I'm perfectly happy to continue the conversation. — fishfry

Great. And if it seems like you're no longer making debatable points or asking questions, I'll take that as a hint that the conversation has reached its end. :D

I'm only saying that you might be disappointed if you hope to convert me to your degree of passion, even on items where I agree with your point of view. — fishfry

Might? As in there is still a chance? [said like a clueless teen not getting the hint from repeated rejections from his crush. Lol.]

I'm sure poor old Zeno is getting a sufficient workout in the staircase thread. — fishfry

Yeah, let's keep Zeno to that thread. I'm glad to see you couldn't resist joining in, though. :)

If you and I agree on something but I just don't allocate it the same percentage of my overall interest and passion as you do, that's ok, right? We basically agree on Zeno, I just don't give it much thought. I've given it some thought over the years. But I truly never cared about it in the sense that you do. And I hope you can make your peace with that, because you seemed to be saying that you wanted to convert me not only to your point of view, but also to your level of passion. And that may not be productive. — fishfry

Time is valuable, and it's perfectly fine for you to express that you're not interested in continuing our conversation; we can leave it at that. If you choose to end the discussion but also mention that you agree with me, that's a nice extra, though not necessary. Regarding converting you to my point of view, I do want to do that and will seize any opportunity that comes up. I thought that since you provided your resolution to Zeno's paradoxes that you invited further discussion, but it seems I may have misinterpreted your intentions.

ZENO'S PARADOX


Quantum Jump - Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. This is not an acceptable solution to Zeno's Paradox. I agree with you that Zeno's assumptions about motion are flawed, but you haven't offered an alternative premise that holds up. The whole point of his paradox was to highlight that the standard view of motion was flawed. Additionally, it's not definitively established that physical space is discrete. It's possible that only our measurement of space is discrete. This latter perspective is my belief which I'll expand on in a couple of paragraphs.


Zeno Non-sequitur fallacy - I agree that the conclusion doesn't logically follow from the premise, but that's only true if you interpret the paradox from a fresh perspective. From the traditional understanding of motion, the conclusion indeed seems to follow logically. This is precisely Zeno's point.


I wrote the following in a different thread but it's relevant here. Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. Zeno’s famous parables about Achilles' incremental pursuit are illustrative of (and an attack on) this presentist perspective. However, Zeno himself subscribed to the opposite belief, which we now call eternalism. This philosophy asserts that past, present, and future coexist as a single, unchanging "block universe." From a vantage point outside this block, everything would appear static; thus, in this comprehensive perspective, motion is impossible. One could argue that in his perspective, the only movement is in the gaze of God, and wherever God looks becomes the present (I use God here not to push a religious view, but for simplicity). The discreteness that are looking for is not in space but in measurement/observation. In other words, God's fundamentally cannot watch everything. This actually should come as no surprise since the Quantum Zeno Effect demonstrates that an observed system cannot evolve.

Zeno was remarkably prescient. The concept of eternalism and the block universe gained serious traction only after Einstein introduced theories that showed eternalism to be more consistent with the principles of relativity. Yet, the narrative is still unfolding, as the singularities in classical black holes demonstrated that relativity is not the ultimate explanation of physical reality. Enter QM and the importance of observation/measurement.

STAIRCASE PARADOX


A minute cannot pass - This scenario involves an infinitely large object (the staircase), an infinitely complex task (traversing the entire staircase), and the passage of one minute. You're suggesting that the issue lies in the impossibility of a minute passing? It seems you may have labeled the most logical and uncontroversial element in the paradox as illogical. If you think the problem has to do with Icarus's steps then frame your solution in that context.
No end to the staircase but the end is reached - Yes, this is the very issue I'm trying to highlight. And this has nothing to do with continuous acceleration or motion. Could it be that supertasks are impossible?
restricting ourselves to the physical world - The physical world is not the only realm that exists; there's also the abstract world, which operates under its own set of rules. For instance, in an abstract world I can define, it's perfectly valid to set the speed of light at 100 m/s. This isn't incorrect—it's simply a different premise. However, I do believe in a kind of symmetry where truths in the physical world often find parallels in the abstract world.


