I was only offering a possible distinction between statements and propositions. — Cartesian trigger-puppets

Of course. It's important in the study of propositional logic that we understand that 'statement' is considered only in the sense of declarative statements.

There is variation though among logicians as to the meanings of 'proposition' and 'statement'. (But not so much extending to including interrogatives and such.) What is meant by 'proposition' and 'statement' may depend on the particular logician's or philosopher's framework.

Usually, we take 'sentence' to mean the syntactical object - the string of symbols.

Then it's a question whether we take 'proposition' as a synonym for 'sentence' or whether we take 'proposition' for what is expressed by a sentence.

Same for 'statement' - whether it just means a sentence or whether it means what is expressed by a sentence.

So, 'sentence', 'statement', 'proposition'. We just have to be careful what we mean in context.

/

For example, Church takes 'sentence' in the usual syntactical sense, but for him a proposition is a different abstract object. ('Introduction To Mathematical Logic, pg 26)

I said nothing to suggest that I was criticizing you. You mentioned that there are expressions other than declaratives, which is of course true. (And, one can devise systems of logic for interrogatives too.) But since the context has been propositional logic, and to answer any potential question whether propositional logic considers expressions other than declaratives, I noted that it does not.

I should have explained explicitly what I meant when I wrote "(Cantor)" as you interpreted my intention backwards. — keystone

I think I see now. You didn't mean that Cantor claims that we can list the points in the line, but rather Cantor showed that we can't do that?

If you let me know that the above is correct, then I should retract what I said earlier.

IF there is any merit to my view, then the hard work hasn't even begun. — keystone

Of course, non-infinitistic systematizations for mathematics are interesting and of real mathematical and philosophical import. And there are many systems that have been developed. Personally though, I am also interested in comparisons not just on the basis of having achieved the thing, but also in how complicated the systems are to work with, the aesthetics, and whether fulfilling the philosophical motivations are worth the costs in complication and aesthetics.

I feel like you could give me a little more slack here on my phrasing. — keystone

Your phrasing struck me as polemical and misleading by saying "magic" and "leap", which does not do justice to the fact that set theory is axiomatic, and while the set of naturals is given by axiomatic "fiat", the development of the integers, rationals and reals is done from the set of naturals in a rigorous construction.

My point is that your description is not an accurate or even reasonable simplification of how set theory proves that there is a complete ordered field and a total ordering of its carrier set. (The carrier set is the set of real numbers and the total ordering is the standard less-than relation on the set of real numbers.)

Keep in mind that no contradiction has been found in ZFC.
— TonesInDeepFreeze

Most notably Hilbert's paradox of the Grand Hotel, but also the following:

Gabriel's horn
Galileo's paradox
Ross–Littlewood paradox
Thomson's lamp
Zeno's paradoxes
Cantor's paradox
Dartboard paradox — keystone

Yes, those are paradoxes. But my point is that they are not contradictions in ZFC* (and I'm not claiming that you claimed that they are contradictions in ZFC).

Zeno's paradox is actually resolved thanks to ZFC (I mean thanks to ZFC for providing a rigorous axiomatization for late 19th century analysis).

Cantor's paradox was met by ZFC by not adopting unrestricted comprehension.

Galileo's paradox strikes me a "nothing burger". I am not disquieted that there is a 1-1 between the squares and the naturals.

Gabriel's horn. I don't know enough about it.

Ross-Littlewood. Another limit problem. Doesn't bother me. Indeed, set theory provides a framework for rigorously distinguishing between terms of a sequence and the limit of the sequence .

Thompson's lamp. A non-converging sequence, if I recall. Again, rather than this being a problem for set theory, it's a problem that set theory (as an axiomatization of analysis) avoids.

Dartboard paradox. I don't know enough about it.

/

Tarski-Banach. I'm not expert on it. But my impression is that it strikes as paradoxical only when we overlook that points are not physical things. Points are abstract, used in a conceptual armature to "model" physical things but we don't contend that those physical things are actually made of abstract points. At least that's my naive layman's take on it.

/

But Lowenheim-Skolem. The problem for me is that I'm not sure that my write-up to myself about it is correct in all details. But even with the technical explanation, it does raise for me some puzzlement.

/

* A contradiction in ZFC would be a theorem of the form:

P & ~P

No such theorem has been shown in ZFC.

Is it possible for a continuum to exist and be defined mathematically without relying on numbers? — keystone

I don't know whether a geometric theory can pick out one particular line and definite it as 'the continuum'? I'm too rusty on the subject.

I'm referring to a curve (1D continuum), surface (2D continuum), etc. — keystone

Thanks.

The hotel simply has actually infinite rooms. Do you think it's a gross misrepresentation of infinite sets? — keystone

Hilbert's Hotel is an imaginary analogy that seems fine to me.

Grice defended material implication as a faithful representation of conditional reasoning in natural languages — Srap Tasmaner

That's interesting. I'd like to understand more about that.

This is my guess how (most?) English speakers think (contrary to the material conditional) about 'if then' in everyday life:

(1a) "If London is in England, then vodka is a beverage."

False, because there is no relation between the true antecedent and the true consequent.

(1b) "If London is in England, then Westminster Abbey is in England."

True, because the true antecedent implies the true consequent.

(2a) "If London is in England, then marble is soft."

False, because there is no relation between the true antecedent and the false consequent, and the consequent is false anyway.

(2b) If The Beatles recorded "Help", then Eric Clapton played on it.

False. There is a relation between the antecedent and the consequent, but the sentence is false because The Beatles having recorded "Help" doesn't imply that Eric Clapton played on it, and it's false anyway that Eric Clapton played on "Help".

