The trouble comes like many such Zenos paradoxes. You wish to speak and reason in the realm of actual infinities when you cannot do such a thing. Reasoning fails there. So your tool of reasoning is the wrong tool. Well done.

The potential infinity realm can still use reasoning.

But that requires a currently bounded scenario.

So, you loaded the question and that is not nice.

In any case the example is horrific as well with each side being half the pervious. It is not neat. It's not even really that interesting. The sides should be of the same length. And since infinity extends in both directions, or all directions, and not just one direction your arbitrary single bound of natural numbers is yet another nonsensical limit that does not help in any way. The absolute value and zero included as a set is more tenable as an infinity and it ruins your nonsense. Lead be thou gold!

I know you are but what am I, ... infinity!

: First off, thanks so much for your insightful and kind response!

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. — fishfry

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. 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?

This interesting puzzler has a clear and unambiguous mathematical resolution. — fishfry

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.

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. — fishfry

Let me propose a third, perhaps controversial, scenario:

3) The probability of each event is 1.

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. While this notion may seem preposterous, similar issues emerged with calculus, which were resolved using limits. For instance, finding the tangent by dividing by zero results in a singularity, yet one can sensibly approach a zero denominator. In a similar vein, I argue that dealing with the set of all natural numbers also results in a singularity, but probabilities can be sensibly managed by approaching an infinite set. In other words, infinite sets as completed objects do not truly exist.

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

What do you think?

You wish to speak and reason in the realm of actual infinities when you cannot do such a thing. Reasoning fails there. So your tool of reasoning is the wrong tool. Well done. — Chet Hawkins

I don't think you understand my position. I'm playing in the "paradise" which Cantor created (involving infinite sets) not because I believe in it but because I want to convince others that it's a mirage (at least in my view).

The sides should be of the same length. — Chet Hawkins

Isn't the concept of an infinite-sided die that could fit in your hand intriguing? It’s impossible to construct a die with finite volume if you insist that all sides must be of equal length.

And since infinity extends in both directions, or all directions, and not just one direction your arbitrary single bound of natural numbers is yet another nonsensical limit that does not help in any way. — Chet Hawkins

Do you not believe in natural numbers being bounded by 0 (or 1) on one end? And regarding time, isn't it widely believed that time had a beginning (meaning one boundary of time is t=0)?

There are two types of infinitely sided dice; those whose set of sides is actually infinite - meaning Dedekind-infinite as in the set of dice sides possessing a countably infinite proper subset, versus those dice whose set of sides is potentially infinite - meaning Dedekind-finite but without having an a priori finite upper-bound on their number of sides.

You need a non-standard probability theory to express the idea of an infinitely sided fair die, but the idea only makes sense for Dedekind-finite dice.

Rather than assigning a real number to a set of outcomes to denote the probability of the set, which doesn't work in the case of infinitely sided dice, we can in the case of a fair and infinite die eliminate the distinction between sets of outcomes and their probabilities, because in this case the probability of a set of outcomes is equivalent to the set itself.

So let P(N) = N

- We simply drop the normalisation factor since it is a constant, and let N directly denote both the set of natural numbers and the probability of choosing a natural number (i.e the value of probability one) .

Let N/a denote the set of natural numbers that is the complement of the subset of natural numbers a. Then

P(N/a) = N/a

P(a) = a

P(a OR N/a) = N/a + a = N

- If N/a is cofinite, meaning that it's complement is finite, then its interpretation as a probability value is larger than the probability value for any finite set of naturals, but is nevertheless smaller than N.

- if a is finite but non-empty, then it's interpretation as a probability value is smaller than the size of any cofinite set and any infinite set, but is nevertheless larger than zero.

- If both N and a are infinite, then their corresponding probability values are of intermediate magnitudes between the two previous cases.

I think this is pretty much all that is needed for a basic non-standard probability treatment of a fair infnitely sided dice, and the resulting measure is both countably additive and normalizes to 'one'.

Lastly, the assumption of Dedekind finiteness is important, because we don't want to derive
P(N) < P(N).

