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
This interesting puzzler has a clear and unambiguous mathematical resolution. — fishfry
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
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
The sides should be of the same length. — Chet Hawkins
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
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
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 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
3) The probability of each event is 1. — keystone
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
While this notion may seem preposterous, — keystone
similar issues emerged with calculus, which were resolved using limits. — keystone
For instance, finding the tangent by dividing by zero results in a singularity, yet one can sensibly approach a zero denominator. — keystone
In a similar vein, I argue that dealing with the set of all natural numbers also results in a singularity — keystone
, but probabilities can be sensibly managed by approaching an infinite set. — keystone
In other words, infinite sets as completed objects do not truly exist. — keystone
Although my proposed resolution has significant implications, I believe none of these are insurmountable. — keystone
What do you think? — keystone
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
(Which is one reason you cannot even in theory randomize across the natural numbers with uniform probability.)
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
You're right, it's a paradox. — fishfry
I don't know what a singularity is in math or set theory. — fishfry
I have no idea what you are trying to say. — fishfry
I think you are making an analogy [with calculus] where there isn't one. — fishfry
Uh oh. You contradicted the game. You can't make a random choice from a bag that never contains all the numbers. — fishfry
If you reject completed infinite sets, you can't play the game in the first place. Right? — fishfry
It occurs to me that perhaps you're getting at infinitesimal probability theory. — fishfry
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
Of course God, being God, has an infinite roll of tickets. — fishfry
I wouldn't venture to disagree with Kolmogorov on this matter. — keystone
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
I'm not saying I've convinced you, — keystone
but no one—especially a mathematician—has ever responded like this to my mathematical/philosophical thoughts. — keystone
It makes me feel a bit less out of my mind. Thank you. — keystone
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
Does it still seem like the information content of my idea is nil? — keystone
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
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
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
I subscribe to the concept of completed infinite sets, — keystone
but with a twist: I believe they encompass just one unique element. — keystone
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
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
Ew. Actual infinitesimals are no better than actual infinities. — keystone
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
I assume you understand my general stance, — keystone
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
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
You definitely need to see the next Avatar movie in the theater in 3D. — keystone
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
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
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
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
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
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
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 — sime
, unless the Axiom of Countable Choice is assumed. — sime
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
The natural numbers are well ordered in their usual order. — 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 — sime
But then what of the idea of rolling said die an actually infinite number of times? — sime
That surely is equivalent to choice, — sime
You reject the modern theory of limits? — fishfry
If there are no infinite sets there's no game in the first place. — fishfry
Hilbert's hotel is completed. — fishfry
Now suppose we play the same game, but with rationals. — fishfry
If we were playing with a real-number sided die, there's an obvious strategy. — 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. — keystone
The more faces a die has, the more it needs to bounce around to ensure fairness. — keystone
Infinite faces - The game never concludes since the dice continue bouncing indefinitely. — keystone
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
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
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
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
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
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
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
It's impossible to place uncountably many numbers on countably many sides. — keystone
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
Additionally, how does this reconcile with Kolmogorov's axiom that the sum of the probabilities of all possible events must equal 1? — keystone
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 thought you were trying to put a sensible probability measure on N
that formalizes the obvious intuitive correctness of Adam always switching. — fishfry
But physical infinite-sided dice seems to be what you are interested in. I am confused. — fishfry
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
The formal definition of a limit, the epsilon-delta definition, is perfect rigorous and leaves no room for metaphysical ambiguity. — fishfry
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
I'm glad if I said something you found useful. — fishfry
I'll close with this xkcd, which I just ran into yesterday. — fishfry
I agree with you that a uniform probability measure on N is impossible. I think the resolution to this paradox lies elsewhere. — keystone
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 see two possible resolutions to the paradox: — keystone
1) The game never starts because infinite objects don't exist.
2) The game never ends because infinite processes never terminate. — keystone
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
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
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
Please consider my version of the Stern-Brocot Tree: — keystone
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
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
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
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
However, noncomputable real numbers exist, and they do not have algorithms. — fishfry
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
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
You know, there's a thing called the counting measure. — fishfry
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