Here's what we have:

A putative description of an imaginary world (which is not a physical world).

The description is not coherent, since it posits that there is a last state for a process that does not have a last state.

Since the description is not coherent it does not specify even an imaginary world. Perforce, not a world that is a model of set theory.

Set theory is consistent*.

Set theory does provide a mathematical version of infinitely many steps. But not with a last step that is the successor to the previous step.

It's not the fault of set theory that it doesn't have a version of an impossible world. Indeed, it is a virtue of set theory that it doesn't have a version of an impossible word.

It is a fail to claim that Thomson's lamp impugns set theory. Indeed, if Thomson's lamp imgugns anything, it's the supertask that is described. Just as set theory does not assert that there exists such a supertask.

* Presumably it is consistent, (1) since no inconsistency has been found, and (2) by arguments regarding its hierarchical nature (see Boolos).

Ha! I hadn't read this until just now:

https://plato.stanford.edu/entries/spacetime-supertasks

"Benacerraf (1962) pointed out [that the] description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to “complete” the description of Thomson’s lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency."

And that is what my own writeup says too.

I just now recognized that it's 'Thomson' not 'Thompson'.

Okay, I see. I forgot the details of the Thompson's Lamp paradox. f(n) corresponds to the incremental time of light switching, not the incremental distance Achilles travelled. To me that is a moot point, — keystone

It's an important point, since it and the alternating states are what makes Thomson's lamp a different problem from Zeno's paradox.

Do you agree that your formal definition describes the informal notion that there exists a complete table (having no last term) as described below?

Step #[n], incremental time [f], current state of lamp [g]
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc. — keystone

No last entry. But keeping in mind that 'time' and 'current state' and 'lamp' are not in the mathematics itself.

Also, if you look at the Wikipedia page (https://en.wikipedia.org/wiki/Thomson%27s_lamp) you will see a table which is more closely aligned with the paradox:

Step #, cumulative time, current state of lamp
0, 1, on
1, 1+1/2, off
2, 1+1/2+1/4, on
3, 1+1/2+1/4+1/8, off
etc.

Do you think that the incremental time table and the cumulative time table convey the same information, just in a different format? — keystone

We can add whatever math you want to my writeup. Define:

s(0) = 1
s(n+1) = s(n)+(s(n)/2)

j(n) = <n s(n) g(n)>

And still my point about the writeup stands. We have an infinite sequence. There is no last entry in that sequence. (And we can also throw in the infinite series too, though it doesn't change the point). There is no contradiction there. Thomson's lamp is not a description of physical events. And it's not even model abstract set theory. Thomson's lamp does not show that set theory is inconsistent nor that set theory fails to provide mathematics for the sciences.

ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty".
— TonesInDeepFreeze

In ZFC, is the equation 1+1=2 a definition, a theorem, or something else? My understanding is that if ZFC were inconsistent then one could prove both that the natural numbers are distinct and that they are equal. — keystone

You tendered the notion that infinite sets are empty. I said that's a contradiction (more exactly, it's inconsistent). Then you replied that if set theory were inconsistent then set theory has that infinite sets are empty. And above you quoted me yourself instructing you that if set theory is inconsistent then still "infinite sets are empty" is inconsistent. (!!!)

So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on.
— TonesInDeepFreeze

I dispute a continuum composed of points. — keystone

Non responsive. You say there is no continuum, but in the imaginary world you describe, you have a ruler that you say is the continuum. Have cake or eat it. Choose one.

I take it that you consider the points to equal the continuum. — keystone

No, I never said that. I posted explicitly (in this thread or another that I think you were in) what the continuum is. The continuum is:

<R L> = {x | x in R} where L = the standard less than relation on R

In other context, it's okay to say that the continuum is:

<P M> where P = {<x 0> | x in R> and M = {<t s> | t in P & s in P & Exy(t = <x 0> & s =<y 0> & Lxy)}

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them.
— keystone

Wrong. Look up the math sometime.
— TonesInDeepFreeze

I know the standard construction, starting with natural numbers then integers then rationals then reals, etc. And often we say that the naturals are defined as nested sets of sets. I am disturbed by this approach but I know in another thread you are already debating the definition of a set so let's leave it at that — keystone

No, let's not leave it at that.

(1) I did not debate the definition of 'set'.

(2) That you are "disturbed" doesn't change the fact that in set theory, distinctness of natural numbers doesn't require consideration of a continuum. You are just plain flat out wrong.

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).
— TonesInDeepFreeze

Then maybe ZFC is inadequate for analysis. — keystone

That is an idiotic non sequitur.

Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness.
— TonesInDeepFreeze

I see points as emergent from distance — keystone

Goody, your undefined 'nothingness' is justified by your undefined 'emergent from distance'.

I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse.
— TonesInDeepFreeze

Care to try? — keystone

At SAG-AFTRA rates. For that matter, I should already be charging you at least AFT rates for the instruction I'm giving you.

We both think the other is not listening or being reasonable. — keystone

We both think that, but you're wrong about it and I'm right about it. I have mulled over your remarks a pretty fair amount. I have turned them around in mind, including from what I understand to be your point of view, and I have responded on point to them as exhaustively as feasible for me. And I do have at least a little exposure to finitistic approaches and systems, and an interest in learning more about them.

You, on the other hand, just slide across what I say, not even taking a moment to understand, and instead getting it quite wrong, quite confused, and, at key points, essentially putting words in my mouth. And without a mote of intellectual curiosity to learn even the very first things about set theory.

And I don't insist on putting finitistic alternatives into the framework of set theory. But the core of your arguments, repeated over and over in same form or with variations, is to to insist on putting set theory into your finitistic frameworks.

I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse.

So nope, it's not a parity.

Of course, I'm not saying that the natural numbers are actually equal. I'm saying that natural numbers as defined in an inconsistent system can be easily proven to be equal. IF ZFC were inconsistent and IF someone proved that to be the case then wouldn't this be the celebrated conclusion? — keystone

You don't know what you're saying.

ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty".

I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist.
— TonesInDeepFreeze

It's funny how you criticize me if I don't respond to some of your points but then you criticize me if you don't respond to some of my points. — keystone

What in the world? I don't criticize you if I don't respond to some of your points.

Anyway, my point stands: Your circle bit is yet another variation on your theme. I've responded over and over to such variations, even if not to the circle in particular. I've done enough.

the wooden stick upon which tic marks are placed is the continuum. — keystone

So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on.

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them. — keystone

Wrong. Look up the math sometime.

Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly.
— keystone

What on earth are you talking about?
— TonesInDeepFreeze

I'm referring to the objects of set theory being beyond our grasp. — keystone

You are very kind to your unresponsive dogmatism to just now omit the key point in my reply:

The mathematicians don't just assume the theory will work. Rather, they prove that it does, by deriving the existence of the real number system, then proving the theorems of mathematics used by the sciences. — TonesInDeepFreeze

I've answered that and answered it and answered it already. The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it?
— TonesInDeepFreeze

I get what you're saying, but I don't agree with it. — TonesInDeepFreeze

You don't get to disagree with it. It's a plain fact, no matter anyone's philosophy.

I'll explain it yet again for you: Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).

You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see?
— TonesInDeepFreeze

Sometimes there's not enough room for two conflicting ideas. — TonesInDeepFreeze

Don't worry about two. There's not enough room in your mind for even one coherent idea.

Anyway, my point stands that it is poor thinking to always try to jam the sense of one framework into another one incompatible with the first.

Points are not "nothingness".
— TonesInDeepFreeze

It occupies zero space. — TonesInDeepFreeze

Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness.

You're adding things into what I wrote that are not there.
— TonesInDeepFreeze

I'm confused why you embedded a geometric series into the definition. — TonesInDeepFreeze

Another doozy by you. I said that you add things into what I wrote, and you reply by adding it again!

There is no geometric series in my writeup about Thompson's lamp. Period. No geometric series. Look at it again, hopefully this time not hallucinating, and you will see that there is no geometric series there.

Let's start with my main claim - Nothingness (i.e. points) cannot be assembled to form a something (i.e. a continuum), no matter how much of it you have. Even infinite nothingness is still nothingess. — keystone

Points are not "nothingness".

I honestly thought you were covering both Thompson's Lamp and Zeno's Paradox simultaneously since you included the geometric series in your description — keystone

I exactly stated it as regarding Thompson's lamp.

There is no geometric series in my writeup.

Step #, incremental distance, current state of lamp, dist. travelled
0, 1, on, 1
1, 1/2, off, 1+1/2
2, 1/4, on, 1+1/2+1/4
3, 1/8, off, 1+1/2+1/4+1/8
etc. — keystone

I said nothing whatsoever about "distance" or "travel". And I said nothing whatsoever about sums.

You're adding things into what I wrote that are not there.

I recommend that you read what I wrote without imposing your preconceptions about it.

I argue that any valid demonstration uses a parts-from-whole (points-from-continuum) construction. — keystone

Meanwhile I showed you math.

With the mathematics, we have that certain functions are continuous. This allows that mathematics is modelled by such things as Achilles crossing the finish line. And we know that Achilles does cross the finish line. So the mathematics works. The paradox though is a model in which Achilles does not finish the race. So we're not so attracted to that model, and indeed thankful that it is not a model of the mathematics.
— TonesInDeepFreeze

Aha! Your answer rests upon continua, not points! — keystone

Aha! You're making stuff up again!

There is no notion of continuity mentioned whatsoever in my writeup.

The mathematics works because you're considering the journey as a whole. The complete journey exists first and only then can we choose to talk about what happens at t=0.5 or t=0.95 or any other instant. But we cannot talk about the journey starting from t=0 proceeding to the adjacent instant.....because there is no adjacent instant. Atalanta cannot even begin her journey. — keystone

There is no "journey" mentioned in my writeup. The writeup has nothing to do with "journeys".

I suggest that you clear your mind for just one moment and read my writeup free of incorrect preconceptions about it.

You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see?

You complained that I skipped some of your arguments — keystone

Where 'some' includes the most important ones.

in set theory, infinite sets differ in this salient way from finite sets.
— TonesInDeepFreeze

The way I would phrase it is that we already know that in set theory, infinite sets don't conform at all to the intuitions we've developed from all sets that we've actually worked with direction. — keystone

"We". There are mathematicians and philosophers for whom set theory is intuitive. You don't get to declare for all "we".

There's no law of the universe that one can't just chug along proving theorems without declaring a philosophical position.
— TonesInDeepFreeze

[...] philosophy of mathematics we can refine our intuitions and apply them in our quest to understand our universe. — keystone

Of course. And I said that mathmematicians and philosophers may choose among many philosophies. You left that out. Probably you didn't even take in that I said it.

paradoxes are the most important guidepost in our quest to see truth. — keystone

I don't know that they are the most important subject, but they have been at the very heart of philosophy of mathematics. It's instructive then how mathematical thoeries are presented to avoid contradiction.

so you get a contradiction but outside the original terms you set: an imaginary world corresponding to set theory.
— TonesInDeepFreeze

I believe all worlds (physical and abstract) are simulated by computers. Not necessarily a computer made of silicon, just an entity that computes. One might say that 'God is a mathematician'. I'm not trying to squeeze set theory into a specific world, I'm just trying to imagine a computer with enough capacity to hold it. If you say that set theory is a finite set of rules, axioms, properties, etc. from which theorems could be derived then Set Theory could exist on a finite computer and all is good.

However, if you say that set theory describes the behavior and asserts the existence of another computer which directly works with the sets themselves then such a computer would necessarily be infinite in capacity (and infinite in speed as well to complete infinite operations in finite time). I'm willing to begin by assuming that such a computer exists, and then explore whether it would explode. I do not have the skill to prove that the computer that holds set theory will explode. However, I think I have the skill to at least discuss whether the computer that holds the infinity paradoxes will explode. — keystone

I have been pointing out that your main arguments are to try to make set theory fit models that are not models of set theory. And your response to that? Another writeup in which you view set theory per a model that is not a model of set theory!

You require that sets can be "built" only in finite "processes".
— TonesInDeepFreeze

I'm willing to explore computers that perform supertasks, like one that simulates Zeno's Paradox where he performs infinite steps in finite time. And IF that computer explodes and IF I can form a 1-to-1 correspondence between the processes on that computer and the processes on the computer that holds the infinite sets then it's reasonable to say that that computer also explodes. If you think I haven't proven this yet...well that is true! — keystone

I'm glad it's not my job to reconstruct verbiage like that so that it makes sense. In the meantime, why don't you learn something about computability theory?

In Set Theory we say 'There exists a set...'. What do we mean by this?
— keystone

When I say something exists I mean there is a computer where it is in memory. [...] I just can't envision any computer holding even just the natural numbers without exploding.

So when you write "For a given model M, ExPx is true in M iff there is at least one member of the domain such that that member of the domain is also in the subset of the domain that is the denotation of 'P'", are you assuming the existence of an infinite computer? — keystone

Wow. You just cannot help yourself from continuing to ask me to make set theory fit your Prorcustean beds.

And you took not a millisecond to think about my own answer.

Some of your quote links are not going to the posts in which the quotes occur.

