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.

After 60 seconds 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. — Michael

I see that I misunderstood your idea. You are counting time backward. Ok I'll respond to that. But just wondering, when you realized I misunderstood you earlier, why didn't you point that out?

Ok. Suppose that I start at 1 and count backward through 1/2, 1/4, 1/8, ...

Clearly I say all the numbers. at 1 I say 1, at 1/2 I say 2, at 1/8 I say 3, and in general at $\frac{1}{2^n}$ I say n.

It's perfectly clear that I say all the numbers, and iterate through all the negative powers of 2. This is elementary. What number don't I say?

What natural number did I not say? — Michael

There is no natural number that wasn't said. Therefore they were all said.

You can't answer, therefore it is metaphysically possible to have recited the natural numbers in descending order. — Michael

It's perfectly obvious that an infinite sequence is infinite at one end. So you can iterate in one direction and not the other. I can't for the life of me imagine why you think that means anything important.

Now look at the sequence 1, 1/2, 1/4, 1/8, ... again. Graph the points on the real number line. You start at 1, then move leftward to 1/2, then leftware to 1/4, and so on.

The sequence has the well-known limit 0.

Now if you were to start at 0 and move any positive length to the right, no matter how small, you would necessarily jump over all but finitely many elements of the sequence. That's inherent in the meaning and definition of a limit point.

It's exactly the same as 1/2, 3/4, 7/8, ..., whose limit is 1. In fact it's the exact same situation but with the order relation reversed.

Now you want to impose some kind of Newtonian understanding of time, call 0 a time, and say this proves something. It proves nothing but ... and I don't know any other way to say this ... it proves nothing but your own lack of clear thinking around the nature of limits of sequences.

So main points;

* If you iterate through 1, 1/2, 1/4, 1/8, ... while vocalizing "0", "1", "2", "3", etc., you will iterate through ALL the elements of the sequence and you will vocalize ALL of the natural numbers. After all, what member of the sequence do you think is missing? What natural number won't be vocalized?

* Secondly, it's perfectly clear that an infinite sequence starts at one element and continues indefinitely, with no last element. So of course you can't iterate an infinite sequence "from the end." But this is a triviality, it has no significance.

Obviously the above is fallacious. — Michael

Fallacious! Non sequitur! Metaphysically impossible! Nonsense!

These are words. They are not arguments.

It is metaphysically impossible — Michael

There you go again, as Ronald Reagan once said to Jimmy Carter.

to have recited the natural numbers in descending order. — Michael

Sure, because an infinite sequence has no end. You seem to think this elementary and trivial fact has deep meaning. It does not.

The fact that we can sum an infinite series with terms that match the described and implied time intervals is irrelevant. The premise begs the question. And the same is true of your version of the argument. — Michael

I'm disappointed that you won't engage with the argument I'm making. I'll add "begs the question" to your list of buzzphrases used in lieu of substantive argument.

I found that discussion very helpful. — Ludwig V

Glad to hear that.

But in the staircase problem, if 1 is "walker is on the step" and 0 otherwise, then we have the sequence 1, 1, 1, 1, ... which has the limit 1. So 1, the walker is on the step, is the natural state at the end of the sequence.
— fishfry
Have I understood right, that 0 means "walker is not on the step", and that "the step" means "the step that is relevant at this point" - which could be 10, or 2,436? So 0 would be appropriate if the walker is on the floor from which the staircase starts (up or down)
My instinct would have been to assign 0 also to being on the floor at which the staircase finishes (up or down). It makes the whole thing symmetrical and so more satisfying. — Ludwig V

Could be. Truth be told I got lost in the OP involving many non-relevant fairy tale elements and probably don't even understand what the staircase question is.

That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is.
— fishfry
I don't like that way of putting it, at least in the paradoxes. Doesn't the arrow paradox kick in when you set off in the.reverse direction? Or perhaps you are just thinking of the numbers as members of a set, not of what the number might be measuring. I suppose that's what "ordinal" means? — Ludwig V

Ordinals are [ul=https://en.wikipedia.org/wiki/Well-order]well-ordered sets[/url].

As with my standard example, if you take the ordered set <1/2, 3/4, 7/8, ..., 1>, where I'm now using angle brackets to denote ordered sets, suppose you start from 1 and take a step back. Since 1 has no immediate predecessor, any step back necessarily jumps over all but finitely many members of the sequence. It's a counterintutive quirk of limit ordinals. Any path back to the beginning involves only finitely many steps, because the first step back makes such a jump.

Michael's way of putting the point is, IMO, a bit dramatic. — Ludwig V

Yeah. "Metaphysically impossible!" "Non sequitur!" "Nonsense!" Never an actual argument. Tagging @Michael so as not to disparage him behind his back.

The boring truth for me, is that the supertask exists as a result of the way that you think of the task. If you think of it differently, it isn't a supertask. It's not about reality, but about how you apply mathematics to reality. — Ludwig V

I still don't know if walking across the room is a supertask or not.

Not to mention that, if we take the real numbers as a model of space, we pass through uncountably many points in finite time. That's another mystery.
— fishfry
Well, if you insist on describing things in that way .... I'm not sure what you mean by "model". — Ludwig V

Nobody knows the ultimate metaphysical truth about reality. All we can ever do is model is. Relativity is a mathematical model, as is quantum physics, as was Newtonian physics. All science can ever do is build models that fix the experimental data to a reasonable degree of approximation. That's all I mean by model.

I think of what we are doing as applying a process of measuring and counting to space - or not actually to space itself, but to objects in space. — Ludwig V

Only to our latest conceptual model of space. We can't know ultimate reality. Or if we can, we don't as of yet.

