Of course it does. I can't wait to see how it all plays out.
Though there is at least one case where the idea got transformed and returned with a vengeance. I mean the Pythagoras' and Plato's idea that ultimate reality is mathematical, meaning the only reality is the mathematical as opposed to the physical, world, returns as the idea that the physical world is mathematical. Weird. — Ludwig V

Tegmark's trolling. And the world is mathematical to us just as it's sound to a bat. The world does whatever it's doing. We do the math. The world is described by the math to a good degree of approximation. It's a metaphysical hypothesis that the world "follows" the math. Clearly the world did not stop following Newtonian gravitation when Einstein came along. Both theories are just approximations to something deeper ... or perhaps nothing at all. Nobody knows.

That it explains nothing? I agree. Like saying "God did it." Or saying the Great Sky Computer (GSC) did it. Except that God is not restricted to being a computation, whereas the GSC is, making God a less unreasonable hypothesis.
— fishfry
My word, there's a discovery! A hypothesis that is more unreasonable than God! This should get a Nobel prize of some sort. — Ludwig V

The computational theory of the world requires that the world is a computation. That is indeed more restrictive than the hypothesis that God did it. Computable functions are a tiny subset of all possible functions. There is no reason at all for the world to be computable. I find it unlikely.

That is (to repeat myself): The street corner preacher says that all of us are created in the image of God. The TED talker says that we are all created by the Great Simulator, who operates as a Turing machine. That is a most restrictive stipulation. Far less likely than God. It's ironic that the intellectual hipsters mock God and flock to simulation theory, which is a far less likely hypothesis.

In your opinion, how is simulation theory any less magical and unrealistic than God? And why should God be restricted to be a Turing machine? I never understand this point.

If you allow the transfinite ordinals, the sequence 1, 2, 3, ... has the limit ω. And even if this seems unfamiliar, it's structurally identical to the sequence 1/2, 3/4, 7/8, ... having the limit 1, which is much more familiar.
— fishfry
Yes, I do remember our earlier discussion of this. I don't pretend I understand them, but I do admit they exist - my allowing them or not is irrelevant. — Ludwig V

That's why I prefer the 1/2, 3/4, 7/8, ... example. Same structure in more familiar clothing.

What is the starting point of no axioms? It's like playing chess with no rules.
— fishfry
Did someone mention a starting-point of no axioms? It would be indeed be like playing chess with no rules - or discussing infinity. — Ludwig V

Mathematicians have incredibly precise rules for infinity. The rules are the axioms of ZF or ZFC set theory.

But if my consciousness itself is simulated, then the simulation argument requires that consciousness is computational, a point I strenuously disagree with, with Penrose and Searle on my side.
— fishfry

Why do you think it's not computational? — RogueAI

Searle. Bit flipping lacks intentionality.

Is your web browser passing judgment on the opinions you post to this site? Does Ms. Pac-Man experience pleasure eating white dots, and terror being gobbled by monsters? The ideas are ridiculous on their face. The onus is on those who claim that a digital circuit could be self-aware. And to anticipate a common objection, the brain does not operate by the same principles as a Turing machine.

Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right?
— fishfry
I start with a few.
1) It's not all a lie. I mean, I can't know that, but if it's all crap, then I can know nothing regardless of how I interpret the lies, so I have no choice but to give weight to the empirical.
2) It's not about me. If I am the center of the universe, the rest is probably a lie. So I pretty much find that any view that puts me, humanity, Earth, the universe itself, as the center of something larger, to be unproductive. — noAxioms

So you DO have axioms :-)

Descartes apparently worried about it all being a lie. I reject that road only because it is untravelable, not because it is wrong. But it seems that modern science has thrown a cold paid of doubt on the validity of "I think therefore I am". — noAxioms

It works for me, as an objection to the VR aspect of the simulation argument. Even if I'm living in a realistic VR, that doesn't explain the "I" that's being deceived.

Not sure what you mean by empirical testing here.
As I said, one can empirically examine the causal chain that makes the body walk for instance. In a VR, it does not originate in the brain of the avatar, but external, from the mind controlling the body. Say you're playing tomb raider. Open up Lara Croft's head. No brain in there, or if there is, it's just a prop. None of the stuff she does has its cause originating from there.
Why does nobody pursue such investigations? Is technogoly still so backwards that it can't be done? They already have machines that can detect a decision having been made before you are aware of having done so yourself. — noAxioms

I think Bostrom is trolling us and can't believe so many otherwise smart people take him seriously. Likewise Tegmark's mathematical universe. An even more obvious troll.

Trendy, yes. Kind of dumbs down the validity of any scientific discovery. Why would a simulation choose to display CMB anisotropy if that isn't what a real universe would look like? — noAxioms

Why does Ms. Pacman have to eat those silly pellets if we, the simulators, have a much wider variety of nutritious and tasty food? In fact video games are the counterexample to the claim that our simulators' world must be similar to our own.

I think that example was being used as an illustration of Moore's law, and not as support for a VR hypothesis. — noAxioms

Hmmm. Moore's law is just a heuristic, and is already failing. It's not a law of nature. You could be right, perhaps I'm misremembering where I saw the argument [that video games have advanced greatly therefore simulations will eventually be indistinguishable from reality].

In any event, if you regard our media environment as a simulation, it's already taken for reality by billions of us. You know the meme going around. The Amish didn't contract covid, because they don't watch tv.

But you are the one saying that you only have rationals.
— fishfry
No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say that 0.25 lies in the middle of that interval, I will say no, 0.25 lies between (0,0.25) and (0.25, 0.5). And what this amounts to is cutting (0,0.5) such that it no longer exists anymore. In its place I have (0,0.25) U [0.25] U (0.25,0.5). — keystone

I'm afraid I don't know what a line is, absent the real numbers, unless you mean the original line of Euclid, "A line is breadthless length." I'm not a scholar of Euclid so I really can't say.

I mentioned Peirce to you because it seems to me that you are interested in the "top down" definition of a continuum. I'm deeply unqualified to discuss the matter. I can only give you the standard mathematical interpretation, which is unsatisfying to both of us. I don't know enough about the philosophy of the continuum to comment.

Let's move away from directly using sets to describe the path. Instead, we'll describe the path using a graph, and then define the graph with a set. — keystone

Sigh. Your pictures don't help. What is a line? What does the notation [0, 0.5] mean?

I don't know anything about set theory with urlements.
— fishfry
Urelements are indivisible 'atoms'. The lines that I'm working with are divisible. — keystone

You said that your sets contains things other than sets. You just keep making up your own terminology. I don't think we are making any progress, and I no longer know what we are discussing.

You only believe in rationals. Where are you getting these things?
— fishfry
That is not what I believe. I can define a line using no rationals: (-inf,+inf). — keystone

That directly contradicts what you said earlier. And I don't know what your notation means.

I see this line as a single object (a line). — keystone

What is a line?

It is not populated by rational points. It is not populated by any points for that matter. — keystone

It's empty? We're going in circles.

I've drawn it for you below in between points at -inf and +inf. To walk this path from -inf to +inf you don't need limits, you just walk the corresponding graph from vertex 0 to vertex 1 to vertex 2. — keystone

Ok.

You would call this green line the 'real number line'. You see this as 2^aleph_0 points. You believe that to walk any length on this green line you need limits. I understand what you're saying. We're just starting from different starting points. You're starting from the bottom and I'm starting from the top. — keystone

Ok. We're going in circles. I have no idea what you're talking about.

Yes, I mean endpoints. I used the term 'bounds' because it is a more general term that applies to higher dimensional analogues. I'm searching for a way to keep this conversation going so it doesn't end prematurely out of frustration. — keystone

I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers.

Currently, I don't believe I can persuade you that a continuum can exist without points. — keystone

I'm perfectly willing to believe it, I just don't know anything about it.

However, I've come to realize that convincing you of this isn't necessary. Here’s my new approach:

1) Start at the bottom
2) Build up to the top
3) 'Start' at the top
4) Approach the 'bottom' from the top — keystone

Was this supposed to be helpful?

I see this equivalent to starting at the bottom of the S-B tree -> working our way to the top of the tree -> then proceeding back down to approach the bottom. I know you won't see it that way, which is fine. But to be clear, I still believe things are very ugly at the bottom filled with pumpkins. Nevertheless I do understand how mathematicians think things work at the bottom and if starting at the bottom is the only way you'll allow me to get to the top then I'll go with it. I understand your criticisms of starting at the top, I just don't accept them. Once you allow me to get to (3) I endeavor to show you that (3) and (4) alone fully satisfy our needs and if I'm careful (e.g. by only allowing for rational endpoints) that (1) and (2) are not only superfluous but undesirable. Is that a fair approach? — keystone

Jeez man ...

But now only the endpoints are rational, leaving me baffled as to what those objects are.
— fishfry
Yes, the endpoints are rational, — keystone

Two seconds ago you denied this.

and the object between any pair of endpoints is simply a line. — keystone

What is a line?

It doesn't go deeper than that. I understand you see that line as a composite object consisting of 2^aleph_0 points. — keystone

I did not say that, and there are other characteristics that a line must have. I am perfectly willing to adopt your ontology, if only you will state it clearly.