'Infinite sets are empty' is a contradiction....
— TonesInDeepFreeze

How is that a contradiction?
— keystone

I'm sorry, but are you serious?
— TonesInDeepFreeze

If I write N = {1, 2, 3, ...} it seems that N has infinite elements. But appearances can be decieving. If someone proved that 1=2=3=... then N actually only contains one element. — keystone

You answered the question. You're not serious.

Here's another answer: Because if it had only finitely many points, we wouldn't be able talk about points greater than any of the positive points you chose or less than any of the negative points you chose. And we wouldn't be able to talk about those that are in between any two adjacent points.
— TonesInDeepFreeze — keystone

I'm not proposing that the real line is composed of an unchanging, finite set of points. I'm proposing that the real line has infinite potential to 'give birth' to points as they are needed. — keystone

You asked me about finitely many points, not about potentially infinitely many points. Be clear.

You never did respond to my post where I drew a circle for you. That post depicts what I mean, but in any case, I think the following paragraph will as well. — keystone

I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist.

If you're a teacher and you tell the students to measure a table, you wouldn't hand them a bunch of nothing (points) to make the measurements. You would hand them a ruler (continuum) which has perhaps tic marks every 1/8 inch. If the table lines up halfway in between the tic marks, then and only then do you add a new tic mark based on the average of the adjacent tic marks. And if matter weren't discrete, you would have the potential to endlessly add tic marks to the ruler. It's just that you never would complete the job. — keystone

You didn't have to waste your time typing that. I know your notion of potential infinity.

And you egregiously obfuscate the terminology. 1/8 increments is not a continuum. You could at least give the consideration of not appropriating terminology in a blatantly incorrect way.

And in fact, if you somehow were able to completely populate the ruler with tic marks the ruler becomes useless. It's just one big black tic mark. In this example, The distinction between numbers is lost and the set of real numbers is no more useful than a set containing only one element. — keystone

Another of your arguments by analogy. Mathematics doesn't cover rulers with ink. The existence of the set of real numbers doesn't stop you from considering only a finite number of numbers for a given problem. The rest of the infinitely many numbers are not there waiting to spill themselves like ink all over your favorite finite set.

This is the problem: In a context such as this, it's fine to deploy analogies to illustrate intuitions and things like that. But the argumentative force is limited, at best. I shouldn't indulge you more.

Here's my airplane analogy. — keystone

Right. Since you can't be bothered to see the point of my analogy*, you skip it and just jump to your own analogy. Barely read the posts, not taking moments to reflect on what's been said, to ingest, so the posts are just jumping off points for you to say yet again how you think mathematics should be. You've got this down to an art, if not a science.

By the way, my analogy was not offered as an argument but merely lagniappe for you to (hopefully) grasp an idea that is not your own for a change.

Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly. — keystone

What on earth are you talking about? Sounds like a pitch for an opening scene of a movie or something. Who do you have in mind to direct? Spielberg? Soderbergh maybe?

The mathematicians don't just assume the theory will work. Rather, they prove that it does, by deriving the existence of the real number system, then proving the theorems of mathematics used by the sciences.

Again, we go around in circles, because you keep re-insisting on points that had long ago been answered. You are a sinkhole.

I'm acknowledging that they're good engineers and they've built a good plane but is that ritual really necessary? Is an actually infinity of points really necessary? — keystone

For heaven's sake! I've answered that and answered it and answered it already.

The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it?

But that does not preclude that one can devise a different system that yields the theorems for mathematics for the sciences, with different axioms, and if needed a different logic, in which we don't have the theorem that there exist infinite sets. Got it?

So if you really are interested in a mathematics without infinite sets, then go look at the other systems already!

I said that one my choose whatever axioms one wants. That does not imply that there are not many mathematicians who choose axioms on the basis of their truth. Among mathematicians and philosophers of mathematics, there are different notions of how axioms ought to be chosen.

Tarski in The Semantic Conception of Truth and the Foundations of Semantics uses denote for one or more items.

For example, he wrote:
1) The expression "the father of his country" designates (denotes) George Washington.
2) We have seen that this conception essentially consists in regarding the sentence "X is true" as equivalent to the sentence denoted by 'X' (where 'X' stands for a name of a sentence of the object language).
3) While the words "designates," "satisfies," and "defines" express relations (between certain expressions and the objects "referred to" by these expressions)
4) We should reconcile ourselves with the fact that we are confronted, not with one concept, but with several different concepts which are denoted by one word — RussellA

And that is misleading.

First, at least it is misleading formatting. The items are from very different parts of the paper, and not a list he made, as it looks from your formatting, of considerations about 'denote'.

Second, (4) is not about the meaning of the word 'denote' but rather it's about the meaning of 'true'. That 'true' has different conceptions.

(Thanks to bongo fury for catching all that and causing me to notice it too.)

However, Tarski does mention elsewhere that there are different senses of 'denote', but it's a highly technical matter he's addressing. Usually, he uses 'denotes' or 'names' in the very ordinary sense of the words.

It's ludicrous that you bring Umberto Eco into this. Tarski in 1931 and 1944 is concerned with mathematical logic, not literary criticism. He uses 'denotes' in a very ordinary sense, not anticipating what Umberto Eco might say decades later.

And to drive the point home about what Tarski means by 'denote', he gives the formulation of the method of models in which symbols are mapped to individuals, functions and relations. That is a formalization of the idea of symbols denoting those individuals, functions, relations.

It's the very simple idea:

'Chicago' maps to Chicago.

'Carl Sagan' maps to Carl Sagan.

'Cats' map to cats.

'maps to in an interpretation', 'denotes', 'names', 'refers to'. Different words doing the same job here.

If you want to understand Tarski and not be bogged down in misunderstandings, then you'd do well to start there, and to refrain from dripping goop all over by ridiculously dragging Umberto Eco into it.

My goal is to understand Tarksi's Semantic Theory of Truth — RussellA

That's good. You started out in this thread pretty confused. Hopefully some of the comments from posters have helped you.

not get bogged down in unimportant detail and misunderstandings — RussellA

.

The details I have mentioned are not unimportant.

What misunderstandings? Your own? Or posters misunderstanding one another? In either case, misunderstandings deserve to be remedied. I have misunderstood things in this thread. I corrected myself.

I wrote "Denotes infers points to, and "snow" is doing more than pointing to snow."
You wrote "That is not correct. The word 'denotes' doesn't infer. People infer; words don't infer."
Of course I am not suggesting that the word "denote" is doing the inferring. — RussellA

You seem not to know what you've written, even immediately upon quoting yourself: "Denotes infers [...]". That which infers is doing the inferring.