So the intuition stated in the OP, that switching is always the best decision, is represented in terms of the magnitudes of cofinite sets of natural numbers as always being larger than the magnitudes for finite sets, but without succumbing to the false conclusion that the probability of getting a better result is 1, which is a 'bug' of classical probability theory caused by it's insistence upon using standard models of arithmetic.

Thanks @sime for the mathematical treatment!

I'm quite fond of this potential infinity solution and believe it may be the correct direction to pursue.

However, the die in the paradox possesses an actually infinite number of sides (the set of sides is Dedekind-infinite). What more needs to be said to argue that such a die cannot exist?

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

I'm quite fond of this potential infinity solution and believe it may be the correct direction to pursue.

However, the die in the paradox possesses an actually infinite number of sides (the set of sides is Dedekind-infinite). What more needs to be said to argue that such a die cannot exist? — keystone

The first problem is one logical inconsistency. In Kolmogorov's treatment, the axioms exclude the proposition; if one introduced such a die as a new axiom, the system wouldn't be consistent. Whereas in my above (very rough) proposal, A Dedekind infinite set is measured directly in terms of its definition rather than in terms of it's cardinality,but which in turn implies that it has lower probability than its subsets, violating additivity, (Here I am assuming that we want to use standard rules for mapping distributions from one set to another. I'm not actually sure if there might be some other workaround than banning Dedekind-infiniteness).

The second problem is one of motive. Is the motive good enough? Consider what it means to say that the Natural Numbers are Dedekind infinite. In type theory, it refers to an object N with an arrow
1 + N --> N that has an inverse ( here 1 denotes zero, and + indicates disjoint union, and the arrow is the successor function). A standard computational reading of this arrow is that it conveys the fact that one can count upwards from zero to an arbitrary finite number of one's choosing and then count downwards to return to zero. In a nonstandard reading one is also allowed to count from an arbitrary position that cannot be reached from zero. But in either case, the arrow doesn't have the extensional significance that set theorists like to assume. That is to say, the arrow doesn't imply that "every member of the natural numbers exists prior to it being counted" , rather the arrow is used to construct as many members as one desires. In summary, we can say that Dedekind-infiniteness is a type of rule that can be used to generate Dedekind-finite extensions of any size that can be freely extended as and when one desires, by applying the rule once more.

In the case of an infinitely sided die, if the die can only be rolled a finite number of times, then its trajectory of outcomes is equivalent to the trajectory of some Dedekind-finite die that by definition is guaranteed to possess an arbitrary but finite number of unrolled sides after the final roll of the die. Is rolling the die a Dedekind-infinite number of times extensionally meaningful? Not according to the functional interpretation of Dedekind-infiniteness, which deems the previous analysis sufficient for the philosophical analysis of the fall of man paradox.

I get an occasional email from Quora about interesting questions and answers of the day. By pure serendipity this came up today:

https://www.quora.com/Whats-the-craziest-mathematical-fact-you-know/answer/Brian-Overland-1?ch=15&oid=164968393&share=3b2e848f&srid=o1wj&target_type=answer

The whole answer is related to this thread, but the second to last paragraph is of particular interest to me in regards to this theoretical dice:

(Which is one reason you cannot even in theory randomize across the natural numbers with uniform probability.)

This leads me to think that maybe, after all, it's not safe to assume god could prepare such a dice.

And here's a stack exchange conversation on why.

https://math.stackexchange.com/questions/14777/why-isnt-there-a-uniform-probability-distribution-over-the-positive-real-number

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. — fishfry

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

You're right, it's a paradox. — fishfry

I'm not saying I've convinced you, but no one—especially a mathematician—has ever responded like this to my mathematical/philosophical thoughts. It makes me feel a bit less out of my mind. Thank you.

I don't know what a singularity is in math or set theory. — fishfry

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.

I have no idea what you are trying to say. — fishfry

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

I think you are making an analogy [with calculus] where there isn't one. — fishfry

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.

Uh oh. You contradicted the game. You can't make a random choice from a bag that never contains all the numbers. — fishfry

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. 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.