What is a line?

However, I view the line as a primitive object. — keystone

Ok. Euclid again?

Clearly, our starting points differ. To move the discussion forward, could we agree to a compromise where we refer to a line as a "composite" object? This way, by including composite it's evident that I recognize your perspective, but the quotes indicate that my viewpoint doesn't necessitate this classification. — keystone

You could move this forward by telling me what a line is. But I don't think I'm helping anything by sniping at your ideas in frustration.

A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.
— fishfry
I agree with this point. The issue has been the lack of viable alternatives. I see that Peirce was suggesting the use of infinitesimals, and you're aware of my stance on those—the one from the comment where you thought I was just trolling. — keystone

Just giving a reference to what you seem to be getting at. A continuum that can't be divided into points.

You are right that the historical contingency should make us suspicious. (Descartes, by the way, has a description of statues "animated" by a hidden hydraulic system - I think in Versailles). But I don't think the process is simply over-enthusiastic. It seems reasonable to try to apply a new discovery as widely as possible. That way, one discovers its limitations. — Ludwig V

Yes ok, but that supports the possibility that in the future, our current preoccupation with "mind as computer" will look as dated as "mind as waterworks" of the Romans.

So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is.
— fishfry
That's more or less one Ryle's favourite arguments against dualism. — Ludwig V

That it explains nothing? I agree. Like saying "God did it." Or saying the Great Sky Computer (GSC) did it. Except that God is not restricted to being a computation, whereas the GSC is, making God a less unreasonable hypothesis.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of Michael's example of counting backwards.
— fishfry
Clearly "<divide by> 2" is not applicable at 0. — Ludwig V

Well you never "reach" 0, but 0 is the limit.

Would it be right to say that "+1" begins at 0 and has no bound and no limit, and that "<divide by> 2" begins at 1 and has no bound, but does have a limit? But they both they have a defined start and no defined end. — Ludwig V

If you allow the transfinite ordinals, the sequence 1, 2, 3, ... has the limit $\omega$. And even if this seems unfamiliar, it's structurally identical to the sequence 1/2, 3/4, 7/8, ... having the limit 1, which is much more familiar.

Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right?
— fishfry
Yes. The difficulty is how to evaluate a starting-point. True or false isn't always relevant. Which means that it can be difficult to decide between lines of reasoning that have different starting-points. — Ludwig V

What is the starting point of no axioms? It's like playing chess with no rules.

This is what I mean by reciting backwards:

If I recite the natural numbers <= 10 backwards then I recite 10, then 9, then 8, etc.
If I recite the natural numbers <= 100 backwards then I recite 100, then 99, then 98, etc.

If I recite all the natural numbers backwards then I recite ... ?

It's self-evidently impossible. There's no first (largest) natural number for me to start with. — Michael

I don't think you and I are making progress.

I have agreed repeatedly that we can't "count all the natural numbers backwards" since an infinite sequence has no last element.

It is Achilles' run but with time reversed: https://plato.stanford.edu/entries/spacetime-supertasks/#MissFinaInitStepZenoWalk — Lionino

Any initial step necessarily leaps over all but finitely elements of the sequence. Same reason that any neighborhood of the limit of a sequence contains all but finitely many elements of a sequence.

I accept this:

P1. If we can recite forward 1, 2, 3, ... at successively halved intervals of time then we can recite all natural numbers in finite time

But I reject these:

P2. We can recite forward 1, 2, 3, ... at successively halved intervals of time
C1. We can recite all natural numbers in finite time — Michael

Given P2, what is the first natural number not recited? I seem to remember having asked you this several times already.

If you want to claim that C1 is true then you must prove that P2 is true. You haven't done so. — Echarmion

What is the first number not recited?

I would put things differently. We have clearly made tremendous progress in simulating all manner of physical processes, including those happening inside brains. Where we have made no progress is in developing a conceptual framework for connecting such physical processes with the subjective experience of consciousness. — Echarmion

But you are agreeing with me. We have made zero progress in simulating or implementing consciousness.

We are already able to create systems that appear like a conscious subject on a passing glance (though humans also occasionally ascribe consciousness to anything from cats to rocks, so perhaps that's not surprising). — Echarmion

Yes, the humans are the weak point in the Turing test. And LLMs are not conscious or intelligent, they're just "stochastic parrots."

It seems likely that we'll be able to create artificial systems which are indistinguishable from conscious subjects in a number of circumstances in the near future. — Echarmion

I'll take the other side of that bet, having observed the AI hype cycle since the 1970s. And even if I'm wrong about that, we still haven't implemented consciousness. And you agree with me.

Perhaps this will bring us closer to understanding our own consciousness, but perhaps not. — Echarmion

Most likely not. The current mania for LLMs is literally silly. Their utility is already fading as we've run out of training data, and they're starting to feed on their own online output.

Does Bostrom actually address this distinction?
— fishfry
Bostrom seems to presume that consciousness is computational, and leaves it undefended.
In such a simulation, nobody is being fooled. — noAxioms

I've seen the argument -- perhaps this wasn't in the original Bostrom paper, I don't recall -- that we should consider Pong, the original video game. versus the amazingly realistic video games of today. The argument is that in the far future, our video game technology will be indistinguishable from reality.

That might be true.

But if the Great Simulator in the Sky (and exactly how is that any different than God?) is implementing my consciousness as well as my perceptions, then we have made NO progress since the days of Pong, since we have no idea how to implement or simulate consciousness. So that argument fails. That's one of my objections to simulation theory. The "progress in video games" argument" fails. We've made no progress in simulating consciousness.

In a VR, is it a lie to have the subject experience a world that is not the same world as the reality in which the mind exists? If so, most forms of dualism are arguably deceptions. — noAxioms

But it's a commonplace fact that we don't experience reality as it is. There are sounds out there that bats hear and we don't. Flies have those crazy compound eyes. We're back to Plato's cave and Kant's nuomena. There's a reality "out there" and we only experience its shadow, or representations of it mediated, filtered, and distorted by our senses. Huxley's doors of perception. The idea is clearly true. Our vision is terrible. If we had better resolution we could see molecules.

Berkeley had the most parsimonious version of this idea. Since we experience everything through our senses, there's no need for an outside world at all.

The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers.
— fishfry
They can't both be right? — noAxioms

Ok. We are ALL of whatever our latest technology is. Well ... maybe so. Something to be said for that.

You're agreeing with my point.
I think I am, yes. — noAxioms

Ok. But I'm arguing that the simulation theory, or the computational theory of mind, is suspect because of its very timeliness. We invent computers and the philosophers all go, "Ooh we're computers." That's a point against the idea IMO.

Anything analog can be approximated with digital. But anything digital can be perfectly implemented with analog. Searle is perhaps referencing property dualism? I don't know if I got that right. Can't seem to articulate the differences between the variants. — noAxioms

I think Searle was arguing against dualism in this instance. He was saying that mind is not a computation; but it's not something non-physical. Rather, there's something physical about living things that implements consciousness, in a way that rocks and digital circuits can't. I only saw him mention this on video, so perhaps he's added more detail in his writings.

I guess I'm even more skeptical than Descartes. I win! I didn't pick my handle for no reason. I try not to leave anything unquestioned. — noAxioms

Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right?

VR says that all you know is potentially lies. You are not of this universe, but rather you are experiencing it. All very dualistic. The 'brain' in the body (if there is one at all, have you ever checked?) is not what's making any of the decisions.
If you think about it, the view can be empirically tested. Not so much with the simulation hypothesis. — noAxioms

Yes but everyone agrees with that. There's a world "out there," and we experience it through our senses. Not sure what you mean by empirical testing here.

It's always been unclear to me which aspect of simulate/VR Bostrom is arguing.
Definitely the former. But Elon musk is arguing for VR, and references Bostrom's paper to support it, so he has no idea what he's talking about. — noAxioms

Right. And I saw a TED talk where George Smoot, the guy who discovered the cosmic background radiation anisotropy, was enthusiastically advocating simulation theory. Neil deGrasse Tyson too. A lot of people who should know better say trendy things for no reason at all. More arguments against simulation IMO. The pronouncements of celebrity scientists outside their expertise are always suspect.

The comment above (and my reply) belongs in the other topic. I see you posted more or less the same question there. — noAxioms

Yes. Let's talk about this over there.

There is never a final tick in an infinite sequence, even if the sequence has a limit.
or not a first tick. Zeno's dichotomy very much as a final tick. I can make a scenario that has a first and last, and gets singular in the middle somewhere. Just illustrating the classical snippet: Never say never. — noAxioms

Ok. Just talking about standard mathematical sequences. It's a common misunderstanding in this thread. The sequence 1/2, 3/4, 7/8, ... has a limit, namely 1, but no last element.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of @Michael's example of counting backwards.

You’re suggesting that my line, which already extends in space, requires additional points, which themselves individually have no length, to actually have length. I wish you could appreciate the irony in your position. — keystone

Likewise.

If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length. — keystone

But you are the one saying that you only have rationals.