Of course the T-sentence "P" is true IFF P is not a detail. It is extremely important. I never said it was a detail. — RussellA

No, you just protested that you don't like all the detail. I don't know what you mean is included in that. I didn't claim what you think is the unneeded detail. But an important point that you kept not responding to is that the schema is not concerned with only analytic sentences. So I reiterated that point, stressing it's not a mere detail, whether you think it is or not.

I said that in my opinion "snow is white" is an analytic proposition. — RussellA

And now have skipped twice the very exact response I gave about that.

I never said that Tarski said that "P" has to be analytic. — RussellA

Granted. And to be exact, I did not say you had. Rather, you kept saying that you think 'snow is white is analytic' (while skipping my replies about that). I wouldn't know why you think it being analytic is important, unless perhaps you think that Tarski chose it for the reason that it is analytic. So, without claiming that you had said that the schema disallows synthetic instances, I stressed that indeed it does not.

To recap:

'Snow is white' is analytic or not depending on the definition of 'snow'. With a definition of 'snow' that includes 'white' then I'd say 'snow is white' is analytic. But I showed you another definition in which 'white' is not included.

If the analyticity of 'snow is white' does not imply that Tarski chose it because it is analytic and thus that the schema is meant to convey that its only instances should be analytic, then what is the point of going on about the purported analyticity of 'snow is white'?

But now I see that you finally state your agreement that the instances of the schema don't have to be analytic.

So what's next? Most importantly, what's for lunch?

I really appreciate that. It means more to me than you might think. Thank you, jgill.

$ The worst crank on this forum claimed that I lack training in philosophy.

(1) I did not say I have not studied philosophy. In philosophy, I am not a scholar, and I have not retained many of the particulars I learned a long time ago, but I have taken courses in philosophy (and not just philosophy of mathematics) and read books. In the philosophy of mathematics, I am not a scholar, but I have read many books and articles. In mathematical logic and set theory, I'm not a scholar, but I have a good handle on the basics through taking courses and careful study of several textbooks, and I have also compiled a rigorous log of formalizations, definitions and proofs in set theory and the first stages of some other branches of mathematics. Meanwhile, I don't opine in threads on various philosophical discussions where the level of conversation would require me to be more adequately versed.

And whatever particulars I've forgotten about Western philosophy through history, I still retain much of the understanding and appreciation of philosophical methods themselves.

Philosophy of mathematics may be discussed at a general level that does not reference particular mathematical developments, but usually, as in this forum, discussions in the philosophy of mathematics do turn the mathematics itself. In that regard, it is crucial that that mathematics not be mistaken or misrepresented.

Indeed, threads in this forum are often premised as critiques of set theory and classical mathematics. Critiquing set theory and classical mathematics is great and vital. But the critiques need to be based on actually knowing what is being critiqued. A critique from ignorance, confusion, and misrepresentation is intellectual and philosophical garbage.

So what about the crank himself? There is not a hint in what he writes that he knows even the least thing about such basic contexts of philosophy of mathematics as Frege, logicism, Hilbert's formalism, Godel's 'What Is The Continuum Hypothesis?' constructivism, intuitionism, reverse mathematics or anything else really. The crank himself knows not a thing about the first order predicate calculus, second order logic, formal axiomatics, set theory or, as far as I can tell, any mathematics beyond everyday arithmetic. But that doesn't stop the crank from claiming it's all nonsense. Indeed, the crank insults mathematics itself and mathematicians themselves.

Of course, the crank does not fault himself for lack of training in philosophy of mathematics, though he spews his ignorance, illogic and literal nonsense about it regularly, for years.

And what would the crank say about other posters who we can tell are sorely "lacking in training" in the philosophy of mathematics? What is the training in philosophy of mathematics of the original poster of this thread? Let alone what are his philosophical commitments? Does the crank (and another poster) hold that everyone must have a philosophical commitment to qualify for posting? I don't believe posters must.

* The original poster started with a presumably empirical question, then went on top claim or insinuate that infinitistic set theory is wrong, and it is still not clear whether the original poster has clear position as to whether he's an ultrafinitist, even that he is a particular kind of finitist.

The crank pounced on my humble statement that I don't hold to a particular philosophy, saying that I "lack training" in philosophy. The crank shows his pettiness, illogic, and factual incorrectness. And lies about me.

(2) It is not necessarily a fault not to have a philosophy. One can be open minded about many philosophies and employ their virtues.

/

$ The crank says it is annoying for a philosopher to have to see a non-philosopher post in a philosophy of mathematics discussion and insist that philosophers shouldn't discuss the metaphysical bases on which mathematical axioms stand, if they have not first studied mathematics.

(3) The crank shouldn't presume to speak for others.

(4) The crank is lying about me. I have never said that one must study mathematics to talk philosophically about mathematics. What I have said is that one should not spew disinformation and confusion about the mathematics without knowing anything about it. The posts by cranks are not just philosophy but they include claims about the mathematics that is the subject of the philosophy. I don't even say that one shouldn't talk about that mathematics without first studying it; rather that one should not misrepresent it, and that the first step to not misrepresenting it is to learn at least a little bit about it.

$ The crank says that it is philosophy that is being discussed not mathematics.

(5) The cranks lies again. The profuse record of actual posts show that the cranks make many claims about the mathematics and that the discussions reference the mathematics. This very thread is one of them.

The crank himself, at nearly the start of this thread wrote: "Try naming pi to its final decimal place."

The crank lies, directly belying his own posting, when he says the discussions are about philosophy but not also the mathematics itself. How can that be topped?

Does Set Theory model Achilles' journey or not? — keystone

You have it backwards, and show that you didn't bother to read my previous post about modelling.

The realm of the paradox is not a model of the mathematics. Formally, the mathematics is not a model; it's a theory. Models are of theories, not the other way around.

With the mathematics, we have that certain functions are continuous. This allows that mathematics is modelled by such things as Achilles crossing the finish line. And we know that Achilles does cross the finish line. So the mathematics works. The paradox though is a model in which Achilles does not finish the race. So we're not so attracted to that model, and indeed thankful that it is not a model of the mathematics.

Can you see that? Can you be reasonable enough to see that that is reasonable.

If mathematics does not allow for such nonsense then it cannot claim to resolve Zeno's paradox. — keystone

No, because mathematics shows how to calculate that Achilles did finish the race.

You just keep showing that you don't care about understanding any of this but instead want to continue roving among already answered points as if they weren't already answered.

You asked me about Thompson's lamp. I replied about it, now with rigor.

If you then just jump to another subject (and you're incorrect about it also) then I take it you are not in good faith.

Thompson's lamp.

It's a non-converging sequence.

Set theory doesn't have a "final state" with that.

But here's what set theory does have:

Let N = the set of natural numbers.

