see where you're going with that.

No, I'm only talking about topological metric spaces. — keystone

A metric space is typically just called a metric space. There aren't "nontopological" metric spaces. Any metric space can be made into a topological space by defining the open sets in terms of the metric.

I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, they act like topological spaces without a metric), — keystone

This is kind of muddled. Typically we start with a set and put some structure on it -- a metric, a topology, whatever. It makes no sense to talk about "outside" the space till we say what set that is. For example, what's outside the real numbers. Well the complex numbers are, but so are all the animals on Old McDonald's farm. The complement of any set is the entire rest of the universe; and if you don't say what universe you're working in, you run in to the "set of all sets" paradox. The unrestricted complement of a set is not a set. So it would be good if you could clarify this point. What's outside your metric space of interest?

and internally, they have entirely geometric characteristics (meaning internally, they are indistinguishable from metric spaces without the topological aspects). — keystone

Metric spaces are indistinguishable from metric spaces, yes. But isn't that a trivial remark?

And as I said, you will have trouble rigorously defining what you mean by outside of your metric space, unless you first say what the enclosing set is. So please do. By analogy, if you wish to discuss what's outside the real numbers, you have to say if you're talking about the complex numbers, the quaternions, or everything in the entire mathematical universe, which turns out to not be a set. Because the set of all sets that don't contain themselves is a member of the "outside" of the real numbers. Hope I'm making this clear.

Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm referring to a non-trivial universal metric. — keystone

As it happens, the trivial topology is already defined as the opposite idea. The discrete metric has the most possible open sets. The trivial metric has the fewest open sets. Only the empty set and the entire space are open.

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

But you can't just eliminate the one metric that falsifies your idea, there could be other weird ones. You have to say exactly what you mean.

Also I have no idea what the "universal metric" is. You have not communicated that to me.

There's a whole SEP article on holes. Deep stuff.
— fishfry
Wow, it's a deeper topic than I imagined. — keystone

Holes are deep!

It turns out the photos were more helpful to me than to you. You've helped me realize that what I'm actually discussing are metrics. — keystone

Ok.

So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this.
— fishfry
There are two primary methods for creating core mathematical artifacts: — keystone

You just ignored my comment and steamrollered over it. Why do I care which is more fundamental? I don't even know what that means. Sets are fundamental, then you add properties. That's how it works.

Bottom-up Approach:
Starts with tiny building blocks to assemble (or at least define) more complex mathematical objects.
Points are considered fundamental in this approach. — keystone

Sets are fundamental, not points. Elements of sets are sometimes called points, but it's possible to do set theory without elements! All you actually need is to describe the relationships among sets, without regard for the internal contents of the sets.

This method is akin to assemblage art, where separate elements are combined to form a whole.

Top-down Approach:
Begins with a larger, unified block and divides it to produce mathematical objects.
Continua are fundamental in this approach.
Similar to sculpting, where material is removed from a larger mass to reveal the desired form. — keystone

I can't imagine how you would get anything done that way. And you are not getting me to believe you have a coherent idea about it.

I've observed that orthodox mathematics predominantly favors the bottom-up approach. — keystone

Starting from sets, yes. Lot of mindshare the past century and a quarter. There's also type theory, which I imagine you'd see as another bottom up approach. I don't know what a top down approach to mathematical ontology would look like.

However, my informal exploration of the top-down method has revealed — keystone

Not to me. Maybe to you. You have not yet communicated to me what is a top-down development of math. How would you top-down construct or define the real numbers? Unless you mean axiomatically. Is that what you mean?

a perspective where everything seems to fit together perfectly, without any apparent disadvantages, paradoxes, or unresolved issues compared to the bottom-up view. — keystone

Where's the beef? That's handwavy, tells me nothing.

I'd like to share this perspective with you, — keystone

I'd like to hear it. What is a top-down construction of the real numbers? Of the integers? Of the number 6?

so you can either help identify any potential flaws (I don't want to waste my time on a dead end) or guide me further (for example, I've already learned from this discussion that I should be describing them as topological metric spaces rather than elastic rulers). — keystone

A metric space is a metric space. If you are interested in metric spaces there's a large literature on the subject.

I don't get the top-down idea. 'Splain me please.

↪jgill That's true, but that just makes it physically impossible. I think it's stronger: logically impossible. — Relativist

@Michael keeps making the same claim, and I do not understand the argument.

I agree that it's impossible to do infinitely many physical things in finite time according to present physics.

I do not see what the logical impossibility is.

Is there good reason why the Supreme Court should not have already quickly and unequivocally ruled that Trump is not above the law? — Fooloso4

Shouldn't this thread go into the Trump thread, which was specifically created so that people could vent their Trump spleen in one place?

No. An infinite set is not an infinite sequence of events. An infinite sequence of events would be counting every member of an infinite set. It is metaphysically impossible to finish counting them. — Michael

Ok. Clearly this is a matter of semantics.

Mathematically, if I have a set of events $\{e_1, e_2, e_3, \dots \}$, there's no problem whatsoever.

You seem to assign some meaning to the word "event" that I don't understand. Must an event be physical? In probability theory we have events that need not be physical, such as the probability of choosing a random real number between 0 and 1/3 from the unit interval. That's an event with no physical meaning at all.

An infinite sequence of events from the set I defined above would be $e_1, e_2, e_3, \dots$. No muss no fuss. That's an infinite sequence of events.

Perhaps you can tell me what an event is, bearing in mind that event is a technical term in probability theory that does not imply physicality.

https://en.wikipedia.org/wiki/Event_(probability_theory)

That's not relevant to the claim I'm making. — Michael

The claim you're making is not one I'm disputing.