Our problem is that you are only allowing points in your sets. — keystone

In standard set theory, the only thing that sets can contain is other sets. We can call them points but that's only a word used to convey geometric intuition. Actually sets don't contain points, they contain other sets.

Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set? — keystone

I don't know anything about set theory with urlements.

Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?
— fishfry
I wanted to show you that even if I cut my unit line to contain all rational points between 0 and 1 that there would still be stuff in between the points -- continua. Perhaps I used the wrong tactic by talking about an idea which I don't support. I did say at the start of the paragraph that it was impossible but maybe I could have been clearer. — keystone

Only adds to my annoyance level. But that's a low bar so no worries.

Yes, you believe in continua, but not as 'objects in and of themselves'. You believe that continua can't exist in the absence of points. Please confirm. — keystone

Too deep for me. I don't even know what that means.

My preference is that you accept non-points into sets, — keystone

So set theory with urelements? I don't know much about that subject past the definition.

however, if you're unwilling to do that then here's an alternate approach. To move this conversation forward, let's say that when I say 'a line', you can think to yourself that I'm referring to 2^aleph_0 points — keystone

You only believe in rationals. Where are you getting these things?

(which somehow assemble to form a line), and I'll think to myself that I'm simply referring to a line (which cannot be constructed from points). — keystone

If you have a line and you have the rationals, you will get the real numbers by Cauchy-completing the line.

In other words, you can go on thinking that points are fundamental and I'll go on thinking that lines are fundamental. How does that sound to you? — keystone

Your idea is not coherent. If you start with a line (a thing you have declined to define) and say it's populated by the standard rational numbers by cuts, then you can construct the standard real numbers.

All I need from you really is to allow me to restrict my intervals to those whose bounds are rational (or +/- infinity). Could you let that fly? — keystone

By bounds you mean endpoints? So now you are backing off entirely from your last half dozen points, and saying that your ontology consists of intervals with rational endpoints. But the real numbers are indeed present inside the intervals after all? Is that what you are saying?

What are these lines of yours, anyway?

...Just to see how far my position can go in the absence of the explicit use real numbers (I'm fine if in your eyes their use is implied but I just won't ever mention them)... — keystone

You can't get anywhere as far as I can see.

So for example, can you allow me to say that there are 5 objects depicted below? You can go on thinking that 2 of the objects are composite objects and I'll go on thinking that all 5 objects are fundamental (they're either 0D or 1D continua). — keystone

You haven't given a coherent definition of these objects. All along you've been saying they are intervals of rationals. That's at least coherent, even if such intervals lack all properties of being continua.

But now only the endpoints are rational, leaving me baffled as to what those objects are.

ps -- A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.

Perhaps you are getting at some idea like that. Here's one link, you can Google around to find others if this interests you.

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

His ideas on continuity:

https://plato.stanford.edu/entries/peirce/#syn

No, you said that debt was transferred to working-class taxpayers, which is not the case. — Vera Mont

You lost me on the fetish bit.
— fishfry

I can live with that. — Vera Mont

I think we're at the end here. Nice chatting with you.

He doesn't seem to know the difference between the simulation argument (Bostrom is a good example of this) and a virtual reality argument (the Matrix is the typical example). — noAxioms

It's never been clear to me whether Bostrom himself makes this distinction.

If the point is merely that we're being fooled by the simulator, this is just Descartes's clever Deceiver. Since my consciousness is outside the the simulation (or as Descartes puts it, even if I'm deceived, there's still an I that's being deceived) the simulation argument explains nothing. The mystery of consciousness remains.

But if my consciousness itself is simulated, then the simulation argument requires that consciousness is computational, a point I strenuously disagree with, with Penrose and Searle on my side.

Does Bostrom actually address this distinction?

The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too.
— fishfry
I am so sorry. I started a hare by mistake. — Ludwig V

No worries. Like a certain Supreme court justice, I am not a biologist.

The horse first appeared in this comment
Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said.
— Ludwig V
So a horse here is shorthand for whatever physical object one is trying to put into mathematical harness. Zeno's horse is the tortoise, or Achilles, or both. — Ludwig V

Ok. I figured that out, just couldn't remember anything about a horse. I agree that a hare or a tortoise or Achilles does just as well. Thanks.

I fully understand your criticism. The problem is that you are missing my point (or perhaps I should say you are missing my 'continua'). — keystone

I'm not missing your point, I'm challenging it. The length of your rational intervals is zero. That causes a problem for your argument.

Let's continue to work with the path defined as [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] as depicted below. — keystone

The length of that union is zero, if the intervals are restricted to rationals. Do you agree with that point?

I say that (0,0.5) and (0.5,1) contain no points so you think I'm only working with three objects - the points as depicted below. The length of all points within my system is 0 so you think the objects I'm working with have zero length. — keystone

No points. So they're all the empty set? I'm not supposed to push back on this?

I say that (0,0.5) and (0.5,1) describe continua so I say I'm working with 5 objects as depicted below. The length of all points within my system is indeed 0 but the length of the continua within my system add up to 1. — keystone

No countable additivity? What then? How do your segments add up to 1 if they only contain rationals.

I hate to be so pedantic about this but I don't understand how an interval of rationals can have a nonzero length.

I prefer working with such simple paths as described above but let's do the impossible and say that somehow I could cut my unit line aleph-0 times such that there is a point for each rational number between 0 and 1. — keystone

The length would still be zero. Countable additivity again.

And it's not a "unit line" if it only contains rationals. This is where your intuition is failing you.

You say that the length of all these rational points adds up to 0. I agree. — keystone

Ok good.

You say that there are gaps between these points. I disagree. In between each pair of neighbouring points would lie an infinitesimally small continua. — keystone

There are no "neighboring points." Bad intuition again. Between any two rational numbers are infinitely many more distinct rationals.

If I add up the lengths of all of these tiny continua it would add up to 1. These infinitesimally small continua are indivisible. — keystone

There are no "infinitely small continua." You're just making all this up out of bad intuitions about the nature of the real numbers.

I'm not fond of discussing impossible scenarios as they tend to lead to incorrect conclusions. Indeed, rational points do not have neighbors, and continua are inherently divisible (unless we're treating points as 0D continua, in which case they are indivisible). Therefore, we shouldn't lend too much credence to this example, but I thought it was necessary to address your points more directly. — keystone

Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?

The problem is that you're not allowing continua to be valid objects in themselves. — keystone

Of course I do. I very much believe in the continuum, which is pretty well modeled by the standard real numbers. I say pretty well because there are other models such as the constructive real line and the hyperreal line, but those lines are not Cauchy-complete. [The constructivists wave their hands at this with some technicalities].

It is as if you are only allowing points to be valid objects. — keystone

As opposed to what? You keep hypothesizing things that do not exist, like empty continua.

[==== your second post ====]

So I figured out a better way to talk about this instead of using metric spaces. Instead, it is better to use Graph Theory.

... [stuff omitted]

To travel from vertex 0 to vertex 4 we simply walk the connected path. One nice thing about this view is that it's clear that no limits are required to walk these graphs. — keystone

Ok.

I am at an utter loss as to what you have been getting at all this time. Can you get to the bottom line on all this? So far I get that your "continua" are either empty or have length 0. Or that they somehow have length 1, despite being composed of only rationals.

I doubt that consciousness is computable
— fishfry
what, because consciousness is not a physical process, or that physical processes cannot be simulated? You seem to be in the former camp. If that's the case, then no, it probably isn't computable. — noAxioms

Consciousness could perfectly well be a physical process, but not computable. So, what kind of process is physical but not computable? A task for some future genius to elucidate. FWIW Penrose believes that consciousness is not computable. He may be wrong, but he's Sir Roger and the rest of us are not.

After all if we're computations, what are the odds we'd figure that out right when we're in the age of computation?
Pretty much 1-1 odds. That's when the terminology became part of our language. You describe yourself in terms of the things you know. — noAxioms

That's my point. The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers. The historical contingency is an argument against the theory, not for it.

We are water. The vast majority of mass would be lost (as would consciousness) if the water was taken away. Lots of pipes going here and there. It's a pretty good description for the Roman days. — noAxioms

You're agreeing with my point.

Because if so, then where is the conscious mind? In the pencil? In the paper? In the air? In a neural network?
In the process. — noAxioms

I agree with you that IF consciousness is a computation, then it could be implemented with pencil and paper. I regard that as an argument against the premise.

Yes, I saw a domino logic gate on Youtube a while back.
Gawd, I spelled it 'Turning' machine. More typos.
Anyway, yes, the discussion was inspired by that. Any moron can create a domino or gate, but creating a nor gate gets tricky. Any gate can only be used once, so it's impossible to create say a flip flop, normally a trivial thing created with a pair of nor gates.