This is a paradox I've come up with myself. But as Michael has mentioned it's very similar to Thomson's lamp. Where do you see problems with it?


Focus first step up, not last step down- Unfortunately, the stairs are numbered in ascending order from the top down, so the first step up wouldn't be numbered 1.


Non-standard numbers-I'm certain you're a strong mathematician, but I also feel like you're overcomplicating things. This reminds me of an Einstein quote: “If you can't explain it to a six year old, you don't understand it yourself.”

,
Only a potential infinity-My purpose in presenting this paradox is to underscore the problems associated with the concept of actual infinity.


Thomson's Lamp-Indeed, the Staircase Paradox shares significant similarities with Thomson's Lamp Paradox, particularly in that both scenarios lead to states considered invalid by conventional logic after one minute has elapsed. In the Staircase Paradox, we are left unsatisfied by claims that the staircase either exists or does not exist. Similarly, in Thomson's Lamp Paradox, we find it unsatisfactory to definitively say whether the lamp is on or off. The difference is that, supertasks aside, Thomson's Lamp is a critique of infinite series whereas the Staircase Paradox is a critique of N.


Trip from 0 to 1-I don't get it.


[0,60)-Your point is valid, for brevity I didn't explicitly state that the first instant he passes the stairs he arrives on the ground. However, as the poem indicates, my view is that at that instant, he actually arrives at a singularity, similar to what one might encounter at the center of a classical black hole.

PARADOX OF THE GODS


As Michael noted, your barrier paradox is Bernadete's Paradox of the Gods. I find this paradox intriguing. In the realm of physics, I think quantum tunneling offers a solution to this issue.

Okay, now I'm wondering if it was me...it is possible...and likely...

I've delete my message from here and posted it in the correct thread.

@Metaphysician Undercover, @fishfry, @andrewk, please move your related posts as well if that's not too much trouble.

EDIT: I think I made a mistake by incorrectly posting in this thread a message that was meant for another thread. I've since deleted that post.

Oh YOU messed the threads up? — fishfry

No it wasn't me. That was the Canadian in me saying sorry!

All of my responses were to messages on this thread! — keystone

This statement was incorrect. I said it not knowing that the threads got mixed up.

EDIT: I think I made a mistake by incorrectly posting in this thread a message that was meant for another thread. I've since deleted that post.

@fishfry

Oh you're right...this got messed up. Let me reach out to the moderators. Sorry!

EDIT: I think I made a mistake by incorrectly posting in this thread a message that was meant for another thread. I've since deleted that post.


All of my responses were to messages on this thread!

Zeno wasn't attempting to prove that motion itself is impossible; rather, he aimed to demonstrate that motion, as understood by the prevailing theories of his time, was impossible. — keystone

@fishfry: Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. Zeno’s famous parables about Achilles' incremental pursuit are illustrative of (and an attack on) this presentist perspective. However, Zeno himself subscribed to the opposite belief, which we now call eternalism. This philosophy asserts that past, present, and future coexist as a single, unchanging "block universe." From a vantage point outside this block, everything would appear static; thus, in this comprehensive perspective, motion is impossible. One could argue that in his perspective, the only movement is in the gaze of God, and wherever God looks becomes the present.

Zeno was remarkably prescient. The concept of eternalism and the block universe gained serious traction only after Einstein introduced theories that showed eternalism to be more consistent with the principles of relativity. Yet, the narrative is still unfolding, as the singularities in classical black holes demonstrated that relativity is not the ultimate explanation of physical reality. Enter QM...

Wow this was a good post. I understood everything you're saying and I agree with much of it. Even in parts where I disagree, we're still talking about the same thing. Thanks for this. — fishfry

Great! It does feel nice to feel heard.

This is different than the others. A four-sided triangle is impossible simply by virtue of the meaning of the words. I thought that since you called googolplex abstract and possible, then you would use the transfinite ordinals and cardinals as examples of abstract and impossible things.