A geometrical point has no dimensions at all. So it is easy to see how we can pass infinitely many points in a finite time. (I'm not quite sure how this would apply to numbers, but they do not have any dimensions either.) This doesn't apply to the paradoxes we are considering, which involve measurable lengths, but it may help to think of them differently. — Ludwig V

The unit interval [0,1] has length 1 and is composed of uncountably many zero-length points. That's a mystery.

Name the first one that's not. It's a trivial exercise to identify the exact time at which each natural number is spoken. "1" is spoken at 60, "2" at 90, "3" at 105, "4" at 112.5, and so forth.
I did not "simply assert" all the numbers are spoken. I proved it logically. Induction works in the Peano axioms, I don't even need set theory.
— fishfry
Yes, but you didn't speak all the natural numbers, and indeed, if induction means what I think it means, your argument avoids the need to deal with each natural number in turn and sequence. — Ludwig V

I apparently misunderstood @Michael's backward counting example, I'll be addressing that shortly as I slog through my mentions.

But if I count forward at successively halved intervals: Saying "1" at time 1, "2" at time 1/2, "3" at time 1/4, etc., I will certainly count all the numbers. You can't name the first one I don't say. And we can calculate exactly what time I'll say 47, or Googolplex, or Graham's number.

Obviously the above is fallacious. It is metaphysically impossible to have recited the natural numbers in descending order. — Michael

I already agreed with this, because limit ordinals do not have immediate predecessors.

The fact that we can sum such an infinite series is irrelevant. And the same is true of your version of the argument. — Michael

If you would engage in your private time with the 60 second puzzle, you would see that each number is spoken at a specific, calculable time; that there is no first number that's not spoken; and therefore every number is spoken.

It's not productive for me to give a high-school level inductive argument and for you to say "nonsense" and "metaphysically impossible" without ever engaging with the argument.

Please read the Wiki page on mathematical induction and ask questions as necessary, and challenge yourself to engage with the argument.

Ask yourself: What is the first number not spoken? If you ask yourself that enough times, you may have an epiphany.

No you haven't. Your premise begs the question and simply asserts that all the natural numbers have been recited within 60 seconds. — Michael

Name the first one that's not. It's a trivial exercise to identify the exact time at which each natural number is spoken. "1" is spoken at 60, "2" at 90, "3" at 105, "4" at 112.5, and so forth.

Can you not see that we can calculate the exact time at which each number is spoken?

I did not "simply assert" all the numbers are spoken. I proved it logically. Induction works in the Peano axioms, I don't even need set theory.

If you work through this example you will obtain insight.

No, we're reciting the numbers in descending order. It's impossible to do, even in principle. The fact that we can assert that I recite the first number in N seconds and the second number in N/2 seconds and the third number in N/4 seconds, and so on ad infinitum, and the fact that the sum of this infinite series is 2N, doesn't then entail that the supertask is possible.

That we can sum such an infinite series is a red herring. — Michael

You're right that we can't "name all the numbers" going backward. but that's obvious. There's no largest number and limit ordinals don't have immediate predecessors.

It's pointless for you to snap back a minute later arguing with well established mathematical facts. I gave a solid inductive argument that by the premises of your 60 second puzzle, all the numbers will be spoken. That's because there's no first number that won't be spoken. If you doubt that, then name a number that's not spoken.

I ask you to read carefully what I'm writing, and think about it.

Did you ever learn mathematical induction? If yes, I gave a standard inductive argument. If no, that's a good starting point and I'll be happy to give a summary. I gave you the Wiki link. I can't argue well established facts with you.

In the puzzle you gave, every number must be spoken. In fact we could calculate, if we cared to, the exact time at which it's spoken.

Please give this some thought.

What number won't be spoken?

It begs the question. Your premise is necessarily false. Such a supertask is impossible, even in principle, to start. — Michael

Did you learn mathematical induction in school? Please review that. Please take the time to understand the argument I made.

Under the premises of the problem you posted, there is no number that does not get spoken.

It's imperative that you understand that. It's pointless for you to disagree. You must show that there's a number that did not get spoken. If you can't do that, then every number gets spoken.

You just listed five rational numbers and are claiming that this is proof of you reciting all the natural numbers in descending order? — Michael

I did not make that claim. I said I counted backward from a limit ordinal. That's easy. It's always a finite number of steps.

You're talking nonsense. — Michael

I'm counting backward from a limit ordinal. Very standard math.

What number do you recite after 1? — Michael

7/8 will do just fine. I necessarily had to jump over all but finitely members of the sequence.

Of course I can not count ALL the numbers backward. That's impossible. That's because limit ordinals do not have predecessors. That's the definition of a limit ordinal, an ordinal that does not have an immediate predecessor. So it's your challenge that's nonsense.

But please, I'm asking you to sit down and think about the inductive argument I made.

Counting forward with your 60 second idea, which number won't be spoken?

We can certainly say "1". And if we say n, we can say n + 1. This is high school mathematical induction. Please tell me you learned this. If not, that would explain your confusion. But if you made it through high school math (do they still teach induction in high school? What do I know) then you have the means to understand the argument.

Please take the time to think it through. What number can't be spoken?

ps -- I looked it up. Perhaps induction is not universally taught in high school, and it doesn't come up in calculus.

Do you know mathematical induction? It's a row of dominos.

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

Ok. See also ↪J 's other thread. — Banno

Thanks much.

Because it begs the question. — Michael

I go 1 at 60, 2 at 30, etc.

