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The Story
God warned Adam, "You may look upon any die with a finite number of sides, but you must never gaze upon the die with infinite sides, for doing so will lead to your certain death."

Yet the serpent, cunning in its ways, tempted Adam, suggesting, "Let's gamble with the die of infinite sides. Should you outroll me, divinity will be yours. But should your number be lower, death will be your fate. Rest assured, the dice are fair."

Faced with a seemingly even chance of victory, Adam agreed to the serpent's game, all the while keeping his eyes shut as he rolled his die to adhere to God's command. "Look at your die" urged the serpent, "and if your number displeases you, take mine, and I shall roll once more."

Considering the logic of smaller dice games, where checking one's roll can help one decide whether it's advantageous to swap, Adam reasoned it would be wise to apply the same logic to the infinite-sided dice game.

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

Choosing life over death, Adam opted for the serpent's number, revealed to be 42. The serpent rolled again, extending the same offer. This cycle continued for hours until an epiphany struck Adam: by never concluding their game, he could elude death. Adam fled the Garden of Eden and took refuge in the comprehensibility of the finite world until, ultimately, his newfound finiteness overtook him.

My questions
Is this truly a paradox? If not, why not? If the probability of winning falls from 50% to 0% as described, at what point does this shift occur? Considering that observing his die leads to the same conclusion in every case, he could have reached that conclusion without ever looking at his die. This scenario seems to indicate a problem with the concept of an infinite-sided die, possibly even suggesting that such a die cannot exist. What are your thoughts?
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This scenario seems to indicate a problem with the concept of an infinite-sided die, possibly even suggesting that such a die cannot exist.

"Infinite" is not a number as an amount of sides but is that which cannot be gotten to since it never completes, but to play along, the die is ever becoming more of a higher and higher resolution smoother sphere.
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There would be no way to read the numbers because any of them could appear to be unlimited and by the physical limitations of the universe would therefore be impossible. Most numbers wouldn't fit in our universe.

If the numbers cannot manifest, if they aren't allowed, then the dice cannot be infinite.

I guess you could use compact symbols though but you'd come up against some limitation of discernment, where what conveys one number is indistinguishable from what conveys another. The capacity for subtle discernment would have to be infinite also.

The serpent would end up disagreeing with you what number is on the face of the dice and you'd have to bring in a third party (God, if you trust him) to verify.
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"Infinite" is not a number as an amount of sides but is that which cannot be gotten to since it never completes, but to play along, the die is ever becoming more of a higher and higher resolution smoother sphere.

No, the die in the story remains unchanging and complete with a side for every natural number. This distinction matters because the potentially infinite die you describe doesn't concern me.
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There would be no way to read the numbers because any of them could appear to be unlimited and by the physical limitations of the universe would therefore be impossible.

I concur that this narrative couldn't unfold in our physical reality, but your argument doesn’t address the core of the paradox. The inclusion of God and the Garden of Eden in the story was specifically to lift us beyond our finite limitations.
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Is this truly a paradox? If not, why not? [...] This scenario seems to indicate a problem with the concept of an infinite-sided die, possibly even suggesting that such a die cannot exist. What are your thoughts?

Just as a polygon with infinite sides of equal length would be empirically indiscernible form a perfect circle—this irrespective of how close one observes it, say with a quantum microscope—so too would a die with infinite sides of equal length be empirically indiscernible from a perfect sphere, even when analyzed with a quantum microscope.

Hence, having the numeral “42”, or any other, viewable on any one of the die's infinite sides would be a logical impossibility considering the nature of empirical reality—an empirical reality which both the serpent and Adam utilize to look at the die.

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only a finite number of outcomes were lower, yet an infinite number were higher, making the serpent's probability of beating him exactly 100%.

We are just concerned with the finite numbers that appear on both dice and which is greater than the other. Sounds like 50/50 chance.

Once the player knows what the serpent threw, there will be an infinite amount of numbers greater than that number and a finite amount less than, and therefore chance of winning against the serpent approaches 0. The limitation of possible representation is still an issue here; there must be a finite maximum magnitude for the game. Infinity cannot be a stipulation.

You could fix the problem with doing away the limit of natural numbers and including the negative integers to infinity. Chance of 50/50 seems locked in here, even after the player becomes aware of the serpent's number, unless the chance is actually undefined here. Maybe it's undefined for the natural numbers also.

If the serpent pulled 42 on a number line of integers what is the chance of throwing a higher or lower number? Does it even make sense to ask?
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It probably doesn't make sense to roll an infinite sided dice in the first place
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Just as a polygon with infinite sides of equal length would be empirically indiscernible form a perfect circle—this irrespective of how close one observes it, say with a quantum microscope—so too would a die with infinite sides of equal length be empirically indiscernible from a perfect sphere, even when analyzed with a quantum microscope.

Your argument merely skims the surface, focusing only on the impossibility of the story occurring in our physical world. Let's remember, this is a die created by God, and the story takes place in the Garden of Eden, where ordinary physical laws don't hold. Nonetheless, I'll engage with your point since you're considering mine.