I've not seen the video, but mention of it inspired me to design a Turing machine with the technology. Can dominos be used to run a physical simulation? I think it's possible since I found not obvious roadblocks. I'm tempted to start a topic on it, but not here since it isn't a philosophy topic at all.[/quoet]

I don't know about dominos. The pencil and paper argument is stronger.
— noAxioms

Perhaps it's some kind of analog computation, but that's not the same thing.
I've also programmed analog computers in school, never on the job. It's a different sort of thing, I tell ya. — noAxioms

I've seen Searle argue that consciousness is physical but not computational. Some kind of secret sauce found in living things and not in digital circuits. Don't know much about analog computation with respect to consciousness.

ps -- I checked out the Simulation thread and from there, saw your initial post in the "What is the Simulation Hypothesis" thread, and I agree with everything you said. I especially appreciated the distinction between simulation and VR, which is something a lot of the simulation discussions miss.
Your view of consciousness is modelled by a VR. One big distinction is that a VR cannot be implemented with paper and pencil (or dominos). — noAxioms

The illusion can. But my consciousness can't. As Descartes noted, I may be deceived, but there is an I who is being deceived. So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is.

It's always been unclear to me which aspect of simulate/VR Bostrom is arguing.

I was imagining a clock that speeds up in its ticking to ape a convergent geometric series.
— fdrake
OK, that would be pretty much what has been the topic of discussion this whole thread. If it completes in finite time, it's a supertask. Don't forget the inverse case where the clock starts fast and slows down to its final tick. — noAxioms

There is never a final tick in an infinite sequence, even if the sequence has a limit.

Example please?
— fishfry

Capital, the fetishistic worship thereof. — Vera Mont

This was in response to your saying, "And thirdly, because you pretend that government is responsible for everything it cannot possibly control."

You lost me on the fetish bit. Anyway all I said originally was that Biden's cynical election year loan "forgiveness" is

a) illegal, which is not only my position, but that of the Supreme court; and

b) transfers over five hundred billion dollars of debt to the taxpayers.

I stand by both those assertions.

Maybe I'm not being clear, so I'll try one more time. — Michael

You've been perfectly clear, and I've clearly responded to your points several times already.

If you want to argue that the first supertask can end ... — Michael

I have never made any such statement. I've repeatedly challenged you to name the first number not verbalized when we count forward 1, 2, 3, ... at successively halved intervals of time.

I ask you once again to tell me whether you appreciate the point that any interval containing the limit of a sequence must necessarily contain all but finitely elements of the sequence.

If you understand that, it addressed your counting backward argument. If not, let's discuss it.

So I ask again: can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? — Michael

Halving the time is your own thought experiment. It's not mine. Once I accept your own premise, I then work out the logical consequences. Halving the time is not my premise.

No. I'm talking about computability theory.
— fishfry
Gotcha. No argument then. As I already pointed out, you had referenced power instead of computability: "there's no difference in computational power between parallel and serial processing." and I took it as a statement of work over time. — noAxioms

Ok fair enough. I think of power as computability but perhaps I could have been more clear.No. I'm

I brought this up in my simulation-theory topic. A simulation of Earth to a precision sufficient for consciousness can be done by pencil and paper, or by dominos falling, — noAxioms

I doubt that consciousness is computable, nor is the universe, and I utterly reject the notion that consciousness can be simulated by any computational device. Period. Consciousness is not a computable phenomenon.

We are not computers no matter how many TED talkers declare it so. Just as we aren't Newtonian machines, as we thought we were in the Newtonian machine age; nor are we flowing fluids as the Romans thought, in the age of their great waterworks. After all if we're computations, what are the odds we'd figure that out right when we're in the age of computation? It's historical relativism or whatever the phrase is.

But I do agree that if consciousness were computable, then the computation could be carried out by pencil and paper. Glad you made that point. Because if so, then where is the conscious mind? In the pencil? In the paper? In the air? In a neural network? I reject the idea.

Do people really think their web browsers or word processors are having subjective experiences? I know they think their neural networks are. The idea's absurd. Don't get me started :-) I would characterize myself as a mysterian.

The latter is really interesting: set up dominos so that you get the function of a Turning machine. Not easy, but it seem that it can be done. — noAxioms

Yes, I saw a domino logic gate on Youtube a while back. Any physical substrate will do. For computation, not consciousness. For consciousness you need some secret sauce not yet understood. It will turn out to be something other than a digital computation.

Perhaps it's some kind of analog computation, but that's not the same thing.

Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.
Plank length is not a physical limit, only a limit of significance. If I have it right, any pair of points separated by a distance smaller than that is not meaningfully/measurably distinguishable from just the two being the same point. It doesn't mean that the two points are necessarily the same point.
But I gave some QM examples that suggest a non-continuous model of reality. — noAxioms

If reality is not continuous that goes a long way to solving Zeno. If it is continuous then walking across the room is a supertask.

The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too.
Yes. Search for 'horse' in the last 20 posts or so. — noAxioms

I have the bad habit of only responding to my mentions, but I'll take a look. Thanks.

ps -- I checked out the Simulation thread and from there, saw your initial post in the "What is the Simulation Hypothesis" thread, and I agree with everything you said. I especially appreciated the distinction between simulation and VR, which is something a lot of the simulation discussions miss.

I don't wish for poor kids to be deprived of an education.
— fishfry

Only because you seem to be so vehemently against letting them off some of the accumulated compound interest on their student loans. — Vera Mont

I would not say vehemently, it's not a core concern of mine. But these are legal loans that students signed for. But the one doesn't follow from the other. I can be opposed to Biden's illegal bailout without saying I want the poor kids to go back into the coal mines and be grateful for their bowl of gruel.

You, on the other hand, would prefer to have aborted them long ago, solving the problem that way. Why not just kill them now?

And maybe because you seem hell-bent on putting an unfair burden of putative working class taxpayers. — Vera Mont

Excuse me? I'm trying to spare the taxpayers. You keep misrepresenting (aka lying about) my positions.

And thirdly, because you pretend that government is responsible for everything it cannot possibly control. — Vera Mont

Example please? I don't know what you are referring to.

And lastly, because you appear to have a peculiarly skewed view of the working class, even as you advocate for its supposed interest. — Vera Mont

Not wanting the tax burden of irresponsible college kids dump on their heads is having a "particularly skewed view" of them?

You just made four lies about my positions. I think you must not have an argument.

I would prefer if Congress would pass a law to have high income earners fund college costs.
— fishfry

Well, who wouldn't? — Vera Mont

Point being that Biden's debt "forgiveness" is illegal.

But Congress and Senate are protecting high earners - perhaps because they themselves are high earners?
Both the Senate and the House have now passed a bill to block President Joe Biden’s student loan forgiveness program, which promises to cancel up to $20,000 of debt for millions of borrowers but has been held up by courts. CNN
So you'll probably get your wish: no matter how poor they are, educated people will be crippled with debt before they even get started. — Vera Mont

My wish? My wish is for the president to follow the law. Well that hasn't happened since before the Nixon administration, and maybe not ever. Presidents are notorious law breakers.

I don't wish for poor kids to be deprived of an education. Why do you keep saying I do?

You're the one who (in another thread) wants to abort the poor. That would solve the problem of funding their college aspirations. You said it, I didn't.

But you know, the government has caused the cost of higher education to grow much faster than inflation in general. First you have the government guarantee student loans. Next, banks freely lend money to students whose majors show that they'll never be able to pay back the loans. The banks don't care because the government (ie the taxpayers) backstop the loans. Then colleges have no reason to control costs, because the schools are getting paid by the banks, backstopped by the taxpayers.

That's why higher education costs are out of control. In fact if we abolished government guarantees of student loans, the banks would be more careful with who they lend money to, and the schools would work harder to control costs, and college would be more affordable.

It's another problem caused by the government claiming to address the problem.

But you haven't got a continuum if your intervals contain only rational numbers.
— fishfry

Ok, this was an excellent post! — keystone

Yay! Thanks.

I better understand your criticism. It lies in the fact that I'm using the term 'interval' in an unorthodox manner. I use the term interval to describe the objects (whatever they may be) lying between the upper and lower bounds. — keystone

I have no problem with intervals. But intervals of rationals make terrible continua.

Let's consider the interval (0,0.5). — keystone

By this I take you to mean the set of some but not all rational numbers between those values, inclusive. Yes? I say some but not all because you have said that yourself.

From a bottom-up perspective, the objects within the interval are aleph-1 actual points. — keystone

Not if the interval contains only rationals. It depends on how you define your notation.

As a side remark, there are $2^{\aleph_0}$ real numbers, and the question of whether that happens to be equal to $\aleph_1$ is the continuum hypothesis.

From a top-down perspective, the object within the interval is a single continua. — keystone

How can it be if it contains only rationals? I have challenged you on this point several times already without your providing satisfactory explanation.

It doesn't contain the rational points between 0 and 0.5, it contains no points. — keystone

No points. It's empty?

However it holds the potential for rational number points between 0 and 0.5. — keystone

Even if I accept that, its length would be zero and it would be full of holes, hardly a continuum.

It's only deep from a bottom-up perspective. From the top-down perspective it is elementary. — keystone

So far you have expressed strange and unjustified ideas about continua. Such as that they are empty or have length 0.

Do you believe in the number 1/3 then?
— fishfry

I believe that I could use the Stern-Brocot algorithm to generate a 3 layer tree whose third layer will contain a node described by LL and having all the properties that we generally attribute to 1/3. — keystone

Um ... ok I guess that's a yes ...