Name the first number that I fail to count

Third time I'm asking you the question. (At ever decreasing intervals of time!)

This is a standard inductive argument. If it's impossible to name the first natural number at which a property fails to hold, the property must hold for all natural numbers. Think back to when you learned inductive proofs in school. I can name 1. And if I name n, I can name n + 1. Therefore I can name all the numbers. Counterintuitive though it may be, it's true. You learned this in high school.

Please give this argument some thought.

That's not counting down from infinity. — Michael

You have no proof or evidence. On the contrary, the mathematics is clear.

I don't know what you mean that supertasks are nonterminating by definition.
— fishfry

Tasks are performed ad infinitum. I never stop counting. There's always another number to count. — Michael

Did I not move you, surprise you, convince you, that if you count 1, 2, 3, ... successively halving the time intervals, that you will indeed count every single natural number in finite time? If not, why not?

I am still waiting for you to name the first number I didn't count.

This is a standard inductive argument. To prove that a property holds for all natural numbers, I show the impossibility of there being a first number where the property fails.

I'm talking about reciting the numbers. So imagine someone reciting the natural numbers up to infinity. Now imagine that process in reverse. That's what I mean by someone counting down from infinity. — Michael

But counting backward from infinity is always finite! I showed you how that works, counting backward from 1 in the ordered set <1/2, 3/4, 7/8, ..., 1>

In fact this is true of all the transfinite ordinals. It's only finitely many steps backward from any transfinite ordinal, no matter how large. That's because stepping back from any limit ordinal (defined as an ordinal without an immediate predecessor) necessarily jumps over all but finitely many elements of the sequence that led up to it.

It is a non sequitur to argue that because we can sum an infinite series with terms that match the proposed time intervals that it is possible to have counted down from infinity. It is impossible, even in principle, to start such a count. The maths doesn't change this. — Michael

It's easy, I'll count backward from infinity right here on a public Internet forum, in plain view of the world.

1, 15/16, 7/8, 3/4, 1/2. Done. My first step necessarily jumped over all but finitely many elements of the infinite sequence. It must be that way.

That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is.

Counting backward from infinity is easy, and always finite!

Davidson is just the ubiquitous On the very idea of a conceptual scheme. — Banno

Not ubiquitous to me, I'm a philosophical dummy. I'll Google around.

There's a prima facie disagreement here, but I think it is on the surface only, that Midgley is espousing something not too dissimilar to Davidson's anomalism of the mental. — Banno

I'm way out of my depth. I will do some surfing and maybe glean some clues.

A quick Google search yielded:

What is Davidson's summary of the very idea of a conceptual scheme?

Davidson attacks the intelligibility of conceptual relativism, i.e. of truth relative to a conceptual scheme. He defines the notion of a conceptual scheme as something ordering, organizing, and rendering intelligible empirical content, and calls the position that employs both notions scheme‐content dualism.

and my eyes glazed over. I'll check out the thread you linked. Thanks for the pointers.

Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition?
— fishfry

If supertasks are impossible and motion is possible then motion isn't a supertask. — Michael

I don't find that satisfactory. It only casts doubt on the premise "if supertasks are impossible."

I agree with you that the lamp and staircase and other related puzzles are qualitatively different than Zeno's paradoxes of motion, so perhaps in that sense you want to reserve the word supertask for the former. But your definition is "completing a countably infinite number of tasks in finite time," and walking across the room seems to satisfy that definition.

Not to mention that, if we take the real numbers as a model of space, we pass through uncountably many points in finite time. That's another mystery.

* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction.
— fishfry

By definition supertasks are non-terminating processes, therefore you've gone wrong somewhere if you conclude that they can terminate after 2N seconds. — Michael

I don't know what you mean that supertasks are nonterminating by definition. Just thinking mathematically for a moment, limits "terminate" a sequence in the sense that 1 is the terminus of the sequence 1/2, 3/4, 7/8, ... The limit 1 is not part of the sequence, but we can imagine the 1 stuck at the end of an ordered set, as I have been doing, and it's perfectly sensible.

In other words supertasks are nonterminating, but they definitely may have a terminal state; just as a convergent mathematical sequence has no final term, yet has a limit. Is my analogy unsatisfactory with respect to your conception of supertasks?

Also I think the clearest example I gave was that of having counted down from infinity. We can assert (explaining what happened in reverse) that I recited 0 after 60 seconds, recited 1 after 30 seconds, recited 2 after 15 seconds, recited 3 after 7.5 seconds, etc., and we can say that we can sum an infinite series with terms that match the described (and implied) time intervals, but it doesn't then follow that we can have counted down from infinity; we can't even start such a count. — Michael

I don't follow how you are counting down from infinity. In fact when you count down from infinity, it's always only finitely many steps back. If I take the ordered set <1/2, 3/4, 7/8, ..., 1> and I start at 1, my first step backwards jumps over all but finitely many elements of the sequence, and it's always only finitely many steps back from 1 to 1/2.

[Per my recent convo w/@No Axioms I am using angle brackets to denote ordered sets].

You did lose me when you said that counting 0, 1, 2, ... is "counting down from infinity." I did not understand that example when you gave it earlier. Mathematically, the ordered set <1, 2, 3, ...> exists, all at once. Its counting is completed the moment it's invoked into existence by the axiom of infinity.

But let me ask you this. Suppose I say 0 at 60 seconds, and 1 at 30 seconds, and 3 at 15 seconds, and so forth.

Now I claim that after 120 seconds (the sum of the series) I have counted all the natural numbers!

Yes I claim that. And as proof, I challenge you to name the first number that I did not count.