Let f be a function.

Let dom(f) = N

Let for all n in dom(f), f(n) = 1/(2^n)

So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...

0 is not in ran(f).

Let g be a function.

Let dom(g) = ran(f)

Let ran(g) = {"off", "on"}

Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)

So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ...

So I've mathematically "translated" Thompson's lamp.

What about the "final state"? There is no final state. The mathematics doesn't have a "final state" here, exactly because the mathematics doesn't have nonsense like "an infinite process with a final state". Of course you get a paradox from having an infinite process with a final state. It is the very point that mathematics is not capable of such nonsense.

But this:

Let (h) = g u {<0 "off">}

So the "final state" of h is "off". No contradiction.

Let j = g u {<0 "on">}

So the "final state" for j is "on". No contradiction.

Choose whichever "final state" you like - the "final state" with h or the "final state" with j. But neither is determined by g. — TonesInDeepFreeze

Basically, you have your concepts of how the statements of set theory should be understood, so you posit a realm to embody those concepts. But your concepts are not analogues of any actual statements in set theory. So it's a cheat.
— TonesInDeepFreeze

I don't think you're being reasonable here. — keystone

You're not being reasonable. Do you want to inform and enlighten yourself about the subject, or you just want to raise objections premised in not knowing anything about it? Better to know the thing, then you could critique it fairly.

You say that many of the paradoxes are resolved by Set Theory — keystone

I said that, and it's okay. But in the last posts, I realized that the more pertinent point is the one I'm making now.

essentially it is you who is making the link between the two — keystone

What are you talking about? You are mentioning the paradoxes to impugn set theory. Then I reply: (1) The paradoxes don't show a contradiction in set theory, (2) the set theoretical, mathematical methods don't end in contradiction, so, in that way they "solve" the paradoxes; not that the paradoxes dissolve on their own, but rather when we do actual mathematics instead, we don't incur contradiction, (3) the realms in the paradoxes are not models of set theory anyway.

Indeed, if no one mentioned a connection between the paradoxes and set theory, I'd have no objection at all!

but then you say that criticisms of the paradox can't touch Set Theory. You can't have it both ways. — keystone

You are completely missing the point. And now your lily pad jumping from Thompson's lamp to Zeno.

I gave you a rigorous answer regarding Thompson's lamp, but instead of comprehending that, you skip it as if it doesn't exist and frog hop to another pad.

You're not in good faith.

Zeno's Paradox — keystone

I answered about Zeno's paradox many posts ago (in this thread or another one).

How about having some attention span and look at my rigorous response to Thompson's lamp?

I'm referring to the fact that set theory proves there exists a complete ordered field and a total ordering of its carrier set. And then I highlighted that the carrier set is the set of real number and the total ordering is the standard less-than relation on the set of real numbers.
— TonesInDeepFreeze

You've repeated this a lot so it's clearly important. — keystone

I think I've said it about two or three times. It is important because it is the very heart of the matter, which is that set theory axiomatizes the mathematics of the real numbers. But, for some reason, I don't see the passage in the post you linked to. If you look at where I said it, then you'll see how it was a response to a comment or question of yours.

'Infinite sets are empty' is a contradiction. And set theory does not have that contradiction. So if it's your intuition, then set theory is not for you.
— TonesInDeepFreeze

How is that a contradiction? — keystone

I'm sorry, but are you serious?

where we need to assume that the real line is composed of infinite points — keystone

[bold original]

I've answered that before. (In this thread or another one.)

Here's another answer: Because if it had only finitely many points, we wouldn't be able talk about points greater than any of the positive points you chose or less than any of the negative points you chose. And we wouldn't be able to talk about those that are in between any two adjacent points.

If you're the teacher and you tell the students there are only these certain points, then the student comes to class the next day and says, "I wanted to use mathematics to figure out how to build a table, but my mom she needs an overhang between 1/8 inch and 1/16 inch, but there is no number between 1/8 and 1/16 among the numbers you'll allow me to use; it's not in your list of all the points."

Infinitely many points are needed so that we can speak in greatest generality, no matter what degree of tolerance you might think is the greatest degree needed.

The infinitude of the set of real numbers is a consequence of the definition of 'real number'. Set theory axiomatizes that courtesy of the axiom of infinity. If we took out the axiom of infinity, then we wouldn't have the set theoretical construction of the complete ordered field that is the system of the reals.

Think of it this way: I show you an airplane.

You say, "Why do you need that big plate at the bottom; couldn't you make an airplane without that kind of thing?"

I say, "I don't know whether you can, but it is needed for this airplane. And it's not that just that it has a specific purpose, but that if you took it away, then all the rest of the parts wouldn't work together; the whole thing is an interdependence. If you want to build an airplane without that kind of plate, then go ahead and do it."

I reject your productivity and outcomes argument, whatever rubric it correctly falls under.
— TonesInDeepFreeze

[...] your actions fall short of moral excellence (in that indignancy fail prudence and justice) — Kuro

[...] in not acting rightly one retracts from the virtue of their character, which is less so a property of any particular fault or flaw in action rather than those habits which become second nature to them and come to form their decisions & methodology (which is what the discussion came about, the generalized implications versus the particular one, hence the relevance of the quote). — Kuro

If I understand, your view is that by doing things that are not right, one become habitual in doing things that are not right, thus harming one's character.

My point is that I don't agree that it is not right to decry the egregiousness of cranks.

Here you apply an outcomes/productivity argument:

I find myself as frequently frustrated as Mr. Tones with respect to the mathematical, logical or other formal/technical errors that are somewhat frequent on this forum. However, a rude attitude seldom yields anything productive: there is the option of politely leaving a discussion, perhaps at no fault of your own but the inadequacy of your interlocutor, or explaining their mistake at a reasonable level.

"Fanning the flame", so to speak, is unnecessary in whole. [...] — Kuro

So, if I understand, you base the claim that the act is not right, and thus harms character, on the claim that the act causes bad outcomes and is unproductive. Also, you think the act is wrong because it is acting from indignancy which is not prudent and just.

(1) I think that the act has a better outcome than not decrying the egregiousness of cranks, and that it is productive. Not productive toward having bonhomie with the crank, but productive in another way.

(2) I don't think it is wrong for posters to post for reasons other than certain outcomes or productivity, and I don't limit my own reasons that way.

(3) I don't view virtue ethics as determining, not even the main determinant. I see that virtue ethics, in my limited understanding, offers a lot, but I don't see that it should be the sole, or even main, framework to be used alongside a balance among other frameworks.