Consider one of your rational intervals [0,1]. What is its length?
— fishfry

The length of continuum (a,b) is b-a. So consider the continuum defined by interval (0,0.3). It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone

But that interval contains only rational numbers in your notation. Its length is zero by the countable additivity of measures.

You have not grappled with this problem yet.

In fact all the length of an interval is carried by the irrationals. There aren't enough rationals to have any length at all.

Not sure what you mean by potential cardinality.
— fishfry
Pick a number, say 27. I believe it has been shown that there exists a set the cardinality of which is 27, if that's valid terminology. — noAxioms

Yes that's true. There's a set of cardinality 27. One such is the set {0, 1, 2, 3, ..., 26}. There are others, of course.

One could also reference aleph-26, — noAxioms

$\aleph_{26}$ is vastly larger than 27. It's infinite, for one thing, whereas 27 is finite. Not sure where you're going with this.

but I'm not sure that one can prove that no sets exist with cardinalities between the ones labeled 1 through 27. — noAxioms

Of course many such sets exist, as shown by the von Neumann encoding of the natural numbers.

I am not understanding your point. Of course there are sets of all finite cardinalities. And since cardinal numbers are themselves defined as particular sets, there are sets of all cardinalities.

Point being that you get no increase in computational power from parallelization.

I beg to differ. A 16 processor machine can sustain a far greater work load than a single-processor machine. The Cray machines were highly parallelized (SIMD architecture) in which thousands of floating point operations were performed by every instruction. These machines were great for stuff like weather simulation. — noAxioms

No. I'm talking about computability theory. A Cray supercomputer has no more computational power than I do using pencil and an arbitrarily large sheet of paper, which which I can implement a Turing machine.

You are thinking of complexity theory, in which the time and space resources of computations are important.

But in computability theory, a function is either computable or not. If it is, it can be computed by pencil and paper (taking a very long time, of course). If it's not, no supercomputer will help.

As an example, consider the Euclidean algorithm to compute the least common divisor of two integers. It's a simple algorithm that can be executed using pencil and paper. If I had two trillion-digit numbers, I could not feasibly do the computation with pencil and paper; but I could still do it in principle.

Complexity theory is about what can be done feasibly. Computability theory (Turing machines etc.) is about what can be done by algorithms, whether the computation is feasible or not.

A Cray supercomputer can not compute anything that I can't compute with pencil and paper. But it can do so much more feasibly once the inputs become large.

I found a SEP article on the subject.

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

No function is computable by a parallel process that's not already computable by a linear process.
With that I agree. But that same function can also be done by paper & pencil. You said 'powerful', a reference to how fast the work is completed, and more processors helps with that. — noAxioms

Computational power. With the complexity/computability distinction, I believe we're in agreement.

Coloring the steps reduces to the lamp.
I notice that any scenario with a contradiction involves invoking magic. Suppose this physically impossible thing (infinite gods, stairs requiring faster-than-light speed, lamp switches that operate without delay. No magical measurement of something nonexistent. Zeno doesn't do that. No magic invoked, and the first premise thus produces no paradox. — noAxioms

Ok. Not sure where we're going with this. I'm happy to do a Zeno supertask by walking across the room. Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.

My Quora feed gives me a lot of cute cat pics lately. Makes me happy. Quora certainly used to be a lot better.
Oh it serves its purpose, but correct answers are not promoted above the others, and apparently a great deal of their posters don't know what they're talking about when it comes to stuff like this. — noAxioms

Lot of troll accounts on the site and lots of people who don't know what they're talking about. Believe it's something to do with the Quora owners trying to make some money. Sadly I'm a bit addicted to the site.

Zeno's horse is quite real. Almost none of the others are. — noAxioms

The Zeno Wiki page doesn't mention a horse. Did I miss something? @Ludwig V mentioned a horse too.

So there is a common understanding of what the issue is. Your disagreement is about different ways of responding to it. Don't you think? — Ludwig V

You tagged three people before that quote so I'm not sure if this is for me. But I did say I wasn't sure I understood the staircase problem so if I've got that wrong, so be it.

Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said. — Ludwig V

I'm for that. The mathematics could not be more clear. Once you start talking about mythical lamps and staircases, the examples have all the moral force of Cinderella's coach. Fairy tales. Thanks for the quote.

But this problem makes me think that they were wrong. One issue that comes to mind is the issue of making a 2-dimensional map of a 3-dimensional sphere. Euclid doesn't work (accurately). But the problem is resolved by developing a different geometry, which breaks some of Euclid's rules. (I realize I'm oversimplifying here, but I hope I'm not hopelessly mistaken.) — Ludwig V

Yes, Gauss and Riemann et. al. Not sure how that helps us with the lamp, the staircase, or Cindarella's coach.

One point to take into account here. This is a thought experiment, so, while the mathematics is real, the horse is not physical, but imaginary, and the difficulty is to work out what rules apply to that in-between context. — Ludwig V

The horse. Uh oh did I miss a story about a horse? Is that a horse of a different color?

The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:

I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

How does one start such a supertask? — Michael

I have explained this repeatedly. If you have the sequence 1, 1/2, 1/4, 1/8, ... on the number line, the points go right to left. If you start at 0, the limit of the sequence, and move to the right by any nonzero amount, no matter how small, you necessarily jump over all but finitely members of the sequence. That's by virtue of the fact that 0 is the limit.

It's certainly true, and I've agreed to this many times, that an infinite sequence has a beginning but no end; so that you can not iterate through it in reverse. How you get from this utterly trivial fact to some kind of cosmic conclusion, I can not fathom.

From Tasks, Super-Tasks, and the Modern Eleatics:

What conclusions are we to draw from this rather heady mixture of genies, machines, lamps, and fair and foul numbers? In particular, has it been shown that super-tasks are really possible – that, in Russell's words, they are at most medically and not logically impossible? Of course not. In a part of his paper that I did not discuss, Thomson does a nice job of destroying the arguments of those who claim to prove that super-tasks are logically possible; had there been time I should have examined them. In the preceding section I tried to do the same for Thomson's own neo-Eleatic arguments. I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his (supposing that I did destroy them). — Michael

Ok. I asked for a reference. Now I have no idea what I'm supposed to conclude from this. That one person thinks supertasks are impossible and another does.

Can you answer a specific question that I've asked you?

Do you understand that mathematically, if you take a step, no matter how small, from 0, you necessarily pass over all but finitely many elements of the sequence 1, 1/2, 1/4, 1/8, ...?

It's really important me to know if you at least understand this mathematical fact.

@keystone, One more thought that occurred to me, and I'm putting it in a separate post because it's short, and important.

Consider one of your rational intervals [0,1]. What is its length?

Well, if it were an interval of reals, it would have length 1. That's how length is defined in the real numbers.

But if it's an interval containing only rationals, then its length is 0. Why? Because of my very first post in this thread. It's countable additivity again. The length of a point is zero; and the length of an interval composed of countably many points is still 0.

That's another problem with your rational continuum. I said it's full of holes. And the problem is that all the length is stored in the holes!

Student borrowers are taxpayers. The question is, which taxpayers are having to pay more? — Vera Mont

I would prefer if Congress would pass a law to have high income earners fund college costs. That at least would have the virtue of being legal.

You say the working class; I say the high earners. — Vera Mont

Ok, so if the tax rates are progressive enough, you say that would help the students and spare the middle class. I don't necessarily disagree. So let Congress pass a law. It's Congress that sets tax policy, not the president.

Would it be so very terrible if people making over $400,000 a year (many of whom are in the money-lending business) had to pay a little more so that the children of orderlies and fish-packers could get an education? — Vera Mont

No, that would be fine. So let Congress pass a law to that effect. We already have a steeply progressive income tax system. The wealthy already pay a lot more.

But no law allows Biden to transfer by fiat half a trillion dollars in debt from the students who signed for it, to the taxpayers -- wealthy or not -- who didn't.

You accept some rational numbers. Not much of a continuum you have there. You understand that, right?
— fishfry
I concur that rational numbers alone, represented as points, are insufficient for constructing a continuum. That's not the argument I'm making. You keep thinking I'm trying to build a continuum. No, I'm starting with a continuum, defined by the interval notation we have discussed, and working my way down to create points. — keystone

But you haven't got a continuum if your intervals contain only rational numbers

How can you say you exclude the real numbers, then write down an interval and call it a continuum?

There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3
— fishfry

Oh no, the classic debate about whether 0.9=1. — keystone

No that is not what I said at all and it has nothing to do with that.

I know you dislike the S-B tree but it makes the top-down and bottom-up views very clear. Maybe use some eyedrops? :P — keystone

Just gonna skip it. Can't relate, don't see its relevance. I'm more focussed on what you just said: that you are "starting with a continuum" that does not include the real numbers.

I'm afraid I can't comprehend that at all.

Bottom-up view: Using a supertask, — keystone

Time is not an aspect of the tree, there is no supertask.

I'm pretty sure that you won't like my depiction of the bottom-up view as I frame it in a way that make's it clearly problematic. I'm fine with not investing further on this specific topic at this time as it really will just be a distraction from the main topic. — keystone

I don't even dislike it. I don't get the relevance of the entire subject. Tell me more about your continuum made up of only rational numbers. If we could get to the bottom of just one thing ...