The die doesn't have to be polyhedral; it simply needs to be fair. Imagine a line segmented as follows: the first half represents roll 1, the first half of the remaining line represents roll 2, the first half of what's left represents roll 3, and so forth.

If we roll this line into a faceted arc, we achieve an object with infinite sides. All that's left is a little divine magic to ensure it rolls fairly, and extraordinary vision for the players to discern the minuscule markings on those higher rolls.
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The limitation of possible representation is still an issue here; there must be a finite maximum magnitude for the game. Infinity cannot be a stipulation.

Do you believe that the set of all natural numbers exists? Is it reasonable to stipulate its existence? If infinite sets can exist in some realm, why can infinite dice not exist in God's realm?

You could fix the problem with doing away the limit of natural numbers and including the negative integers to infinity.

Addressing a problem by focusing on a completely different issue is like sweeping the original problem under the rug.

Maybe it's undefined for the natural numbers also.

But at what point does the math fail? The calculations don't seem to get much simpler.
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All that's left is a little divine magic to ensure it rolls fairly, and extraordinary vision for the players to discern the minuscule markings on those higher rolls.

In that case, I’ll let this thread be. But I confess, to me, this conflux of mathematical thought with “magic” and “extraordinary vision” not bound by empirical sight so far equates to the question of: What defined result obtains when one divides three and a quarter invisible unicorns by zero? Just saying.
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It probably doesn't make sense to roll an infinite sided dice in the first place

Why not? In a recent post to Javra, I described an object of finite size with infinite sides that could function as a die. Adam could certainly hold it in his hand. We would just need to trust God to ensure its fairness. Just to clarify, I'm not aiming for a religious debate here. I mention God in the story simply because it seems to me that objects with infinite properties could only exist within a supernatural realm.
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In that case, I’ll let this thread be. But I confess, to me, this conflux of mathematical thought with “magic” and “extraordinary vision” not bound by empirical sight so far equates to the question of: What defined result obtains when one divides three and a quarter invisible unicorns by zero? Just saying.

No problem, and thank you for the discussion. I will say that, in my view, the conflux of mathematics and magical thinking was formalized by Georg Cantor and has been nearly universally adopted in modern mathematics. If you believe that infinite sets cannot exist, then I am preaching to the choir.
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I guess you mean this post. This doesn't seem like a paradox to me. Where do you think the paradox is?
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Cool. Thanks.

If you believe that infinite sets cannot exist, then I am preaching to the choir.

Just to be clear: when it comes to the realm of pure mathematical thought, I can see it both ways and so remain on the fence; but when it comes to empirical reality as in the form of a die, a member of the just stated choir I am. :grin:

Interesting way of expressing the issue, btw.
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I guess you mean this post. This doesn't seem like a paradox to me. Where do you think the paradox is?

Yes, this post. To me, the story presents a paradox because the probability of Adam winning appears to drop from 50% to 0% simply by observing his die. Yet, observing his die provides no new information that he couldn't have figured out beforehand. So, does looking actually make a difference, and if so, why? Or was he doomed to lose from the beginning?
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Just to be clear: when it comes to the realm of pure mathematical thought, I can see it both ways and so remain on the fence; but when it comes to empirical reality as in the form of a die, a member of the just stated choir I am.

Although I presented the 'paradox' in the form of a story to make it engaging, it could just as easily have been described purely as a mathematical problem within the realm of abstract mathematical thought. Indeed, when you questioned the feasibility of constructing such a die, it seemed you were addressing the narrative element of the paradox, leaving the core mathematical issue untouched.
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Do you believe that the set of all natural numbers exists? Is it reasonable to stipulate its existence? If infinite sets can exist in some realm, why can infinite dice not exist in God's realm?

Infinity is an unreasonable stipulation and whether or not numbers "exist" is not all that relevant to me. The natural numbers definitely have a limit of physical representation and there'd be an infinite amount of numbers beyond that limit.
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Infinity is an unreasonable stipulation and whether or not numbers "exist" is not all that relevant to me. The natural numbers definitely have a limit of physical representation and there'd be an infinite amount of numbers beyond that limit.

Your message begins stating that "infinity is an unreasonable stipulation" and concludes by stating that there are an "infinite amount of numbers". It seems that you're open to infinities in some contexts. Let's place this paradox in that context where you welcome infinities.
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[...] it could just as easily have been described purely as a mathematical problem within the realm of abstract mathematical thought. Indeed, when you questioned the feasibility of constructing such a die, it seemed you were addressing the narrative element of the paradox, leaving the core mathematical issue untouched.

Well, the issue to me was and remains the conflux of pure mathematical thought and empirical reality. There often (although not always) is no tidy cohesion between the two—as exemplified by an infinite-sided die whose individual sides need to be looked at to be discerned. One can simplify the very same issue by addressing the purely mathematical concept of a perfect circle and its nonexistence in empirical reality.