Since you can not do that, I have indeed counted all the natural numbers.

The mathematics is evidently a non sequitur — Michael

I either don't understand what you mean, or I strenuously disagree.

Explain please?

, and it's a non sequitur in the case of having counted up to infinity as well. — Michael

I just proved to you, using a very standard inductive argument, that I can indeed count all the natural numbers as you described, in intervals of 60, 30, etc. Because you can not name any number I can't count. Did I count 47? Sure. Googolplex? Sure. Graham's number? Sure. There is no number that I didn't eventually count. Therefore I counted them all.

It brings out the conflict in my own arguments, between Midgley and Davidson, and provides something of a logical frame for that discussion. No small topic. — Banno

Have you got a reference to Midgley and Davidson? Is there an interesting professional discussion of these issues?

I said I had no problem with any of that.
Is it a belief thing, like it is some kind of religious proposition or something? "Hey, I'm going rogue here and will suspend belief that 7 is a factor of 35". — noAxioms

I'm making the point that you are perfectly willing to accept {1/2, 3/4, 7/8, ..., 1} as a valid set that contains an infinite sequence and its limit; but you are having trouble accepting {1, 2, 3, ..., $\omega$}, simply because it's far less familiar. But in terms of their order, they are exactly the same set. They have very different metric properties; but strictly with respect to order, they are two different representations of the same ordered set.

Treating infinity as a number, something you didn't do in your unionized set above — noAxioms

Transfinite ordinal numbers are numbers. It's just a matter of expanding one's concept of a number. $\omega$ is a number. It's the first transfinite ordinal number. I am casually calling it a "point at infinity," but if that bothers you, just think of it as 1 in the set {1/2, 3/4, 7/8, ..., 1}. It's exactly the same thing wearing a different suit of clothes.

It's an infinite sequence. I stuck the number 1 on the end.
Yea, when it normally is depicted at the beginning. From what I know, a set is a set regardless of the ordering. There must be a different term (ordered set?) that distinguishes two identical sets ordered differently, sort of like {1, 3, 5, 7 --- --- 8, 6, 4, 2} — noAxioms

Yes, ordered set. I have been casually using the curly braces, but you are absolutely correct. {1/2, 3/4, 7/8, ..., 1} has no order, I could stick the 1 in the middle or at the beginning and it would be the same set, but I'd lose the order that I consider important.

Perhaps a notation like <1/2, 3/4, 7/8, ..., 1> would be better, to indicate an ordered set. You are absolutely right. I did not want to add any more complications earlier, but the curly braces are inaccurate in the way I'm using them. I'm speaking of ordered sets. So I'll use angle brackets from now on.

The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set?
It violates thebijunction. You can't say what number comes just before it, which you can for any other element except of course the first. You can do that with any other element. — noAxioms

Well then here yet another representation of the same idea. Suppose I reorder the natural numbers

<1, 2, 3, 4, ...>

by putting 1 at the end, so that I have:

<2, 3, 4, ..., 1>

You can see that I still have a bijection. As you noted, sets don't have order, so it's still the same set.

Note that I no longer have an order-preserving bijection. I merely have a set bijection. I can still correspond 1 to 1, 2 to 2, and so forth. But I can't do it in an order-preserving manner.

But now I have another representation of an ordered set that consists of an infinite sequence followed by a "point at infinity," or a largest element. That largest element does not have a predecessor, you are right about that.

And in fact we have a name for that. In ordinal theory, an ordinal with a predecessor is a successor ordinal. And an ordinal without a predecessor is a limit ordinal. So your intuitions are spot on.

OK, but what problem does it solve? It doesn't solve Zeno's thing because there's no problem with it. It doesn't solve the lamp thing since it still provides no answer to it. — noAxioms

Ah yes, why am I doing all this?

It solves the lamp problem. The lamp state is a function on <1/2, 3/4, 7/8, ..., 1> defined as "on" at 1/2, "off" at 3/4, "on" at 7/8, and so forth.

But now we see (more clearly, IMO) that the state at 1 is simply undefined. The statement of the problem defines the lamp state at each element of the sequence; but does NOT define the state at the limit.

We also note that there is no way to make the sequence 0, 1, 0, 1, ... continuous.

And since you didn't tell me what is the state at 1, and there is no natural way to define the state at 1, I am free to define the state at 1 any way I like. And inspired by Cinderella, I define the state of the lamp at 1 to be a plate of spaghetti. That's the solution to the problem. The final state is anything you like. It doesn't even have to be on or off since it's not a real lamp, just as Cinderella's coach is not a real coach. The lamp problem is every bit as much a fairy tale as Cinderella.

So for many of these supertask problems, the ordered set <1/2, 3/4, 7/8, ..., 1> is the natural setting for the problem.

Note that the staircase is different. The walker is on step 1, on step 2, etc. So the natural, continuous way of completing the sequence is to say that the walker is at the bottom of the stairs. This is totally different than the lamp, which can not be made continuous or sensible in any way at all.

So my entire point is that <1/2, 3/4, 7/8, ..., 1> is the natural way to think about these problems. The question is always: how did we define the state at the elements of the infinite sequence; and then, how are we free to define the final state at the limit.

Nobody's asking the particle to meaningfully discuss (mathematically or not) the step. It only has to get from one side to the other, and it does. Your argument is similar to Michael wanting a person to recite the number of each step, a form of meaningful discussion. — noAxioms

I'm not sure what you mean by referring to the subjective state of the particles. When Newton wrote down his great law of gravitation, he did not care how the masses feel about it. I'm not following your analogy.

Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes.
It would falsify the first premise. Continuous space falsifies the second premise. Zeno posits two mutually contradictory premises, which isn't a paradox, only a par of mutually contradictory premises,. — noAxioms

I confess to not knowing the answer to Zeno. It's a clever argument. 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.

But I can say "for all we know, ....", and then there's no claim. I'm not making the claim you state. I'm simply saying we don't know it's not true. I even put out my opinion that I don't think it's true, but the chessboard thing isn't the alternative. That's even worse. It is a direct violation of all the premises of relativity theory (none of which has been proved). — noAxioms

Some speculative physicists (at least one, I believe) think the world is a large finite grid. It's not out of the realm of possibility as I understand it. I think I read that in Penrose's The Road to Reality. And if Sir Roger thinks it's good enough to put in a book, it must be of interest.

In other words the chessboard universe is not ruled out by any known theory or experiment. And we know that quantum and relativity have yet to be integrated, and perhaps that's a clue.

A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time."
— Michael
Yea, I don't know how that could have been lost. I don't think anybody attempted to redefine it anywhere. — noAxioms

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? I don't understand this point.

I'm not quite sure what you mean by "believe in the rational numbers." — keystone

You confused me a while back. You said you don't believe in the real numbers [or some similar wording].

So I asked you, what are those symbols 0, .5, 1, and so forth? If they're not real numbers, what are they?

That's why I asked you if you believe in the rational numbers. If you do, then you have to also believe in the reals, since the reals are constructed from the rationals. If you don't, then again I ask you what are 0, .5, and 1?

From a top-down perspective, there's no need to assert the existence of either R or Q, especially since all the subsets within the enclosing 'set' are finite. — keystone

You have been freely using the symbols 0, .5, and 1. If they are not real, and they are not rational, then I don't know what those symbols mean. Can you define them?

"... all the subsets within the enclosing 'set' are finite"???? Means what? Lost me there.

If you suggest that this enclosing 'set' is infinite, then we must rethink our definition of what an 'enclosing set' actually is in this context. I was hoping to put this particular discussion aside for now, as it will likely divert attention from our main focus. — keystone

You're the one with some notion of enclosing set. A metric space is a set with a distance function. If it lives in a larger ambient set, then you have to say what that is. You started a long time ago saying something like "the metric doesn't apply outside the metric space." Ok that's true, but what is outside? You have to say what that is.

Regarding Dedekind cuts, they involve splitting the infinite set of rational numbers into two subsets. This presupposes both the existence of an infinite entity (Q) and the completion of an infinite process (the split). If one rejects the concept of actual infinity, then it's questionable whether real numbers necessarily follow from rational numbers. — keystone

Ok fine. You reject the real numbers. You already said that.

So I asked you, do you believe in the rational numbers. And you asked me what I mean by that!

If you use symbols like 0, .5, and 1, you have to say what they are.

So, do you believe in the rational numbers? Is that the number system we're working in?

In which case I have to echo @jgill's excellent question as to whether you accept intervals like [pi, pi + 1], and if not, why not.

However, the discussion about actual infinity and the nature of real numbers could go on endlessly. — keystone

You could bring it to a quick conclusion by saying, "Yes, we are working in the rational numbers."

But you won't even say that! Leaving me totally confused.

I acknowledge that these concepts are crucial for a bottom-up approach, but can we instead focus on seeing how far a top-down perspective—devoid of actual infinities and traditional real numbers—can lead us? In the top-down view, reals hold a special role, just not as conventional numbers. — keystone

Sure. Then what are these funny symbols 0, .5, and 1 that you keep on using? What do your interval notations denote?

If you're working in the rationals that's fine, but when I asked you about it, you asked me what I meant by the question.

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?
You are the one who started at 0, remember?
— fishfry

I believe the confusion arises from the dual meanings of "start" due to there being two timelines: (1) my timeline as the creator of the story and (2) the timeline of the man running from 0 to 1 within the story. — keystone

The stories are very unhelpful to me. As are timelines.

On my timeline, I start by constructing the entire narrative of him running from 0 to 1. — keystone

What is this '0'? What is this '1'? Define your terms.

The journey is complete from the start. I can make additional cuts to, for example, see him at 0.5. Regardless of what I do, the journey is always complete.
On the running man's timeline, he experiences himself starting at 0, travelling towards 1, and later arriving at 1. — keystone

You are using these funny symbols. I know the usual standard mathematical meaning of those symbols, but you have rejected them in favor of your "top down" idea. So what are these symbols? What if we called them "fish" and "bazooka?" Then nothing at all would be clear, but your logic error would be more obvious

You want to reject standard mathematics but freely use symbols like 0, .5, and 1, without defining them.

Do you see the problem?

I think you're trying to build his journey on his timeline, one point at a time. The runner would indeed believe that limits are required for him to advance to 0.5. I want you to look at it from my timeline (outside of his world), where the journey is already complete. If I want to see where he is at 0.5 I just cut his complete journey in half. Does that clarify things? — keystone

No, since I don't know what 0.5 and "half" mean, in the absence of standard bottom-up math.

Do you see your circularity problem? You want to start by rejecting standard math, but then you won't tell me what these symbols mean in your system.

Unlike supertasks, no magic is required to complete the journey with the top-down view. Assuming you accept the Peano Axioms as a conventional framework, — keystone

Ah. That's quite a lot already, for someone claiming to reject infinite processes and standard bottom-up math.