I'm not questioning the mathematics itself, but rather the philosophical underpinnings of the mathematics. For instance, I recognize Cantor's remarkable contributions to math, even though I personally do not subscribe to the concept of infinite sets. His contributions have a valuable top-down interpretation. — keystone

You should renew your subscription :-)

I think you are an intuitionist.
— fishfry

You make a good point. However, I'm not sure about the details of the constructivist approach - my impression is that a typical intuitionist would say that the number 42 permanently exists once we've intuited it. So while I'm hesitant to label myself hastily, I do think that broadly speaking I fit into this camp. — keystone

I don't understand intuitionism, but you said mathematical objects come into existence via the imagination or acts of will of mathematicians (paraphrasing what you said earlier, sorry if I mis-stated it) and that reminded me of intuitionism.

You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...?
— fishfry
I totally accept and am in awe with the algorithm. I just don't think the algorithm can be run to completion to return a number. I also don't think it has to be run to completion to be valuable. — keystone

Do you believe in the number 1/3 then?

If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum.
— fishfry

I agree, but isn't that what I've been doing all along? Doesn't [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] define a continuum? — keystone

Not if there are only rational numbers in the intervals. Do you understand this point? The rationals are full of holes. More holes than points in fact. Swiss cheese continuum.

Maybe it would be valuable if you detail what you think a continuum must be. For example, will you only accept the definition if it is composed solely of points (and no intervals)? — keystone

Me? The continuum is the real number line. Totally workable definition. Avoids all the philosophical overhead. But the nature of a continuum is pretty deep, way beyond my knowledge of philosophy.

I'd like to move forward since we haven't yet reached the most interesting topics — keystone

That's because you refuse to get there.

, but if you believe that I'm not defining a continuum, then there's no point in proceeding further. — keystone

You said you don't have any real numbers in your intervals, only rationals, and not even all the rationals.

Do you understand that your rational continuum is full of holes? At least tell me if you understand what I mean by that. In other words the rationals contain the points 1, 1.4, 1.41, 1.4142, ... but they don't contain the square root of 2. There's a hole there.

The reals are the completion of the rationals. The reals plug up all the holes in the rationals. That's why the reals are a continuum and the rationals aren't.

You are using interval notation but you are not including the reals. Moths ate your continuum.

Perhaps you can explain to me how an interval of rationals can be a continuum in your mind.

Bottom line: Define [0, 0.5]. Because I have no idea what you mean by that notation.

On those very rare occasions in which the subject arises I have felt the two to be more or less alike. But, here is what Wiki has to say:

Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics. — jgill

Right. Constructivism is purely technical. Intuitionism is constructivism plus some kind of psychological motive or mystic woo. That's my understanding.

I'm not going to disagree with you. But I think regarding it as a plot in the standard sense is not the best way to think about it. I think it was the result of a consensus or "group think" - everybody agreed about the basics and so acted in concert without needing to deliberately plan or co-ordinate anything. Another factor that contributed was more complicated. The distinction between communists and Russians was blurred, that it was easy to continue the suspicion and hostility even when the ideological cause of it was removed. Russians were "othered" during the communist years and remained under suspicion even after communism fell. — Ludwig V

That's my point.

We hated the Soviets. The brave Russian people overthrew the wicked Soviets. Did we say, "Yay brave Russian people, let's be friend now." No! Instead we just got everyone to hate the Russians.

That's a psy-op. The eastward encroachment of NATO was started by Clinton and continued through Bush and Obama. In 2014 the CIA and the neocons in Obama's State dept overthrew the Russia-leaning government of Ukraine, and started shelling the Donbas region, killing some 14,000 ethnic Russians. That's how we got to where we are today.

Hence CIA/neocon/neolib psy-op.

But I hadn't been intending to discuss the situation in Ukraine, maybe that's a different thread.

They did so in the wrong way. The banner of free trade was pinned to the eternal search by capital for cheap labour. The irony of it is that the recipient countries didn't benefit all that much. In general, much of the wealth went to a minority of people who formed a new capitalist class in the recipient countries. It was actually a continuation of colonialism in a slightly different format. — Ludwig V

With you there. Serf's up! There's the new global elite, and there's the rest of us. Time for a revolution? Something's brewing. Much discontent in the air.

They seem to lack a sense of bargaining and deal-making. If you regard it as a competition with winners and losers, you have missed the point. It is of the essence that you allow the other side to make its profit. — Ludwig V

There's an alternate history in which the world became a much more peaceful and prosperous place after the fall of the Soviet Union. That was one of the great missed opportunities of history. Remember the "peace dividend?" That never happened. The warmongers ate it.

Yes, "share their wealth" is a lazy way to put it. It already implies taking something away. But see last comment. But my point was not that I expected them to be overcome with generosity, more that it is not in the long-term interest of the wealthy (even of the moderately wealthy) to prevent others from becoming prosperous. It might mean somewhat lower profit margins, but it doesn't necessarily mean actually taking anything away that they already possess. Its like the argument that it doesn't pay to rip off your customers too much, because they won't come back if you do. — Ludwig V

In the covid period, massive government spending went to the top tier of the economy, while main street got crushed. The $600 stimmy checks were all the middle class got. Was this massive transfer of wealth upward from the middle class to the elite just an accident? Or was it all a plan? A crisis that the big players didn't let go to waste.

Just looked it up. $50 trillion over the past several decades. That ain't pocket change.

The Top 1% of Americans Have Taken $50 Trillion From the Bottom 90%—And That’s Made the U.S. Less Secure


https://time.com/5888024/50-trillion-income-inequality-america/

You can't play it in reverse
— fishfry

So you're saying that it's possible to have recited the natural numbers in ascending order and possible to have recorded this on audio but impossible to then replay this audio in reverse? That seems like special pleading. Am I metaphysically incapable of pressing the rewind button? — Michael

If you play the recording in reverse, the very first movement of the tape or recording, no matter how small, must necessarily jump over all but finitely many of the vocalizations. For the same reason I've explained earlier. Cute thought experiment though.

But it's just like stepping backward from the limit of a sequence of real numbers. The first step, no matter how small, jumps over all but finitely members of the sequence. It's the same fact as saying that any circle drawn around a limit point necessarily contains all but finitely many elements of the sequence.

I am presenting two versions of your argument; one in which I have recited the natural numbers in ascending order and one in which I have recited the natural numbers in descending order. I am using the second version to illustrate the flaw in the first version. — Michael

I didn't see any flaw. I didn't go back to look up that post, but I do remember responding to it. I can only ask you to reread my earlier response.

No, once again you recited the natural numbers in ascending order.
— fishfry

No, I'm reciting them in descending order. I'll repeat it again and highlight to make it clear:

I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum – e.g. my recitation ends with me saying "3" at 12:00:07.5 then "2" at 12:00:15 then "1" at 12:00:30 and then "0" at 12:01:00. — Michael

I already responded to this. It's the sequence 1, 1/2, 1/4, 1/8, ..., accompanied by the vocalizations 1, 2, 3, ... Every member of the sequence gets traversed, every natural number gets vocalized.

Since the limit of the sequence is 0, if you start at zero and take even the smallest step forward, you necessarily leap over all but finitely many elements of the sequence.

Do you understand this point? Mathematically I mean, nevermind the element of time, which is a red herring. Do you understand that any interval around the limit point of a sequence must contain all but finitely elements of the sequence? That's the key insight to untangle your example.

Notice that even if the conclusion follows from the premise that the argument fails because the premise is necessarily false. It is impossible, even in principle, for me to have recited the natural numbers in the manner described. — Michael

I've shown several times exactly how to do it, and I've proven that every number gets vocalized.

Even if the conclusion follows from the premise I do not accept that the premise can possibly be true. Like with the previous argument, I think that it's impossible, even in principle, for me to have recited the natural numbers in the manner described. — Michael

I get that you think that. If you would attempt to engage with my argument you might have an insight and develop better intuitions about limits of sequences.

I have attempted at least to explain why this is impossible (e.g. with reference to recording us doing so and then replaying this recording in reverse), but as it stands you haven't yet explained why this is possible. If you're not trying to argue that it's possible – only that I haven't proved that it's impossible – then that's fine, but if you are trying to argue that it's possible then you have yet to actually do so. — Michael

I can't repeat myself again. I have nothing new to say. If you'd read my posts and have yourself a serious think, then come back with a substantive reply, we might get somewhere. We are not making progress.

Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? — Michael

That's the premise of your own example. It's not my premise. That's hilarious. In the end, you are reduced to denying your own premise.

It's not something that we can just assume unless proven otherwise. — Michael

It's your example, not mine.

Even Benacerraf in his criticism of Thomson accepted this. — Michael

Feel free to give a reference, else I can't respond.

√ω has no meaning in the ordinals, but I believe it does have meaning in the Surreal numbers, which I don't know much about.
— fishfry
OK. I'll accept that. I do believe somebody has shown no limit to the potential cardinality of some sets. — noAxioms

Not sure what you mean by potential cardinality.