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

I subscribe to the concept of completed infinite sets, but with a twist: I believe they encompass just one unique element. 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.
Some may argue that this perspective is naïve, but I disagree. 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.

It occurs to me that perhaps you're getting at infinitesimal probability theory. — fishfry

Ew. Actual infinitesimals are no better than actual infinities.

@fishfry : 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. I assume you understand my general stance, so instead of revising those messages, let me point out two comments that really should be adjusted to properly reflect the tie outcome.

Let me propose a third, perhaps controversial, scenario:

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

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

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. — fishfry

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

Of course God, being God, has an infinite roll of tickets. — fishfry

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) —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.

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.

The first problem is one logical inconsistency. In Kolmogorov's treatment, the axioms exclude the proposition; if one introduced such a die as a new axiom, the system wouldn't be consistent. — sime

[EDIT: Sorry this chart is incorrect. I will repost a corrected chart in the coming days. Removing now to avoid confusion.]

[CHART RETRACTED]

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?

(cc @fishfry)

In a nonstandard reading one is also allowed to count from an arbitrary position that cannot be reached from zero. But in either case, the arrow doesn't have the extensional significance that set theorists like to assume. That is to say, the arrow doesn't imply that "every member of the natural numbers exists prior to it being counted" , rather the arrow is used to construct as many members as one desires. — sime

Love it.

In the case of an infinitely sided die, if the die can only be rolled a finite number of times, then its trajectory of outcomes is equivalent to the trajectory of some Dedekind-finite die that by definition is guaranteed to possess an arbitrary but finite number of unrolled sides after the final roll of the die. — sime

But with this approach, probabilities can only be considered in retrospect, which seems insufficient.

(Which is one reason you cannot even in theory randomize across the natural numbers with uniform probability.)

Thanks for sharing that. The Quora post is definitely relevant. However, I believe the chart I just shared demonstrates a different perspective.

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.

@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!

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.

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. — fishfry

I took the idea to mean that the faces of an infinite die isn't a well-ordered set, unless the Axiom of Countable Choice is assumed. 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.

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?

The natural numbers are well ordered in their usual order. — fishfry

Yes, that is true, by Peano's inductive construction of the natural numbers. And a well-order is usually assumed for an infinite sided die, in spite of its construction lacking an inductive specification (for which side should be assigned what number?) - So the assumption of a well-ordered infinite sided die that lacks an inductive definition is the same as a countably infinite set of objects equipped with the axiom of countable choice.

Sorry Fishfry.

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 - since there is an algorithm choice isn't needed). But then what of the idea of rolling said die an actually infinite number of times? That surely is equivalent to choice, assuming the rolls are random.

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.

@fishfry

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.

Let me attempt another explanation, starting from a different angle.

SETUP PART 1
In the game, Adam can be in one of three states: win, lose, and undecided. He starts in the undecided state (i.e., before the dice are thrown or while they are still in motion) and transitions to either win or lose once the dice stop and their values are observed. Here are the abbreviations I'll use:

• Win: (W)
• Lose: (L )
• Undecided: (W or L)

For instance, if Adam rolls a 10 and the serpent rolls a 4, we record Adam's history as (W or L) -> (W).

SETUP PART 2
What I should have clarified in my original post is that the fairness of the game extends beyond just the dice; it includes the rolling process itself. You cannot simply place the die on the table with the number 6 facing up just because you desire that outcome. The die must be tossed to allow each face some time facing upward before settling on a number. The more faces a die has, the more it needs to bounce around to ensure fairness.

MY PROPOSED RESOLUTION
There are 4 distinct groupings of game histories based on the number of faces on the dice.

• 0 Faces - The game doesn’t start since there’s nothing to roll. Adam’s history remains undecided: (W or L).
• 1 Face - They always roll the same number, 1. Adam’s history remains undecided: (W or L).
• Finite number of faces - Assuming that ties lead to a reroll, Adam’s history is either (W or L) -> (W) or (W or L) -> (L ).
• Infinite faces - The game never concludes since the dice continue bouncing indefinitely. Adam’s history remains in an eternal state of undecided: (W or L).