Small quibble anyway. — fishfry

I didn't bring up transfinite numbers as examples of abstract impossibilities because I knew you might disagree. However, you're right that my initial example was trivial. Let's consider a non-trivial one: "This statement is false." This paradox challenges classical logic by appearing both true and false simultaneously. Yet, consider the profound influence it has had. This paradox sparked the development of numerous non-classical logics. Reflect on its siblings like "the set of all sets that do not include themselves" and "the formula with Gödel number ___ cannot be proved". Dismissing such a seemingly abstract and impossible statement would have deprived us of significant philosophical and mathematical advancements. And I believe the revolutionary impact of this paradox is far from over.

OMG my thoughts exactly. The analogy is non-Euclidean geometry, which was thought to be a mathematical curiosity with no practical use when discovered in the 1840s, and then becoming the mathematical formalism for Einstein's general relativity in 1915. — fishfry

I love this example.

My candidate for the next breakthrough like this is the transfinite cardinals, the higher infinite. Nothing more than a mathematical curiosity today, but in 200 years, who knows — fishfry

Fishfry called it here first :)

I don't share your enthusiasm for logical paradoxes as the fulcrum for the next scientific revolution — fishfry

I think you meant to say 'the next mathematical revolution'. Paradoxes, or singularities, have been and continue to be pivotal in sparking scientific revolutions.

But yes, the lack of enthusiasm applies to you and pretty much everyone else. Unfortunately, I lack the mathematical prowess needed to convince you to listen.

As a longtime student of crankology, I disagree. Alternative and novel ideas don't make one a crank. It's a certain lack of the logic gene or a certain basic misunderstanding of the nature of proof and logical argument that separates the cranks from the merely novel thinkers. — fishfry

You're generally correct, but there are exceptions like Norman Wildberger. I hope that one day AI can help cranks build a more compelling argument because I think they aren't completely mistaken.

Ok. I just don't know if the standard logical paradoxes are that important, but time will tell. — fishfry

It’s tempting to just snip off the loose thread and assume everything is fine. After all, how much damage could a small loose thread really do to your sweater, right? Abstract impossibilities are such rare gems I'm saddened that we don't value them.

I have not realized earlier that you are not interested in the interesting question of choosing an arbitrary natural; but rather trying to link this to some kind of paradox. But the relation's a stretch. I still don't see the connections that you've tried to make with dice that roll forever (why gravity but no friction in your world?), quantum physics, and various other topics. — fishfry

I believe it's impossible to choose an arbitrary natural number in N. I understand that my earlier questions about the impact of observation seemed aggressive, so let me answer them instead and see if you have any comments. Before the dice stop rolling, Adam has a 50% chance of winning. Once Adam sees his roll, his chances drop to 0%. If Adam forgets his roll, his chances go back to 50%. If only God sees the roll, God knows Adam's chance of winning is 0%, but Adam still believes it's 50%. If God reveals that he saw the roll, both are aware that Adam's chance is 0%. It's pretty wild, isn't it? Even if we find a way to choose an arbitrary natural number in N, the situation remains just as bizarre. Declaring that there's no uniform probability on the natural numbers is not an answer. It's akin to dismissing "this statement is false" as an invalid statement that can be ignored because it doesn't fit into classical logic. You're snipping off the exposed part of the loose thread.

Much ink spilled over the years on this, but just not an interest of mine. Personal preference. — fishfry

Fair.

[The dartboard paradox] is a genuine paradox of interest. How does a collection of sizeless points make up a length or an area? We have mathematical formalisms but no real explanation. There's really nothing to be done about the basic paradox. — fishfry

I appreciate your acknowledgment that mathematical formalisms don't provide an explanation. However, I strongly disagree with the notion that nothing can be done about it. It just seems you might not be interested in an informal solution, and if that's your stance, I'm a little sad but it's a reasonable one to hold.

For what it's worth, Newton thought of lines as being paths of points through space, so there's no real paradox if you assume space is like the real numbers. Which it almost certainly isn't. — fishfry

I don't understand.