I worked a great deal of my career writing code for multiple processors operating under the same address space. It gets interesting keeping them from collisions, with say two of them trying to write different data to the same location. — noAxioms

Point being that you get no increase in computational power from parallelization.

Anyway, not sure what you mean by your statement. It seems on the surface to say two processors is no more powerful than one, which isn't true, but two also isn't twice as powerful. — noAxioms

No function is computable by a parallel process that's not already computable by a linear process. Talking computability theory, not software engineering.

You didn't read my comment then. Ability to move is a given (an axiom, not something that can be proven). — noAxioms

I proved it at the supermarket today, unless you think my vat programmers fooled me again.

Given that, doing so is a supertask only if Zeno's premise holds, that for any starting point, one must first move halfway to the goal. I can't prove that it holds, but I can't prove that it doesn't hold either. — noAxioms

Well maybe it's all an illusion.

I defined the terminal lamp state as a plate of spaghetti.
Yes, the PoS solution. — noAxioms

LOL

Does 'bottom of the stairs' imply a bottom step? If every other step was black and white, what color is the bottom step? PoS, I know. Same problem from where I stand. — noAxioms

Coloring the steps reduces to the lamp.

I'll look at that. I have all the respect for the PSE guys, who blow everybody else away. Quora stands somewhat at the opposite end of that spectrum. — noAxioms

My Quora feed gives me a lot of cute cat pics lately. Makes me happy. Quora certainly used to be a lot better.

You convinced me. Let's transfer the legally contracted debt of people who signed for it, to those who never took out that debt, never saw any of the money, and are busy working while the kids are partying it up in school.
— fishfry
That's not happening and nobody's planning it. — Vera Mont

That's exactly what's happening. Over $500 billion according to the Wharton School of Economics.

$559 billion transferred from student borrowers to the taxpayers.

How can you sit here and deny reality?

https://budgetmodel.wharton.upenn.edu/issues/2024/4/11/biden-student-loan-debt-relief

You deny the number? You think the debt will be paid by the debt fairy? What on earth can you mean by, "That's not happening and nobody's planning it?"

It IS happening. The Biden administration is planning it. You should get better newspapers.

Five hundred fifty nine billion dollars. That's $3387 for every one of the 164 million taxpayers in the US.

You deny it?

Which is quite reasonable. Plumbers make about $60,000; a welder's average is $47,000. Still not vast, and they don't start out $50,000 in the hole.
If their graduate kids make a little more, they can buy their old parents a cruise of something. — Vera Mont

Ok fine. You convinced me. Let's transfer the legally contracted debt of people who signed for it, to those who never took out that debt, never saw any of the money, and are busy working while the kids are partying it up in school.

So how about mortgage debt? Why don't we transfer all of the mortgage debt in the country to those rwho don't own property? That would be fair too, don't you think?

Also I maxed out my credit card on video games and luxury vacations. Would you please pay off my credit card debt? It's not fair that I can't pay my Visa bill this month. I need another vacation.

You know, I think I'll enjoy living under your rule. Everything free, paid for by someone else.

Student loaninterest forgiveness for low earners. — Vera Mont

Excellent point. Fred has no job or money. He's a low earner. But Fred loves lavish vacations, that's how he maxed out his credit card. By your logic, a frugal person who works and doesn't take vacations should pay off Fred's debt. Fred likes that plan a lot. The person who has to pay off Fred's debt, not so much.

So long as the workers are being oppressed. — Vera Mont

That's empty rhetoric. Everyone can claim to be oppressed, especially if being oppressed gets them nice benefits in your communist paradise.

Once social justice and balance are established, — Vera Mont

LOL. "Come the revolution ..." as we used to say when I was i school. But even then we meant it ironically, mocking those who really believed it.

there are no sides and classes. — Vera Mont

Are you being unintentionally funny?

Everybody shares the resources and contributes to the community. — Vera Mont

From each according to his ability, to each according to his need.

That means, every child has the opportunity to learn as much as he or she is able to and wants to, without penalties. A just society would have no such thing as student debts, or any other kind of debt-load that keeps growing, even while you're paying. A just society would outlaw compound interest and 90% of the other financial legerdemain on Wall street. — Vera Mont

Don't hold your breath for human nature to change. That's the problem with communism. Humans.

You're make a big show of defending the workers - represented by a skilled occupation, the holder of which probably considers himself middle class, anyway - while assuming that the working class is a static, unchangeable entity: nobody in, nobody out, beleaguered forever by white collar workers.
That's as gross a misrepresentation as that of NY crime and that of Biden's policies. — Vera Mont

Right, crime in NY is only a matter of perception. As is Biden's economy. I bet you're a big Paul Krugman fan.

That is the inevitable outcome, every cycle. Boom, growth, consolidation, wealth concentration, political corruption, bust, depression, protest, repression or revolution. — Vera Mont

Yup.

Transfinite ordinal numbers are numbers.
Are they? Does √ω have meaning? — noAxioms

5 is a natural number in the Peano axioms. Does $\sqrt 5$ have meaning? No. You have to extend to a larger number system.

$\sqrt \omega$ has no meaning in the ordinals, but I believe it does have meaning in the Surreal numbers, which I don't know much about.

You can't say "x isn't a number because I can't take its square root." You couldn't take the square root of -1 before someone discovered imaginary numbers.

The question of what is a number is historically contingent. Cantor was the one who discovered the ordinals.

It's sad IMO that everyone has heard of the transfinite cardinals, yet nobody knows about the ordinals. The ordinals are logically prior to the cardinals. These days cardinals are actually defined as particular ordinals.

It does for numbers. It's a serious question. I am no expert on how transfinite ordinal numbers are treated. It seems like a different species, like having a set {1, 2, 3, ... , green} which is also a valid set, and countable. — noAxioms

In standard set theory, elements of sets must be other sets. But if you allow urelements, which are elements of sets that are not themselves sets, then you can put green into a set if you like. It's not forbidden by the rules of set theories that have urelements.

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

But naturals aren't integers which aren't rationals which aren't reals which aren't complex numbers which aren't quaternions. There are lots of different kinds of numbers with different rules, and they were all discovered by the historically contingent work of mathematicians.

Ordering irrelevant. The set supposedly needs to be countable, and it is. Michael's definition of supertask came from wiki, and that definition says it is countable, else it's a hypertask. The SEP definition of supertask omits the 'countable' part and seemingly groups the two categories under one word. — noAxioms

I should read that SEP article, I'd probably get a better understanding of this thread. Wiki giveth and Wiki taketh away. Wiki has many errors.

The definition also includes 'sequential', meaning parallel execution of multiple steps is not allowed. — noAxioms

Hmm, that's interesting. In computer science you can always linearize parallel streams, there's no difference in computational power between parallel and serial processing.

Yes ok but then ... how is walking across the room by first traversing 1/2, then half of the remaining half, etc., not a supertask?
Clearly it isn't a supertask if it is impossible to go only half the remaining distance for some intervals. If that is possible, then it must be a supertask. — noAxioms

Ok, then since walking is commonplace, so are supertasks. I gather @Michael would disagree. I haven't got an opinion.

It violates thebijunction
— noAxioms
I take that back. It doesn't violate the bijection. And I spelled it wrong too. So many errors. — noAxioms

No prob, I figured it out. But there are many many ways to re-order a countably infinite set. Here's one called the even-odd order:

<0, 2, 4, 6, 8, ..., 1, 3, 5, 7, ...>

You can see that this set is still in bijection with the natural numbers, but it's order-isomorphic to two consecutive copies of the naturals. This is a representation of the ordinal $\omega + \omega$.

Note that I no longer have an order-preserving bijection.
That's fine. The rational numbers are both ordered and countable, but they cannot be counted in order. — noAxioms

Yes. Although the rationals don't represent any ordinal. The ordinals only apply to well-ordered sets.

https://en.wikipedia.org/wiki/Well-order

Ah yes, why am I doing all this?

Sounds like the lamp problem is unsolved. It is still 'undefined'. — noAxioms

It's not undefined. Inspired by the story of Cinderella, I defined the terminal lamp state as a plate of spaghetti. I have solved the lamp problem to my satisfaction.

Another note: The paradox of the gods that I occasionally bring up is fun to ponder, but it isn't a supertask since it cannot be completed (or even started). Progress is impossible. Ditto with the grim reaper 'paradox' where I die immediately and cannot complete the task. — noAxioms

So many paradoxes, so little time. I know many philosophers care about these things a lot.

Your ω might help with the stairs. The guy is at 'the bottom' and there is but the one step there, labeled ω. No steps attached to it, but step on that one step and up you go, at some small finite numbered step after any arbitrarily small time. — noAxioms

Right, but unlike the lamp, there IS a naturally preferred solution to the staircase. If the walker is on each step at each time, then defining the walker to be present at the bottom of the stairs preserves the continuity of the path. So the staircase (if I even understood the problem, which I may not have) at least has a natural terminating state. Whereas the lamp definitely doesn't.