So you are willing to start with the Peano axioms? Is that your starting place? Then I know what 0 and 1 are, but I'm still not sure about this 0.5 thing.

you're familiar with the concept of succession, which defines progression from 1 to 2 to 3, and so on. This is essentially what I'm applying as well; on the runner's timeline he progresses in succession from 0 to (0,0.5) to 0.5, — keystone

0.5 is not defined by the Peano axioms. What is it?

and so on. Please take note, this particular succession from 0 to 0.5 involves only 2 steps. No limit is required, just as no limits are employed with the Peano Axioms. — keystone

No idea what 0.5 is. But at least after all this you agreed to stipulate the Peano axioms. That's a start. A start from classical, bottom-up math.

I'll save you some trouble and show you how to build out the rational numbers from the Peano axioms. You extend the natural numbers to the integers, then you do a construction called the field of fractions of an integral domain.

I'm not entirely sure if that construction is legit in Peano without the axiom of infinity, but I can live with it.

So after all this, I think you are working in the rational numbers, and 0.5 has its usual meaning. Is that right?

I can live with that. Although the rational numbers are tragically deficient as a continuum. You know that, right? They're full of holes. They're not continuous in the intuitive sense.

'Vastly' is a big word. By quick look-up, the average welder's pay is $22.55/hr, while the average primary school teacher's is $23.44/hr. The teacher starts working life with a $58,000 student loan; the welder gets certification for $475. — Vera Mont

Cherry picking teachers is misleading. A quick Google search on "how much to college graduates earn?" said that they make $50k their first year. "Average college graduate salary" yielded $67,786.

But still, you said you're a communist. Aren't communists supposed to be on the side of the workers? Why should the welder pay the teacher's debts, or anyone else's debts? Why shouldn't everyone pay their own debts? And again, if Congress wants to change that, let them pass a law. The president is not authorized to transfer billions of dollars of student debt to everyone BUT the people who agreed to pay that debt.

You keep saying it's the working class who will be 'burdened' by educating its children, so that they can still work when all the working-class jobs except home renovation and domestic service are automated out of existence. Why do you think poor people's kids shouldn't have a choice of careers? — Vera Mont

Do you still beat your wife?

What kind of question is that? It has nothing to do with anything. You're just changing the subject. And arguing that the working class should assume the debts of the college grads. Some commie you are! You still haven't explained this to me.

I'm the one on the side of the workers. I'm a better commie than you are and I'm not even a commie. I used to be one, then I learned something about the world.

President Biden will announce plans that, if finalized as proposed, would cancel up to $20,000 of the amount a borrower’s balance has grown due to unpaid interest on their loans after entering repayment, regardless of their income. — Vera Mont

The debt is not cancelled. It's transferred to the taxpayers. Can we at least be clear about that?

Low and middle-income borrowers enrolled in the SAVE plan or any other income-driven repayment (IDR) plan would be eligible for the entire amount their balance has grown since entering repayment to be canceled under the Administration’s plans. This group of borrowers includes single borrowers who earn $120,000 or less and married borrowers who earn $240,000 or less. — Vera Mont

Not a nickel of debt is cancelled. It's transferred to the taxpayers. I'll concede that you seem to have paid more attention to the details of the plan than I have.

If Congress wants to pass subsidies for the debt of low-income people, let him do that. But why stop at college debt? Why not transfer everyone's credit card and mortgage debt to the taxpayers as well? After all, isn't home ownership a social good?

As for transferring the tax burden from the elite to the working class - - - ? I guess it depends what newspaper you're reading. — Vera Mont

I read the ones that say Biden is cancelling some student debt. By definition, that excludes non-students, people who didn't go to college and didn't take out student loans. So the non-students pay (via taxes and inflation due to the additional borrowing required to pay off the banks) the legally contracted debt of the students.

What do your newspapers say?

President Biden’s tax cuts cut child poverty in half in 2021 and are saving millions of people an average of about $800 per year in health insurance premiums today. Going forward, in addition to honoring his pledge not to raise taxes on anyone earning less than $400,000 annually, President Biden’s tax plan would cut taxes for middle- and low-income Americans — Vera Mont

Is this a campaign ad?

Another Google quickie revealed that Biden's inflation has cost the average family $8,508 relative to before Biden took office. We could play this game all day. What do Biden's tax cuts have to do with his illegal student loan forgiveness?

You keep defending that one deluded man, and don't care how his co-workers struggle to give their children a chance in a fucked-up capitalist society. — Vera Mont

Sorry, what? What one deluded man am I defending? Whose co-workers? Fuck capitalism, down with the man, eat the rich, up the revolution!! Can you try to focus on the conversation?

I saw a pretty funny sign last night:
"Did anyone think to unplug America and plug it in again?"
The system's been cracking for a long time; all anyone can do, short of smashing it and starting over, is apply patches here and there. — Vera Mont

I'll grant you that Marx's predictions about late-stage capitalism seem to be coming true. We don't actually have much capitalism anymore, we have an oligarchy causing unsustainable inequality leading to a revolution or a cyber totalitarian nightmare. The system's broken. In fact the economy is only being held up by government borrowing and printing at this point. You and I may be in agreement on some things.

Well between the two of you I have no idea what a supertask is anymore.
— fishfry

A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time." — Michael

Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition?

What I think about supertasks is:

* Either they are already possible in the sense of Zeno, when I walk across the room; or

* They are physically impossible in currently known physics (because of Planck) but may be possible in future physics, by analogy with previous scientific revolutions; and

* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction.

I agree with you. — Ludwig V

I love when people agree with me. It happens so seldom around here :-)

It suits my approach well, in that the existence of the problem is a result of the way it is defined, or not defined. — Ludwig V

I agree with you too!