In fact I would venture to say that the ultimate nature of space or spacetime is nothing at all like the mathematical real numbers. — fishfry

I would venture to say that the ultimate nature of spacetime is very much like the objects that real numbers are intended to model (continua).

[Zeno's paradox:] Already resolved mathematically by the theory of infinite series, and physically by the fact that motion is possible. Also just not a major interest of mine. — fishfry

I strongly disagree on the topic of infinite series, but I won’t delve into it since it doesn’t seem to interest you. Zeno wasn't attempting to prove that motion itself is impossible; rather, he aimed to demonstrate that motion, as understood by the prevailing theories of his time, was impossible. This serves as a prime example of a concept once believed possible, which he identified as both a tangible impossibility and an abstract impossibility. The paradox remains unresolved to this day.

But what you have failed to convince me of is that "the paradox" -- which one of many that you've discussed?? -- is important, either in general or especially to me. — fishfry

I think we're both at fault here. I haven't explained my perspective well, and you haven't been entirely open to hearing it.

In Thompson's lamp, the final state is not defined so you can make it anything you want it to be. — fishfry

All of the major mathematical paradoxes today share a common theme: superposition. The liar's statement is (true or false), Thompson's Lamp is (on or off), the staircase (exists or doesn't), Icarus is (alive or dead), and the state of Adam's game is (win or lose). Unfortunately, I suspect you might dismiss this entire explanation as lacking substance.

But the other ones, Thompson's lamp and the staircase and so forth, arise from the fact that the final state is simply not defined. — fishfry

The universe must ultimately settle on a state because something has to occur. Are you suggesting that God simply flips a coin? All signs, including those from quantum physics, indicate that we need a new state for Thompson's Lamp upon completion of the supertask, one that goes beyond just being (on) or (off).

Here, instead of concluding that a minute cannot pass, as Zeno concluded that Achilles cannot pass the tortoise, keystone changes things up to say that after a minute has passed the infinite number of steps has been reached. — Metaphysician Undercover

Do you truly believe that Achilles is unable to surpass the tortoise? Do you think that Icarus's deeds influence the passage of time? Is there a concrete analogy in which your actions alter how time progresses for me?

Your poetry asserts this, but the reverse can be done There is simply no first step in the process, just like there wasn't a last step on the way down. The sum of the same series in reverse order is also 60 seconds. — noAxioms

How is it possible for him to ascend the stairs if there isn't a first step? Or do you think that he might not be able to fully descend the stairs?

If he would have traversed the staircase in Zeno like fashion, as specified, although he would have stepped on all the steps in a finite amount of time, there would be no definite position along the staircase that he was at immediately before he had arrived at his destination. — Pierre-Normand

What's your take on this? Do you believe he never finishes descending the stairs? If that's the case, then where would he be after one minute has passed?

Infinity minus one equals infinity
Would the above qualify as a paradox — kazan

If that statement is logically unacceptable, then it could be considered a paradox. However, many people today might not see an issue with it, so you would need to provide further explanation to convincingly demonstrate its paradoxical nature.

Can a paradox be conceived in the a&p realm? — kazan

Let me draw an analogy. Historically, our understanding of the world was believed to be tangible and possible. We thought we grasped the truth, whether through Newtonian mechanics or general relativity. Then, some thinkers pointed out inconsistencies in these prevailing views that defied explanation. What was once deemed tangible and possible turned out to be tangible and impossible. As a result, the model of the world was revised, and the new model was then assumed to be tangible and possible. Over the years, this process repeats, gradually bringing us closer to the truth.

I understand you're asking which of the following four scenarios interests me:

1) Tangible and possible - for example, a horse.
2) Tangible and impossible - such as a black hole as described by Relativity, with a singularity at the center.
3) Abstract and possible - like the number googolplex.
4) Abstract and impossible - such as a four-sided triangle.