(4) I don't think it's incumbent to so seriously apply ethics, let alone pretty demanding ethical frameworks, to all aspects of posting. There is a lot of posting that I don't think has good outcomes or is productive that I wouldn't think of cudgeling with application of an ethical theory or even think of objecting to it at all. Posters don't ordinarily think "Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. People usually post at their pleasure, to express themselves, to advocate for their ideas*. And that expression may include voicing of indignation.

* Though there are other motives too, such as learning from other posters, enjoying pleasant agreements (rare, in this forum), etc.

(5) Perhaps I don't know your context, but I don't think that expressing indignation is necessarily imprudent or unjust.

One can propose whatever axioms one wants to propose. That doesn't entail that mathematicians propose axioms willy nilly or in an undisciplined way. Indeed mathematicians very carefully propose axioms on various criteria.

That one can do X, or is permitted to do X, doesn't entail that one does X.

All bijections are injections. So you're confused to begin with

The reason old-fashioned terminology is not so bad. One-to-one onto, etc. Bourbaki may be to blame? — jgill

That's a classic '50s movie, 'Blame It On Bourbaki'. Cary Grant and, if I recall, Anita Ekberg. Grant's character is trying to get Ekberg's character to take some experimental injections for her terminal illness, but she doesn't want the new experimental drug. Her famous line is, "No, sir, jections!"

I'll say it again (as this is certainly not a mere "detail"):

The schema says that for any sentence P, we have:

'P' is true iff P.

He does not say that 'P' has to be analytic.

Look it up. Anywhere.

You are quibbling over details and things I never said. — RussellA

They are critical details. And I also posted about matters that are not mere details. And you went right past the very critical substantive reply to you, as instead you restated your own contention again, ignoring that I had just clearly rebutted it. That is bad faith posting and bad faith to put the onus on your interlocuter in that instance.

And if you think I've misrepresented what you've said, or strawmamned you, then you can say specifically where.

The exact meaning of "denote" is debated, — RussellA

It's just the ordinary sense here.

And that is supported by the fact that it was Tarski himself who specified how symbols of the object language map with an interpretation mapping.

Look it up. Read up on Tarksi and model theory.

Starting with Tones whereby the denotation of 'snow' is: precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C). Remove the expression "formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)", as this describes how "snow" formed rather than what "snow" is. As "snow" is precipitation, can remove the expression precipitation. Therefore, can simplify the denotation of "snow" as small white ice crystals. — RussellA

When you just completely skip what I wrote about that, you are in bad faith as a discussant. Here it is:

You wrote - "snow" is precipitation ..............white...............
You didn't write "snow" is precipitation.........which may or may not be white........
— RussellA

That's a good point. I overlooked that I chose a definition that happened to include 'white' in the definiens of 'snow'. That was a mistake. I don't know whether Tarski even had a scientific definition of 'snow' in mind, and especially one that has 'white' in the definiens. So, I don't know whether Tarski thought of a particular definition so that he regarded 'snow is white' as analytic. I highly doubt that he did.

I should have chosen one such as this:

"precipitation in the form of ice crystals, mainly of intricately branched, hexagonal form and often agglomerated into snowflakes, formed directly from the freezing of the water vapor in the air"

The point of the Tarski schema is not to define 'is true' for just analytic sentences.

"grass: vegetation consisting of typically short plants with long, narrow leaves, growing wild or cultivated on lawns and pasture, and as a fodder crop"

So 'grass is green' is not analytic.

'grass is green' iff grass is green. — TonesInDeepFreeze

Thompson's lamp.

It's a non-converging sequence.

Set theory doesn't have a "final state" with that.

But here's what set theory does have:

Let N = the set of natural numbers.

Let f be a function.

Let dom(f) = N

Let for all n in dom(f), f(n) = 1/(2^n)

So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...

0 is not in ran(f).

Let g be a function.

Let dom(g) = ran(f)

Let ran(g) = {"off", "on"}

Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)

So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ...

So I've mathematically "translated" Thompson's lamp.

What about the "final state"? There is no final state. The mathematics doesn't have a "final state" here, exactly because the mathematics doesn't have nonsense like "an infinite process with a final state". Of course you get a paradox from having an infinite process with a final state. It is the very point that mathematics is not capable of such nonsense.

But this:

Let (h) = g u {<0 "off">}

So the "final state" of h is "off". No contradiction.

Let j = g u {<0 "on">}

So the "final state" for j is "on". No contradiction.

Choose whichever "final state" you like - the "final state" with h or the "final state" with j. But neither is determined by g.

NOTE: I am not claiming that there aren't mathematical treatments of supertasks that, with advanced definitions and construction, formalize notions of "infinite number of steps in finite time" or "final states for infinite processes". But if the treatments are formalized with ZFC, then it can't be the case that they contradict ZFC. I'm not expert in that matter, but I would put my proverbial money on it.

the mainstream view is that it exists in the Platonic Realm — keystone

I don't know whether platonism and/or variations on platonism are the majority view among those who have a view, but I wouldn't bet against it.

While [Thompson's lamp] cannot exist in our world, it should have no problems existing in the Platonic Realm which is infinite. — keystone

In that realm, there is no final state for the lamp. Poof. Done. Still no contradiction.

Your fallacy is in setting up an imaginary world, with states-of-affairs like set theory, but then adding a state that doesn't exist in the set theory and thus rightfully as an analogue not in the imaginary world, so you get a contradiction but outside the original terms you set: an imaginary world corresponding to set theory.

Basically, you have your concepts of how the statements of set theory should be understood, so you posit a realm to embody those concepts. But your concepts are not analogues of any actual statements in set theory. So it's a cheat.

You require that sets can be "built" only in finite "processes". Then, since it's all processes to you, when you see set theory posit objects not built with such processes, you incorrectly saddle set theory as claiming that certain states are achieved only after an infinite process. Your arguments about the realms then are based on conflating your concept with set theory.

By the way, are you capitalizing 'Platonic Realm' as a proper noun to shade discussion, even if just by hair, about mathematical realism?

I apologize if I'm not addressing the substance of your previous replies. It's not intentional, I thought I was. — keystone

If you reread the posts (and in the other thread) then you'll see exactly where.

In the Platonic Realm, can infinite objects exist but never be constructed? — keystone

Set theory has no theorems about mathematical agents constructing or not constructing things. So you can't plop them into the middle of one of these realms you're making and have it be about set theory. (For an approach to mathematics that does have something like agents constructing things, see intuitionism.)

My view of the Platonic Realm is that it's a world where infinite processes can be completed — keystone

Fine. It's not a realm of set theory. You don't know anything about set theory. But you keep burdening it with what you misunderstand it to be. You imagine a realm that has some similarities with set theory, but also has things not corresponding with set theory. Then you blame set theory.