As an aside, I am aware that without the concept of a perfect circle no such thing as quantum mechanics could obtain, for the latter’s uncertainty principle is necessarily dependent on pi, which can only obtain in a perfect circle (that, again, can only be non-empirical). But the basic discrepancy between pure mathematical thought of a perfect circle and its nonexistence in physical, empirical reality remains. This and related subjects, btw, being why I’m on the fence regarding purely mathematical objects of thought being of themselves existents: e.g., if a perfect circle does not exist, then this directly entails that so too is quantum mechanics a fiction … and, yet, QM works rather well to explain what physically, and hence empirically, is. I acknowledge this is a deviation from Cantor, but to me it speaks to the same issue: the relation, if any, between pure mathematics and the empirically perceivable physical world.
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One can simplicity the very same issue by addressing the purely mathematical concept of a perfect circle and its nonexistence in empirical reality.

In our physical universe, a perfect circle cannot exist, nor can pi be fully represented as a decimal number with infinite digits. However, the universe does permit the complete equation of a circle and the full definition of pi as an infinite series, both expressed using a finite number of characters. The real question isn't whether quantum mechanics is fictional (an idea that seems absurd), but rather if quantum mechanics employs the infinite-rooted objects themselves or merely the finite descriptions of these objects. Or perhaps more crucially, the question is whether we need to assert the existence of the infinite-rooted objects at all, or if we can manage perfectly well with just the finite descriptions.
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The real question isn't whether quantum mechanics is fictional (an idea that seems absurd), but rather if quantum mechanics employs the infinite-rooted objects themselves or merely the finite descriptions of

Interesting perspective! My own slant is that in physical reality there can be no absolutely integral physical being as is symbolized by the number one/1 when the issue is addressed more objectively: all individual objects are integral physical things only in relativistic terms, always permeated (by quanta, for one example) and hence entwined with what (we perceive and conceive to be) other and always in flux. Hence, our very understanding of mathematics in full is grounded in conceptual idealizations, rather than physical realities - starting with the idealizations of 0 and 1.

As regards infinities, if you happen to not be familiar with them, you may get a kick out of surreal numbers. They incorporate both infinite and infinitesimal numbers as well as real numbers. (While I'm no mathematician that can properly describe or implement them in mathematical formulations, surreal numbers have always held a rather aesthetic appeal to me.)
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My own slant is that in physical reality there can be no absolutely integral physical being as is symbolized by the number one/1 when the issue is addressed more objectively

As regards infinities, if you happen to not be familiar with them, you may get a kick out of surreal numbers.

I have a tough time accepting the use of limits to define reals as numbers. The idea of applying limits upon limits is even more difficult for me to accept. But like you, I'm not a mathematician.
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Well your mistake is assuming it drops to 0%. If the die is split the way you said, it absolutely does NOT drop to 0% unless he rolled a 1.

You could only make a good argument that it drops to 0% if every number on the die had an equal probability - but the die you laid out does not have that feature.
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Well your mistake is assuming it drops to 0%. If the die is split the way you said, it absolutely does NOT drop to 0% unless he rolled a 1.

In the game with an infinite-sided die, suppose Adam rolls a 1000. What are his chances of winning? Consider the infinite set of larger numbers the serpent could roll. The probability of Adam winning is exactly 0%. However, a 0% probability of winning does not imply that victory is impossible for him. Instead, it means that he will almost surely lose. This distinction from measure theory opens up a whole new can of worms.

You could only make a good argument that it drops to 0% if every number on the die had an equal probability - but the die you laid out does not have that feature.

When I mentioned that the dice with infinite sides are fair, I was specifically referring to each side having an equal chance of being rolled. After all, God is fair. :P
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To be laconic about things: no. But then many an ontological perspective would need to be addressed to back this up - perspectives that would amount either to an objective idealism or, else interpreted, to a neutral monism.

As it is, as previously attempted, I'll try to bow out of this thread's discussion. But it was good discussing with you.
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The probability of Adam winning is exactly 0%.

How? You said 1 takes up a full half of the die - 1 is less than 1000, and the other guy has a 50% of rolling a 1. He has a 25% chance of rolling 2, so that's already up to a 75% chance to win now. Where are you getting 0 from?
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When I mentioned that the dice with infinite sides are fair, I was specifically referring to each side having an equal chance of being rolled. After all, God is fair. :P

Then why did you point me to your post where 1 takes up half the space on the die :P
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Thanks for the discussion!
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Then why did you point me to your post where 1 takes up half the space on the die

In my initial post I wrote "Rest assured, the dice are fair."

In the very post you mention I wrote "All that's left is a little divine magic to ensure it rolls fairly, and extraordinary vision for the players to discern the minuscule markings on those higher rolls."

Let's set aside the question of how God could make fair infinite-sided dice and just assume that he can.

Now, putting trivial matters aside, do you understand that Adam's probability of winning becomes exactly 0% once he sees his roll? If you disagree, what do you calculate his probability to be?
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Is this truly a paradox? If not, why not?

This interesting puzzler has a clear and unambiguous mathematical resolution.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

So you should switch.

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

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

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

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

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

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

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

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

In conclusion:

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

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

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

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

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.
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