I'm saying that if I have finished counting the members of some set then some member must be the final member I counted. — Michael

I disagree. Counting means to place the elements of some set in order-bijective correspondence with the natural numbers, or in a more general context, with some ordinal.

By that definition, we can easily count the natural numbers. The identity map will do.

You seem to think counting is a physical process. That's fine for most contexts, but it's not the only meaning of counting.

For example we have the famous countable/uncountable distinction between infinite sets. A set is countable if it can be placed into bijection with the natural numbers. The natural numbers, the integers, the rational numbers, and the algebraic numbers are all famous examples of countable sets that are infinite.

If you mean to say that we can't physically count the natural numbers, of course I agree. I personally could not get past 13 or 14 or so without losing interest. We could use a supercomputer, but even that has finite capacity. We could use the entire observable universe, but that contains only $10^{78}$ atoms. So sure, physical counting is constrained by resources.

But who's saying otherwise? Perhaps you can explain that to them, since I have never said anything remotely like that.

I understand that as a trained mathematician, you have the ability to articulate complex ideas clearly using descriptive language. I admire that skill, but as an engineer, my strengths lie more in visual thinking. This is particularly true with mathematics, where I sometimes struggle to express my thoughts precisely in words. Consequently, I tend to rely on illustrations to communicate my ideas. I ask for your patience and flexibility in trying to understand the essense of my message. — keystone

I just don't see where you're going with all this. You're pointing out that some topological spaces aren't metrizable. Right?

Instead of saying that there cannot exist a "Unversal Elastic Ruler" what if I say there cannot exist a "Universal Metric"? — keystone

Oh but there is one. For any universe or set, define a metric as follows: d(x,y) = 1 if x and y are different, and 0 if x and y are the same. This is known as the discrete metric.

You can put the discrete metric on any space of points whatsoever.

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

Think of it like this: a hole is an emergent property. To have a hole, you first need an object that can contain a hole. In this sense, the object is more fundamental. We begin with the object, which holds the potential for a hole. Then, once we make a cut, what we have is the same object, but now with an actual hole in it. — keystone

There's a whole SEP article on holes. Deep stuff.

https://plato.stanford.edu/entries/holes/

I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points. — keystone

I don't know what's common. Does it matter?

If you return to my photographs, — keystone

I did not understand the photos.

you will see that I start with a continous object and put cuts in it. I call those cuts points. Just as an object is more fundamental than the hole, with my view a continua is more fundamental than the cuts (i.e. points). I used k-continua and k-points instead of continua and points because I wanted to avoid a debate over what's more fundamental. In my sandbox the continua are more fundamental. If you want to grant me that, then perhaps we can set aside all this 'k-' terminology. — keystone

Why is "what's more fundamental" important? Do you think I hold one view versus the other? What difference does it make?

Okay, this feels like progress. Let's iron out the points discussed above and then I'll give you more details on where this is going.

If it's not obvious, I want you to know that I really appreciate you sticking with me on this. — keystone

Ok I'll keep going as long as I can, but I feel like I'm going down in warm maple syrup.

So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this. Did I argue the contrary at some point?

Because I'm arguing against the possibility of a supertask. You're the one who interjected with talk of mathematical limits. I'm simply responding to explain that this doesn't address the concern I have with supertasks. — Michael

Ok.

I'm not saying that it's the same. I'm saying that as well as being a physical impossibility, supertasks are also a metaphysical impossibility. — Michael

Now that's something I disagree with. But I don't care about supertasks much so it's better if I don't engage.

No physical law can allow for an infinite sequence of events to be completed. — Michael

This is an open question. Of course no physical law currently known allows for supertasks, but you can't say what we will regard as physical law in another couple of centuries.

The very concept of an infinite sequence of events being completed leads to a contradiction. — Michael

You keep repeating that, but you have no evidence or argument.

To claim that it is metaphysically possible to have finished writing out an infinite number of natural numbers but also that there is no final natural number that I wrote is to talk nonsense. — Michael

Do you deny infinite mathematical sets?

If I finished writing out any number of natural numbers than there will be a final natural number and that natural number will be a finite number. This is a metaphysical necessity. — Michael

Mathematically that's not true. The set {1, 2, 3, 4, ...} contains all the natural numbers, but there's no last number.

I already agree with you that there are no infinite collections of physical objects according to currently accepted theories of physics. But you can't claim that there will never be any such theory.

And besides, eternal inflation posits a temporally endless universe. It's speculative, but it's part of cosmology. Serious scientists work on the idea. So at least some scientists are willing to entertain the possibility of a physically instantiated infinity.

This is an example of a supertask:

I write down the first ten natural numbers after 30 seconds, the next ten natural numbers after 15 seconds, the next ten natural numbers after 7.5 seconds, and so on.

According to those who argue that supertasks are possible I can write out infinitely many natural numbers in 60 seconds.

Examples such as Thomson's lamp show that supertasks entail a contradiction. So even though it is true that 30 + 15 + 7.5 + ... = 60, it does not follow that the above supertask is possible.

It makes no sense to claim that I stopped writing out the natural numbers after 60 seconds but that there was no final natural number that I wrote. — Michael

You're continuing to argue against a position I don't hold. Why are you doing this? There's no interesting conversation to be had. Supertasks are not consistent with known physics. We're agreed on that.

I would, however, disagree with you that being inconsistent with known physics is the same as logical impossibility. Known physics changes all the time, sometimes radically.