Basically your realms are like models. But a model of a theory is one in which the vocabulary of the theory is interpreted in the model and every theorem of the theory is true in the model. But you're adding things that are not the interpretation of anything in set theory and even worse, posting states-of-affairs about them don't model the theorems of set theory. Sorry, no go.

In Set Theory we say 'There exists a set...'. What do we mean by this? — keystone

Let your '...' be "such that P".

For a given model M,

ExPx

is true in M iff there is at least one member of the domain such that that member of the domain is also in the subset of the domain that is the denotation of 'P'.

That means, in terms of your "realms" (whatever their ontological or metaphysical character), there is at least one object in that realm that has the "property" denoted by 'P'.

I currently hold an unorthodox view that set theory might not actually be about sets — keystone

Cool. The formulas can be about whatever you want them to be. Hilbert's tables, chairs and beer mugs.

we never actually manipulate the infinite sets directly. — keystone

Depends on what you mean by 'manipulate'.

Set theory, most strictly, is a certain set of formulas. In mathematical logic, when we say 'ZFC' we are referring exactly to the set of formulas.

Of course, those formulas are "read off", or "rendered" in, say, English as if they are English sentences. But they are not English sentences. Also, most people do have in mind that the formulas pretty much "say" what the English renderings are. However, again strictly speaking, the formal meanings are given by the method of models.

In that context, its hard to say what 'manipulate' means. For myself, I do recognize that I manipulate symbol strings. But I don't at all think that somehow, like a puppeteer, I'm manipulating mathematical objects. Rather, I am writing formulas as lines in proofs. I do intuitively think of those formulas as saying something about whatever objects are in the domain of discourse, but I am not manipulating those objects themselves. Rather, I am writing formulas about them.

But what about having not just a pre-philosophical intuition about what set theory "says" but a truly articulated philosophical position about it? Realist, nominalist, structuralist, fictionalist, consequentialist, instrumentalist...? That's for each mathematician to decide for herself or himself. Or not decide. There's no law of the universe that one can't just chug along proving theorems without declaring a philosophical position. Of course, then one can't provide a philosophical justification for set theory. But then, of course, one can say, "I don't need to provide a philosophical justification. The question of a philosophical justification is a fine one indeed, and I like peeking in on the debates now and then. But no matter how the arguments turn out, I'll still enjoy set theory and I'll correctly point out that it is a recursively axiomatized theory that proves the theorems used for the mathematics of the sciences, and such that, at least in principle, all the proofs can be written in complete formality so that their correctness can be algorithmically verified".

Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points? — keystone

Day one of high school Algebra 1:

"Students, we start with the set of real numbers and the real number line."

And you don't have to tell me yet again your finitistic interpretation of that or how we can work with only a finite number of those points, etc. I know that's your view.

I'm answering very quickly, because I'm out of time now.

if we've only proved that the reals are an ordered field, then is it possible that we haven't proved that sqrt(2) is a number? — keystone

We proved they are a complete ordered field.

I answered that in exact detail in a previous post. Pretty much, you're asking for notes from the first week of Calculus 1.

When I think of A being equinumerous to B, I think that there exists a bijection AND no injection between A and B.
When I think of A being more numerous than B, I think that there exists an injection from B to A AND none from A to B.
When I think of A being less numerous than B, I think that there exists an injections from A to B AND none from B to A. — keystone

All bijections are injections. So you're confused to begin with.

What you mean is that you take equinumerous to mean: A and B are equinumerous iff there is bijection between A and B and no injection from one into a proper subset of the other.

Of course, in ZFC that does hold for finite sets but not for infinite sets.

B is more numerous than A iff there is an injection from A into B and no bijection between them. Same in set theory.

B is less numerous than A iff A is more numerous than B. Same thing with set theory.

You're wasting our time. We already know that in set theory, infinite sets differ in this salient way from finite sets. Galileo's paradox, Hilbert's hotel, Dedekind infinitude, and this latest point about equinumerosity are all variations on the same point: infinite sets map 1-1 with proper subsets of themselves. You don't need to keep giving examples. We already agree that infinite sets map 1-1 with proper subsets of themselves. For you, it's counterintuitive and you won't accept it. Fine. I have no motivation to convince you otherwise.

my intuition based on finite sets leads me to believe that infinite sets are all empty. — keystone

'Infinite sets are empty' is a contradiction. And set theory does not have that contradiction. So if it's your intuition, then set theory is not for you.

I'm not saying that set theory is wrong, I'm just proposing that set theory might not be about actually infinite sets, but instead the potentially infinite algorithms that describe the infinite sets. — keystone

I know. You've said that a dozen times.

I just now watched this video by Wildberger:

https://www.youtube.com/watch?v=U75S_ZvnWNk

In that video, he's an intellectually disorganized, sneaky, weaselly lying sophist and a fool. He's an insult to intelligence.

First, the title of the video is "Modern "Set Theory" - is it a religious belief system?" Yet he mentions nothing about that, let alone supporting it. Indeed, it is unsupportable.

Religions involve some combination of (1) a god, gods, deities, angels, demons and other non-physical beings - usually with personalities and utterances attributed to them, and acting causally, and usually miraculously, on the physical world, (2) explanations of the universe and creation, (3) human souls and accounts of what happens to those souls after death, (4) mythic narratives about human history, including religious founders and prophets, (4) divinely mandated moral codes, (5) prayers, chants and incantations that have power upon the physical world, (6) teleologies and (7) eschatologies. And I confined just to aspects of belief, since Wildberger didn't say that set theory is also a religious practice.

So, with that provocative title, coyly cast as an interrogative, what the sneaky sophist does is wet the water slide for thinking that set theory is religious while he does nothing toward engaging the very meme he's disseminating.

/

In the opening he says he's going to "dismantle" "what we currently are doing now".

Surprise: he doesn't.

He says he's going to "examine the details of what we're currently doing". But instead of "details", all we get in the video are atrocious oversimplifications, and not a single one of those articulates an actual problem with set theory.

Then he says the video presents a fair case that mathematics can have a solid foundation. But it turns out his "fair case" is just to wave his magic wand (actually he uses an old drum stick) and say that the natural numbers are the foundation for all the other branches of mathematics. But he says not a single word showing how that would be done except for a chart with 'the natural numbers' as the base of the pyramid of mathematical subjects. As if doing that carries even a mote of an argument. I'm going to make a chart with avocados at the base of a pyramid of the hierarchy of the U.S. federal government. And, presto, the avocado on my table is now the president, head of state and supreme commander of the U.S. armed forces.