Our physical universe, though entirely described by mathematics, appears to have circumvented singularities. Why not look to it for inspiration? In physics, breakthroughs often occur when one identifies something tangible and impossible and rethinks our understanding to shift it to tangible and possible. This approach has driven many major advancements in the frontiers of physics, which is why numerous eminent minds are engaged in quantum gravity research.

The next significant breakthrough in mathematics could occur when someone pinpoints what is currently abstract and impossible yet accepted within modern mathematics, and the community transforms it into something abstract and possible. The arithmetization of analysis is an excellent illustration of such a transformation. While I deeply appreciate the value of what is abstract and possible (acknowledging that mathematical truths are both beautiful and useful), much of it surpasses my grasp, so I can't personally revel in it. However, what really captures my interest is the pursuit of the abstract and impossible in mathematics. Personally, I view it as the most important, thrilling, and accessible area to engage in at the moment. Although most impossibilities in mathematics have been resolved (no serious mathematician is exploring four-sided triangles), I believe paradoxes like the ones we discuss suggest that some impossibilities still remain.

To summarize my interests:

1. Tangible and Possible - This is my day-to-day work as an engineer. I thoroughly enjoy the innovations that stem from exploring this domain, especially my computers.

2. Tangible and Impossible - The physics community already excels in this area. They are actively working to resolve the impossibilities in their theories. Yet, there are still opportunities to influence through philosophical interpretations of quantum mechanics.

3. Abstract and Possible - Mathematicians excel in this field, continually advancing our understanding and capabilities.

4.Abstract and Impossible - Typically, those who challenge the established norms here are labeled as cranks. There is a significant opportunity for philosophers of mathematics to make strides in this area. This is where my interest lies, in exploring and potentially reshaping the abstract impossibilities that still exist in mathematics.

With this in mind, we seem to disagree on whether the paradox I propose is abstract and impossible or abstract and possible. It might be an exaggeration, but from my perspective, this disagreement translates to me seeing it as crucial, whereas you might view it as merely an interesting concept, but nothing more.

Additionally, I believe I have the beginnings of an idea that could transform it from abstract and impossible to abstract and possible. This concept also holds the potential to resolve many other persistent paradoxes, such as the Liar's Paradox, the Dartboard Paradox, and Zeno's Paradox. Yet, I find myself struggling to even convince you that the paradox, which appears possible from a conventional standpoint, is actually abstract and impossible.

What do you think about this?

Perhaps my next paradox will make a stronger impression. Even though this conversation might conclude, please keep in mind that I'm always open to picking it up again if you're interested.

If instead of choosing a random number, what if we just choose an arbitrary one? — fishfry

It appears that an arbitrary number would be relevant in discussing the potential outcomes of Adam's story before or after the event has occurred. However, for the story to progress as it unfolds, in Adam's 'present' a random number would need to be selected. Please correct me if I'm misunderstanding your point.

@fishfry: I just realized I might have misread the tone of your second-to-last post as suggesting we were wrapping up, even though your latest post raised new questions. I'll get back to those questions later, but I want to make it clear that I understand you're not obligated to continue this conversation.

@fishfry: While I would love to continue this conversation, it sounds like you see this as a good endpoint. I'm going to post a new paradox now that this conversation has ended. I hope to hear what you have to say about it. Sorry if I sounded rude at the end, that was not my intention. I recognize that you have been more than charitable with your time. You're a nice person. Thank you so much for this conversation.

@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.

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?

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."

@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.

@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!

(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.

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.

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)

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.

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.

@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

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.

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?

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)?

: 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?

Then why did you point me to your post where 1 takes up half the space on the die — flannel jesus

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?

Thanks for the discussion!

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. — flannel jesus

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. — flannel jesus

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

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 — javra

Doesn't the effectiveness of classical computers contradict your perspective?

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

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.

One can simplicity the very same issue by addressing the purely mathematical concept of a perfect circle and its nonexistence in empirical reality. — javra

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.

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. — Nils Loc

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.

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. — javra

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.

I guess you mean this post. This doesn't seem like a paradox to me. Where do you think the paradox is? — flannel jesus

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?

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. — javra

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.