The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous.
— fishfry
That's the way ω is defined, isn't it? Although I'm not sure what you mean by "continuous" there.
I still feel uncomfortable, because it does get to the bottom of the stairs by placing a foot on each of the stairs, in sequence. But that's exactly the hypnotism of the way the problem is defined. And if an infinite physical staircase is the scenario, then anything goes.. — Ludwig V

Let me see if I can clarify my point.

In the lamp problem, we have the sequence 0, 1, 0, 1, 0, 1, ... We can "complete" the sequence by defining the state at $\omega$ as 0, or 1, or a plate of spaghetti. In no conceivable completion can the sequence be made continuous, because 0, 1, 0, 1, ... simply does not have a limit.

But in the staircase problem, if 1 is "walker is on the step" and 0 otherwise, then we have the sequence 1, 1, 1, 1, ... which has the limit 1. So 1, the walker is on the step, is the natural state at the end of the sequence.

Does that make sense? The staircase has a natural answer; but the lamp has no natural answer. Any completion whatsoever is as equally bad as any other.

Yes. But I have an obstinate feeling that that fact is a reductio of the process that generated it. So I'm not questioning what you say, but rather what we make of it. — Ludwig V

Right. So why is a lamp circuit that can switch states in arbitrarily small slices of time reasonable, and spaghetti isn't? That's one of the cognitive traps of the lamp problem. IMO the final state is simply not defined by the premises of the problem, AND there is no solution that makes the sequence continuous, therefore spaghetti is as sensible as anything else. And I've convinced myself that this is the solution to the problem.

It may be a bad habit to think of applications of a mathematical process. But that's what's going on with the infinite staircase. So it might be relevant to that.
3 minutes ago — Ludwig V

The staircase is different from the lamp. The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous. So it's to be preferred over all other possible solutions.

I understand. It's probably best not to comment any further. — Ludwig V

Thanks, I hope that didn't come out too ... however it came out. J6 is a sore point on both sides of the issue. If I said anything at all I'd be inviting discussion so I'll just refrain. Anyway some of the discussion about education policy was outside my area of expertise and interest, so I haven't got much else to say here.

Ok, I think that I finally have learned my lesson now. I will never try to defeat formalism again. Seriously, this was my last attempt. — Tarskian

So glad I could provide some insight. That was an interesting question. Formalism is saved!

I certainly do not believe that mathematics revolves around the correspondence with the physical universe. By "correspondentist", I actually mean: correspondence with a particular designated preexisting abstract Platonic world, such as the natural numbers. — Tarskian

The natural numbers have no physical instantiation as far as we know. Their existence is only abstract, fictional if you will. Or, to a formalist, purely symbolic.

It is beyond question that the axiom of infinity gives us a model of the Peano axioms, but both structures are equally fictional or equally formal.

Mathematical realism is about the independent existence of such Platonic universes. — Tarskian

Ok ... but ZF is a symbolic system. It doesn't talk about things in the real world, only sets, whose existence is entirely formal or fictional or abstract.

If these Platonic universes do not even exist, why try to investigate the correspondence with a particular theory? It only makes sense if they do exist, independent of mathematics or any other theory. — Tarskian

Maybe that's a good question but I'm not sure. ZF exists independently of PA, but both are symbolic axiomatic systems.

Model theory truly believes that the natural numbers exist independently from mathematics or any of its theories. — Tarskian

I'm not qualified to agree or disagree, but it sounds suspect to me. I don't think that the model theorists every say to the set theorists, "I bet you didn't know that sets are as real as cheeseburgers." Nobody ever says that or believes it. The structures studied in model theory generally live in set theory. And there's nothing real about set theory except by virtue of our imagination and symbolic processing.

ps -- Ok I have a sharper response.

The Wiki article on model theory says:

"In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold)"

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

But then when you click on structure, it says"

"In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it."

https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

So model theory studies the structures that satisfy some axioms; buy the structures themselves are nothing more than formal systems. A set along with a collection of operations and relations.

I think that resolves your concern. One can study a set along with some operations and relations defined on it, without believing such a set is real or has concrete existence or whatever way you are expressing your concern.

In short, the structures can be taken to be every bit as syntactic as the axioms that the structures are models of.

Ok. Perhaps you and Michael could hash this out. He thinks supertasks are metaphysically impossible
— fishfry
Perhaps he does, but he fallaciously keeps submitting cases that need a final step in order to demonstrate the contradiction. I don't. — noAxioms

Well between the two of you I have no idea what a supertask is anymore.

I say they're conditionally physically possible, but the condition is unreasonable. There seems to be a finite number of steps involved for Achilles, and that makes the physical case not a supertask. I cannot prove this. It's an opinion. — noAxioms

I tend to agree with you, that supertasks either (a) may be physically possible via the physics of the future; or (b) are already possible when I go from the living room to the kitchen for a snack, first traversing half the distance, then half of the remaining half, and so forth, and somehow miraculously arriving at my refrigerator. Which keeps things cold in a warm room, in clear violation of the second law of thermodynamics. Truly we live in remarkable times.

Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set.
— Ludwig V
I have no problem with any that.

You can think of 0 as the infinitieth item, but not the infinitieth step.
OK, that's probably a problem. It is treating something that isn't a number as a number. It would suggest a prior element numbered ∞-1. — noAxioms

You believe in limits, you said so. And if you believe even in the very basics of set theory, in the principle that I can always union two sets, then I can adjoin 1 to {1/2, 1/3, 1/4, 1/5, ...} to create the set {1/2, 1/3, 1/4, 1/5, ..., 1}.