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.

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.

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?

ASIDE
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.


Now on to some of the other discussion points....

You reject the modern theory of limits? — fishfry

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.

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.

If there are no infinite sets there's no game in the first place. — fishfry

The most effective way to discredit a system is to follow it to its logical conclusion and show that this outcome is absurd. Using this strategy, I am entirely justified in assuming that infinite sets and the game exists. With this in mind, I'll retract my stance on infinite sets and strive to discuss the topic in more agnostic terms.

Hilbert's hotel is completed. — fishfry

I have concerns with this hotel.

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).

If we were playing with a real-number sided die, there's an obvious strategy. — fishfry

It's impossible to place uncountably many numbers on countably many sides. 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? Additionally, how does this reconcile with Kolmogorov's axiom that the sum of the probabilities of all possible events must equal 1? That being said, I do see the value in Measure Theory and the concept of probabilities on continua. Those aspects make sense to me.

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/

I thought you were trying to put a sensible probability measure on N
that formalizes the obvious intuitive correctness of Adam always switching. — fishfry

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

But physical infinite-sided dice seems to be what you are interested in. I am confused. — fishfry

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:

Objects - My consideration isn't just for objects that exist within our physical universe, but extends to those that could exist in a simulation or program. For example, a die with a googolplex sides is conceivable. Likewise, numbers and sets are considered objects in this context. Whereas a married bachelor is a contradiction so I say that it cannot exist. If one could prove a similar contraction about surreal numbers then I would say that they do not exist.

Process - This term refers to tasks or algorithms that perform an operation with objects (or other processes). For instance, rolling a die or executing a random number generator are both processes.

This distinction is important, as sometimes one can easily mix up the two. For example, the python code that defines the random number generator is an object. The execution of the code is a process. The random number outputted by the program is an object.

Peeling away the story elements of the paradox, it involves an infinite object (N ) and an infinite process (random number generator operating on N) [As you noted, establishing a uniform probability measure on N is unfeasible in any scenario, which implies that a random number generator targeting N would indefinitely continue without halting.]

I see two possible resolutions to the paradox:

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

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). — fishfry

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?

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

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.

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.

Please consider my version of the Stern-Brocot Tree:


There's a lot to unpack here, so let me explain in detail.

The yellow tree primarily represents an extended version of the original Stern-Brocot Tree, now including negative values. Each vertex on this tree corresponds to a rational number, identified by the sequence of left and right turns taken from the top to reach it. The value corresponding to a vertex is calculated by taking the mediant of the vertices above. For example, the number 1/2 is represented by the path 'RL', and -2 by 'LL'. What's fascinating is that every rational number eventually will appear, in reduced form, exactly once on this tree, and they are organized in increasing order from left to right. For instance, the second row lists the numbers [-infinity, -2, -1, -1/2, 0, 1/2, 1, 2, +infinity], incorporating the rational numbers from previous rows such as [-1, 0, 1].

However, this tree, having no endpoint, excludes real numbers, yet intriguingly, it feels as though the reals should be represented here too. For example, the golden ratio, if it were on this tree, would appear at the vertex RRL (where the underline indicates repeating). But such a vertex would exist at row infinity and no such row exists. This raises the question: if RRL isn't an 'object' in the tree, then what is it? I hope you see where I'm going with this...

Switching gears in the spirit of measure theory, which efficiently handles intervals rather than points, let's consider the blue lines in my diagram. Here, instead of the tree branching, each subsequent row splits into intervals. For instance, 'RL' corresponds to the interval (0,1), and 'LL' to the interval (-infinity, -1). Now, consider the following:

[] = (-infinity, +infinity)
R = (0, +infinity)
RR = (1, +infinity)
RRL = (1, 2)
RRLR = (3/2, 2)
...

Those are the intervals corresponding to the first few digits of RRL. In my perspective, the sequence RRL represents a never-ending descent through the rows, marked by a continually narrowing interval between rational numbers. If this 'tree' had an endpoint, the interval would eventually shrink to a point, specifically the golden ratio of approximately 1.618033988749... However, the absence of a bottom means we're perpetually left with an interval. This illustrates that real numbers are better understood as unending processes that involve ever-decreasing intervals, rather than as objects fixed on the tree.