Unless the answer is that we satisfy Zeno and execute a supertask every time we walk across the room. But Michael objects to that, for reasons I don't yet understand.
His assertion isn't justified, I agree. — noAxioms

Well I agree with you there, but I can't seem to get @Michael to agree :-)

Some speculative physicists (at least one, I believe) think the world is a large finite grid
So much for the postulates of relativity then. I kind of thought we demolished that idea with some simple examples. It seems to be a 'finite automata' model, and the first postulate of SR is really hard (impossbile) to implement with such a model, so a whole new theory is needed to explain pretty much everything if you're going to posit something like that. I haven't read it of course, so any criticism I voice is a strawman at best. — noAxioms

Finite discrete universe is pretty obscure. I don't know if it's ruled out by other physics or not.

The chessboard universe sounds very classical, and it's been proven that physics is not classical, so I wonder how this model you speak of gets around that. — noAxioms

No idea. Found a physics.SE thread.

https://physics.stackexchange.com/questions/22769/is-the-universe-finite-and-discrete

If supertasks are impossible and motion is possible then motion isn't a supertask.
— Michael
This evaded the question ask. Sure, we all agree that if supertasks are impossible, then supertasks are impossible. He asked how you justify the impossibility of a supertask. All your arguments seem to hinge on a variant that there isn't a largest natural number. — noAxioms

Yay you're helping me gang up on @Michael :-) He and I have been having this conversation.

I think I'll go read the SEP article on supertasks.

I had a thread on that a while ago if you care — Lionino

Thanks, I'll check that out. Perhaps it will give me insight into what @Michael means by metaphysically impossible.

You may not realize it but you are asking a loaded question. I believe in 'rational numbers' but not 'the rational numbers'. — keystone

Um ... ok ... I think ...

The difference is that 'the rational numbers' corresponds to Q, the complete set of rational numbers. With the top-down view, such completeness isn't essential (rather, consistency is the aim of the top-down approach). When constructing my metric spaces, I find that I only need to traverse a certain depth in the Stern-Brocot tree to encompass all the rational numbers I require. — keystone

Ok whatever. You accept some rational numbers. Not much of a continuum you have there. You understand that, right?

To clarify, I don't believe in the existence of a complete Stern-Brocot tree. Instead, I believe in the existence of the algorithm capable of generating the tree to any arbitrary depth, although not infinitely. No one has ever encountered the entire tree; rather, we've only interacted with the algorithm and finite trees that it creates. Henceforth, let's refer to it as the Stern-Brocot Algorithm to eliminate ambiguity. — keystone

There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3

Pi has a finite representation. All the computable real numbers do.

Equipped with the Stern-Brocot Algorithm, the mathematical symbols of rational numbers retain their conventional meanings. If we could execute the Stern-Brocot Algorithm to its limiting conclusion and produce the entire tree, there would theoretically exist a 'row-omega' containing the real numbers.
This implies that, theoretically, real numbers necessarily follow from the rational numbers and the Stern-Brocot Algorithm. However, it's evident that running the Stern-Brocot Algorithm to completion is impossible. Consequently, the existence of real numbers doesn't necessarily follow from the existence of rational numbers. — keystone

I already understand that your mathematical ontology includes some but not all rational numbers. Can we move past this please?

Again, I have a strong affinity for the Stern-Brocot Algorithm — keystone

Fine whatever. Enough already.

, but I don't assert that it's the exclusive method to assign meaning to rationals. — keystone

Can we move on please?

The difference lies in our perspectives on the existence of mathematical objects. I assume you are with the bottom-up majority who adhere to the belief that all mathematical entities actually exist, accessible when required, and that these objects fit neatly into sets. — keystone

A view that has near universal mindshare, but ok, I'm a brainwashed mathematical sheep if you like.

In contrast, my perspective maintains that no mathematical object inherently exists; it only manifests when a mind conceives of it. Therefore, if no mind currently contemplates the number 42, it does not exist in actuality; it merely holds the potential for existence. — keystone

Ah. Perhaps you would enjoy intuitionism.

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.[/url]
— Wiki

Regarding the enclosing set, I don't subscribe to the notion of its inherent existence. Instead, I endorse an algorithm capable of generating sets to have arbitrarily many elements, albeit not infinite. If you run this algorithm long enough, it will generate the set we're looking for to define our metric space. — keystone

I think you are an intuitionist. I think that's what you're getting at. Can you read the Wiki link and tell me if that's what you're getting at?

Intuitionism is closely related to constructivism, the idea that mathematical objects only exist if there's an algorithm or procedure to construct them. Intuitionism is like constructivism with an extra bit of mysticism that I can never quite grasp.

I refuse to regard pi as a boundary for my intervals because it cannot be generated using the Stern-Brocot Algorithm. Pi does hold significance in my perspective, but I think it's more appropriate to delve into that explanation if/once we move on to two dimensions. — keystone

You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...?

I only referred to the Peano Axioms to point out the concept of succession. When viewed from the top-down perspective, numbers are not constructed from the naturals (I agree, that would imply a classical, bottom-up math start). Natural numbers are only distinctive in that they are positioned on the right-most branch of any tree created with Stern-Brocot Algorithm, which indeed makes them quite unique. — keystone

Ok. My eyes glaze a little more every time you mention the S-B tree, I have no idea why this idea has such a hold on you.

I agree that rational numbers alone cannot model a continuum. With the top-down view, this is equivalent to saying that points alone cannot model a continuum. And that's why I'm starting with a continuum (i.e. using intervals rather than numbers). It's much easier to get points from a continuum than it is to get a continuum from points. — keystone

If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum.

I can sign up to that. It all went wrong in the 1990's, when the West and capitalism indulged in triumphalism instead of recognizing the need to spread prosperity around the world. — Ludwig V

Other way 'round I think. Clinton and the neoliberals did spread prosperity around the world, at the expense of the manufacturing base of America. The 90's is when offshoring really took off. Don't get me wrong, I loves the Clinton economy. The 90s were great. Maybe the last great decade we ever had.

(WTO is supposed to help with this, but does not work - at least, not anything like enough.) They should have started with a Marshall Plan for Russia and then similar plans for all the other underdeveloped areas of the world. Very expensive, but cheaper than yet another world war. — Ludwig V

Oh yes. After the fall of the Soviet Union we should have honored and made friends with the brave Russians who overthrew our great enemy. Instead, we just made Russia the new enemy and pushed NATO ever eastward after promising not to. Leading to the war in Ukraine. A neocon/neoliberal/CIA plot all the way. Exactly what they wanted. Now they're going to blow up the world.

In my opinion, getting Americans to hate the Soviet Union after they had saved our bacon in WWII; and then getting American to hate Russia after they'd thrown over the Soviet Union, is one of the greatest psy-ops in the history of the world. We hated the Soviets and now we hate the Russians, who overthrew the Soviets and just wanted to be friends. It's a terrible thing what's happened. 30 years in the making. Hollowing out the heartland at home and pressing NATO against Russia abroad. Russia asked to join NATO, we turned them down. They just wanted to be friends, but the neocons only want war.

I mention this because it is a case of the general problem posed for this thread and to have an excuse for promoting the argument for enlightened self-interest as a way of breaking through the reluctance of the wealthy to share their wealth (beyond charity, which they remain in control of). — Ludwig V

Share the wealth meaning what? Higher taxes for handouts to their politically connected friends? That's not working very well. Like the covid bailouts. $600 checks for the proles along with a humongous transfer of wealth from the middle to the upper classes. Shutting down mom and pop so Walmart can eat their lunch. Inflation destroying working stiffs.

Not sure I share your trust in the ability of our leaders to "spread the wealth around," as Obama put it. I don't see the Obamas spreading their copious wealth around, do you? "I got mine, Jack," is their mantra.

Argument 1
Premise: I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in ascending order. — Michael

I believe you have agreed with me.

Argument 2
Premise: I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in descending order. — Michael

No, once again you recited the natural numbers in ascending order.

Normalizing the 60 to 1 for simplicity, you iterated though the sequence 1, 1/2, 1/4, 1/8, ..., while reciting the numbers 1, 2, 3, ... etc.

These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed. As it stands you're begging the question. — Michael

Begging the question! Nonsense! Non sequitur! Metaphysically impossible!

Always a buzzphrase, never a substantive argument.

Why don't you engage with the argument I'm making?

Now let's assume that it's metaphysically possible to have recited the natural numbers in ascending order and to have recorded this on video/audio. What happens when we replay this video/audio in reverse? — Michael

Lol. You can't play it in reverse, there is no end to the natural numbers in their usual order. You never get to the end. But there is no number you don't vocalize at time point, so you do vocalize them all.

I know this pushes hard against your intuition. If you'll engage with the argument, you will at some point develop better intuitions.

It's the same as having recited the natural numbers in descending order which you admit is metaphysically impossible. Therefore having recited the natural numbers in ascending order must also be metaphysically impossible. — Michael

Completely false, as I've demonstrated numerous time.

Both Argument 1 and Argument 2 are unsound. The premises are necessarily false. It is impossible in principle for us to recite the natural numbers in the manners described. — Michael

Irrelevant! Non sequitur! Metaphysically impossible! Nonsense! Begging the question!

I think I'll just start speaking your language. Maybe that will work.