It's such a commonplace example, yet you claim to not believe it? Or what is your objection, exactly? It's an infinite sequence. I stuck the number 1 on the end. The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set? It's a perfect description of what's going on. And it's a revealing and insightful way to conceptualize the final state of a supertask. Which is why I'm mentioning it so often in this thread.

Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length.
But you can traverse the space of that step, even when well below the Planck length. — noAxioms

Only mathematically, In terms of known physics as of this writing, we can not sensibly discuss what might be going on below the Planck length. Maybe space is continuous. Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes. Maybe something entirely different and not yet imagined is going on. We just don't know.

But you can't say "you can traverse the space of that step, even when well below the Planck length" because there is no evidence, no theory of physics that supports that claim.

In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation. — noAxioms

Well yes, motion is possible. That's one response to Zeno. Not so satisfactory though. Did I complete a supertask when I got up to go to the kitchen for a snack? I have no idea, even though motion through space within an interval of time is an every day occurrence.

The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.
So it does. Zeno's supertask is not a closed interval, but I agree that closed intervals have first and last points. — noAxioms

Ok. I thought you were claiming supertasks had to related to open intervals.

I said that Congress should pass a law funding college costs if that's what they want.
— fishfry
I think you said quite a lot more than that. — Vera Mont

Such as ...?

I'm not aware that the elite had been paying for student loans. Citation? — Vera Mont

Citation? Jeez I don't have to read you the daily newspapers, do I? The college students having their loans "forgiven" aka transferred to the working class that you apparently don't like very much, will out-earn the working class by millions of dollars over their lives.

I am really surprised to see a self-described communist want to burden the working class with the student debt of people who will vastly out-earn them. I wonder if you could address this point.

Did we discuss restructuring taxation at all? I have some views on capital gains, shell corporations, off-shore accounts and price-gauging that wouldn't affect most union members. — Vera Mont

No, I'm trying to keep it simple. Biden's illegal and quite regressive transfer of student debt from students to blue collar workers.

Trashing the welder.
— fishfry
Just that one. He probably beats his wife and votes for T***p, too. — Vera Mont

With commies like you trying to saddle him with billions in debt, it's no surprise. You just explained Trump's popularity. The left's abandonment of the working class has a lot to do with it.

I don't think you've done anything at all. — Vera Mont

LOL. Ok I guess we're done. Nice chatting with you. Hope you'll give some private thought to why you are defending the transfer of debt to the working class, whom the communists are supposed to have an affinity for. But that was 50 years ago, wasn't it. Now the left loves the deep state, loves the intel agencies, loves the wars, and hates the working class.

That's why the welder loves Trump. Because the Democratic party and apparently even the communists stopped caring long ago.

I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite. — keystone

So you believe in the rational numbers? But then the reals are easily constructed from the rationals as Dedekind cuts or equivalence classes of Cauchy sequences. If you believe in the rationals you have to believe in the reals.

I thought I twice answered your question. Let me try again. What you don't seem to appreciate is that with the top-down view we begin with the journey already complete so halving the journey is no problem. If we already got to 1, then getting to 0.5 is no problem. You can't seem to get your mind out of the bottom-up view where we construct the journey from points, which indeed requires limits. — keystone

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?

You are the one who started at 0, remember?

I don't believe I've said anything to lead you to believe I'm against education.
— fishfry
Only for people who can't afford it. — Vera Mont

But I did not say that. I said that Congress should pass a law funding college costs if that's what they want. Biden's action is illegal. And as a self-described communist, I'm surprised to see you cheering on the transfer of billions of dollars in debt from the elite to the working class. You sure you're a commie? Or are all the commies elitists these days? That's what it seems like.

You said the welders militarized the police.
— fishfry
No i didn't. I said
That welder who'd rather see his taxes go toward militarizing the police is doing his family no favours. — Vera Mont

"There you go again," as Reagan once said to Jimmy Carter. Trashing the welder. You don't think much of the working class? You sure you're a commie? I mean you say you are, but your words say otherwise.

Don't tell me there isn't one single yahoo in the welder's union who wouldn't rather beef up the police than give some pansy a degree in social work. There is. And he's an idiot. — Vera Mont

Yahoos. So either you're a commie with disdain for the working class, or else communism is now a faddish pastime of the elite. Which is exactly what it is these days, at least in the US.

No, I'm anti representing all working class people as thinking like you. — Vera Mont

I couldn't actually parse that except that I must have done something bad.

I take it you're not a fan of analogies. — keystone

I like analogies fine. I don't understand any of yours. I thought we were making progress on at least having the same conversation when we were traversing the unit interval. Instead of engaging you're changing the subject.

0 and 0.5 have distinct positions on the Stern-Brocot tree. — keystone

You're taking that as fundamental?

I like football but these picture posts aren't doing much for me. We were at least having the same conversation about getting from 0 to 1 on the real line. Then you said you don't believe in the real numbers, and then you declined to respond when I asked you twice how you get from (0, .5) to .5 without invoking a limiting process. And now you're changing the subject.

Model theory makes anti-realist views unsustainable. — Tarskian

I don't see how that is. Take as an example the Peano axioms for the natural numbers. Do we have a model of them? Yes, namely $\mathbb N$, the smallest inductive set guaranteed by the axiom of infinity in ZF set theory.

But the latter is just as fictional as the former, is it not? There's no empirical proof of the existence of infinite sets. They're a mathematical abstraction.

It seems to me that models are often purely abstract mathematical entities. One can take a purely formalist view of ZF for example. There are no sets in the real world in the sense of set theory. Show me the set containing the empty set and the set containing the empty set, which is better known by its more familiar name, 2.