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. — fishfry

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).

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

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.

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

Love it. XKCD rocks.

I'll close with a quote from Niels Bohr:
"How wonderful that we have met with a paradox. Now we have some hope of making progress."

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.

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. — fishfry

Now you're introducing narrative elements into our discussion, mentioning God and fishbowls. If we assert that God can do anything, then we could just as easily conclude that God can define a uniform probability measure on N and leave it at that. However, there are limits to even what God can do. As a programmer would understand, creating a true random number generator is incredibly challenging. While theoretically, you might write such a program (using finite lines of code), in practice, it would run indefinitely without halting. Could God create a random number generator for N that actually stops? Or does his magic only work when we talk informally about fishbowls?

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. — fishfry

I don't think you're truly entertaining my propositions. Did you understand what I was saying?

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

While I would really like to continue this tangential discussion, there's no point in addressing this (and other tangential) comments if you aren't going to read my responses simply because they don't directly relate to the original post.

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? — fishfry

I would have appreciated your specific insights on this topic if you had engaged more sincerely in this tangential discussion.

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. — fishfry

My main concern revolves around the concept of completed infinities. R, N, and the process of generating a random number on N all inherently involve completed infinities. They are interrelated. Now, consider this 'paradox':

God created a married bachelor and declared he would kill the man at noon if he was married. Is the man alive at 12:01?

There are different ways to approach this paradox. One method is to seek a logical explanation for God's decision on whether or not to execute the man. Alternatively, and just as validly, one can challenge the premise itself. You are not allowing for this possibility, which seems unfair.

You know, there's a thing called the counting measure. — fishfry

This definitely aligns with Adam's reasoning. However, as you pointed out, the counting measure is not a probability measure, which I find problematic. Regarding the specific paradox, at what point would it be prudent for him to swap rolls with the serpent? Does this decision occur the moment he opens his eyes and makes an observation? What if he only pretends to open his eyes? What if he makes an observation but totally forgets what he observes? What if he keeps his eyes closed, but an ant sees his roll? What if God is watching? What if God sees the roll and informs Adam that he saw his roll but doesn't say what it was? Counting measure does not offer an answer to these questions.

Or will you instead chose not to answer these questions related to observation and simply say that pop quantum theory is not helpful here?

@fishfry

For what it's worth, here's how I would construct a random number generator on N in our physical universe:

1) Employ a quantum event that has a 50% chance of yielding 1 and a 50% chance of yielding 0.
2) Assign the outcome to the first digit of a binary number—1 for a result of 1 and 0 for a result of 0.
3) Continue this process for each subsequent digit.

Two key observations:
1) There is one potential issue with this approach. It's remotely possible that the latter output could be an infinite sequence of 1's. If, hypothetically, this program could be executed as a supertask (completing in finite time), it might return infinity, which does not belong to the set of natural numbers.

2) The program never halts. If you stop it prematurely, you haven't encompassed all natural numbers. Since the program is intended never to halt, it avoids the theoretical problem of returning infinity, rendering the aforementioned flaw negligible.

If we're discussing fishbowls, I'd argue that when God reaches into the bowl and selects the top ticket, it's an unfair draw. He should shuffle the tickets first. However, when dealing with an infinite pile, the shuffle would never conclude. Let's set aside the fishbowl analogy and turn our focus to programming, which offers a more tangible approach to discussing random number generation on N.

Let's reframe this discussion in terms of my concepts of objects and processes:

1) The random number on N (i.e., the output of the RNG function) - an object that cannot feasibly exist.
2) The code defining the RNG function - a finite object that exists.
3) The process of executing the code to completion - an infinite process that cannot be completed.

In mathematics, there is a tendency to treat the output (1) as the fundamental element. However, I contend that the actual code (2) deserves our primary attention. This shift focuses on the tangible aspects of mathematical constructs rather than on abstract, unattainable outputs.