Then he states his gravamen: "Set theory is a logically inadequate foundation for mathematics". But in the entire video he does not cite a single actual problem with the logic of set theory. His only criticism is that it's axiomatic. Yeah, set theory tells you what its axioms are - so you can take them or reject them - and then shows you exactly, step by algorithmically checkable step, the derivations of the theorems. So what would Wildberger's own foundation be if not to present axioms? He gives not a clue, except that pyramid chart.

* He is creative with that drum stick. He uses it not just as a pointer, but he holds it up to illustrate the real number line. And at other times he uses it like a magic he waves to declare that he's shown some supposed problem with set theory, analysis, and mathematics. Like Trump magically declassifying documents.

/

He says about the natural numbers, "as numbers become big, arithmetical problems arise". Okay, what is one them? He mentions no problems. This is from the guy who had a few minutes ago promised to get detailed in the video.

/

He says the problem with the continuum is how do we model what happens when we look closely at it. Well, that's merely a vague question. What do you mean by "model"? What do you mean by "happens"? What do you mean by "look closely at it"? He then asks, "What is the mathematical way of magniying and subdividing and looking more clearly and carefully at smaller and smaller subdivisions?" Um, how about doing that with proofs of theorems from axioms?

He says, "This is hugely problematic". Yes, Norman, your vagueness and lack of the details you promised is kind of a problem."

He mentions the irrationals, with an actual smirk on his face. Argument by facial expression. (The video is full of those kinds of smug faces, while his words are saying, at best, nothing really, if not, at worst, outright lying.) He says we are faced with how to set up an arithmetic for irrational numbers, and "This is hugely problematic. It's hugely problematic." (Like when Trump* says, "It's a terrible thing. A terrible thing", as we're supposed to be convinced by the mere fact that he says it, and twice.) Wildberger simply skips that mathematics does, from axioms, define the operations on the real numbers with proofs about them.

He says that when mathematicians did lay out foundations for the reals, "[This] is really where the difficulties lie. It's really where the difficluties lie." - Norman "Double Assertion" Wildberger.

* No political comparison is intended between Wildberger and Trump. Just that they're both crackpots and liars.

/

Here's the first lie:

He says that ZFC "was not really a framework based on precise defintions and clear theorems".

ZFC is nothing but utterly rigorous definitions and utterly rigorous proofs of theorems. One may reject the axioms, but it is beyond dispute that the defintions are rigorous and the proofs rigorously derived from the axioms.

So that's Wildberger lie #1.

He says the problem is that set theory is axiomatic. Of course it is. Because it's with the axiomatic method that one states rigorous definitions and show rigorous proofs.

Wildberger is not just lying; he's also showing himself to be completely confused.

Then, "It was as if we had abandonded the effort to try to set up this foundational issue very carefully and clearly."

ZFC was exactly an effort to more carefully use set theory than Cantor did. Even if one rejects ZFC, it is beyond dispute that ZFC is a full formalization where Cantor's work was not.

So that's Wildberger lie #2.

He says the set theorists said, "Let's just assume that it works".

The set theory mathematicians didn't just assume that set theory provides a foundation for analysis, they proved that it does, by constructing a complete ordered field and the foundational theorems about it, from which analysis can take over from that point.

So that's Wildberger lie #3. The set theory mathematicians didn't just assume that set theory provides a foundation for analysis, they proved that it does, by constructing a complete ordered field and the starting theorems about it, from which analysis can take over from that point.

Then, "That's really what happened. 'Let's assume that the kind of things we want to be true really are true'.

That's really what did not happen.

So, reiteration of Wildberger lie #3.

Then, "[They say] 'We'll dress this up as if it's an axiomatic framework'".

Argument by characterization rather than substance. They didn't "dress it up" as an axiomatic framework. It is an axiomatic framework.

Then, "The job of framing this all was outsourced to logicians, which is really philosophy, a branch of philosophy that overlaps with mathematics".

He's insinuating that the set theory was not really rigorous because it was done by philosophers not mathematicians.

(1) Actually they were mostly mathematicians. Many of them were not just set theorists but worked in other fields of mathematics too. Some of the greatest names in mathematics were key in the development of the logic, set theory and type theory: Hilbert, Godel, von Neumann, Tarski, Whitehead, et. al (2) Set theory and mathematical logic are mathematics. (3) Many were all three - mathematicians, logicians and philosophers. But mathematical logic is astoundingly rigorous; it's all about rigor

Then, "Its not too much of an exaggeration to say mathematicians outsourced the foundations of their subject to the philosophers".

First "the philosophers", as if there are two distinct sides - the mathematicians and the philosophers. Second, yes many of the logicians were both mathematicians and philosophers. But again, Zermelo, Fraenkel, Bernays, Godel, et. al, where mathematicians no matter what else they were or weren't.

So it's not too much of an exaggeration to say that Wildberger is lying on this point too, and to call it Wildberger lie #4.

Then, "There's kind of an agreement not to examine closely what this logical foundational of set theory is".

You have got to be kidding me! Set theory and its logic have been examined, discussed, critiqued, reworked and reworked until the cows home. In literally library stack upon library stack upon library stacks of books and journals, examining set theory from every angle it can be examined from. I guess Wildberger doesn't have access to even a single library, not even at the university where he teaches.

So that's Wildberger lie #5.

Then he wiggles his hand as he calls set theory, "crazy things".

Argument not just by hand waving, but hand wiggling too!

/

That's just some of it, in just one nineteen minute video.

Just like you can't do very much math without relations. But with just one relation* you can do it all.

* membership

all that money earmarked for space ventures could be spent on more pressing matters — Agent Smith

What could possibly be more pressing than people getting their satellite dishes on the roof to watch reality TV shows of people eating bugs?

If it is consistent, it is not complete; if it is complete, it is inconsistent. — Agent Smith

If it is recursively axiomatized and has enough of arithmetic, then if it is consistent then it is incomplete (which is to say that if it is complete then it is inconsistent).

Many years ago I was watching the Academy Awards (don't ask me why I would waste my time that way). Whoopi Goldberg was at the microphone, and she said something like, "We don't need the space program. We should get rid of it. Nothing ever came out of the space program", as she was being broadcast via satellite. You gotta love it.

we could play around with the axioms to disallow irrationals, oui? — Agent Smith

Si.

I've told you about a million times already, you can have axioms for whatever you want*, even inconsistency if that's your thing.

* 'ceptin sometimes you can't have all of what you want at the same time, as we found out from Godel.

My favorites are cranks who say that mathematical logic, even just sentential logic, is all wrong, as they are typing and reading on computers that are packed to the panels with Boolean programming and would not even exist if not for the invention of modern digital computers in the very crucible of mathematical logic.

Communication is bad because 'communication' is from the same root as 'communism', which is bad because communism is Marxism, which is bad because Richard Marx sang lousy songs.