(2c) "If Paris is on the moon, then I'm a monkey's uncle."

Some people will take that as true, as an idiomatic instance of ex falso quodlibet.

(3) "If New York is in Asia, then vodka is a beverage."

False, because there is no relation between the false antecedent and the true consequent.

(4) "If New York is in Asia, then marble is soft."

False, because there is no relation between the false antecedent and the false consequent, and the consequent is false anyway.

Material implication in classical and intuitionistic logic is a static relationship that holds between sets , as in "Smoking events might cause Cancer events", where the condition always exists ,even after the consequent is arrived at, due to the fact it is talking about timeless sets rather than time contingent states of processes. — sime

Where do you find that explanation of the material conditional?

The material conditional is that the conditional is false when the antecedent is true and the consequent is false, and true otherwise,

A statement being an utterance which expresses a complete idea (not necessarily declarative, possibly interrogative, imperative, etc). — Cartesian trigger-puppets

In propostional logic, we consider only declarative statements.

Your report is quite confused about these notions. I'm afraid that if one wants to properly understand this subject then one has to read a good textbook on it.

Tarski's T-sentence is the Metalanguage — RussellA

The sentence is not the metalanguage. The sentence is written in the metalanguage. (I think that's what you meant.)

The LHS is the Object Language (OL). — RussellA

No, the sentence is a biconditional in the metalanguage. Both sides of the biconditional are in the metalanguage.

In the OL, we can say that the domain — RussellA

No, we don't specify a domain in the object language. In the metalanguage we specify an interpretation of the object language. Part of that interpretation is specification of a domain.

The OL is interpreted in the ML. — RussellA

Right.

The domain of the OL on the LHS of the biconditional is "cooking", "cleaning", "bar", "fog", etc — RussellA

No, 'cooking', 'cleaning', etc. are vocabulary of the object language; they are not in the domain.

The domain of the ML on the RHS of the biconditional is cooking, cleaning, bar, fog, etc — RussellA

No, in the metalanguage we specify interpretations for the object language. Part of an interpretation is specification of a domain. And cooking, cleaning, etc. are predicates over members of the domain.

the T-sentence relates the domain of the OL with the domain of the ML — RussellA

No, as explained above.

1) In the world is the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum. Designate this "white"
2) In the world 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). Designate this as "snow" — RussellA

Right.

3) "Snow is white" is true IFF what has been designated "snow" has what has been designated "white" — RussellA

Hmm, not sure that's the way to put it. 'Snow is white' is true iff what 'snow' stands for has the property that 'white' stands for.

I observe the world and see something cold, white and frozen and a relation between them, the relation snow. — RussellA

In Tarski's sentence, 'snow' is the noun. So 'snow' stands for an object, which, per an interpretation is a member of the domain. So snow is not a relation or even adjective ('is snow') in this particular case. The adjective is 'white' ('is white' or 'has the property of whiteness')

The T-sentence is a biconditional, meaning that the truth of the proposition "snow is white" is conditional on something. — RussellA

Hmm, okay. A biconditional is just the conjunction of a conditional with the converse of that conditional. The Tarski sentence says:

If 'snow is white' is true, then snow is white. And if snow is white, then 'snow is white' is true.

So, yes, the condition for 'snow is white' being true is that snow is white. And the condition for snow being white is that 'snow is white' is true.

"snow" being "white" — RussellA

That doesn't enter into it at all. Of course the word 'snow' is not the word 'white'. And of course the word 'white' is not an adjective regarding the word 'snow'.



.

I'm not employing a utilitarian framework here — Kuro

My main point is that I don't accept your argument of posting in terms of what is "productive" or outcomes ("fanned flames"). Evaluation per what is productive or outcomes, as I would think in that framework, productive outcomes are good and unproductive outcomes are at least less good. That is what I meant by a utilitarian point of view. My comment about the quote was that I agreed with that part of it that I quoted (no comment on the rest of it), but also I don't accept the "utilitarian" (the productivity and outcomes of posing) part of your argument; I did not all intend to suggest the quote itself is from a utilitarian point of view. Indeed, I would have no reason to think it is.

Is that clear? There's the part of the quote that I quote, which I agree with. And there was another part of your argument that I don't agree with. I don't take the quote to be from a utilitarian point of view. But I find an productivity and outcomes argument to be, at least generally put, utilitarian.

But I accept that you don't consider 'utilitarian' as a correct description of your productivity and outcomes argument. Indeed, my point doesn't rely on the particular rubric 'utilitarian' but rather that I reject your productivity and outcomes argument, whatever rubric it correctly falls under.

Also, I don't claim that evaluation of the merits or advisability of posts cannot include productivity and outcomes as a factor. Just that I don't think it is in and of itself determinative and may be trumped by other factors including even a poster's own prerogative to express himself and as a post may offer others access to expressive and cogent and/or well articulated thoughts even if they are not productive in the immediate context of a particular posting exchange. Even further, sometimes it is simply fitting that a poster gets stuff off his chest.

Isn't this why you keep admonishing AS to get some learnin' on the subject? — Real Gone Cat

I say he should learn about the subject so that he'll desist from spewing misinformation about it. He fancies himself a critic of classical mathematics but he persistently makes false claims about it. One might as well go into a biology discussion group and say, "Academic biology is all wrong. Just look at the concept of a cell. They say a cell is an organ of the body. That makes no sense. Look at their concept of carbon. They say carbon is the fluid that runs through veins. How much more wrong could they be?"

is infinity necessary? — Agent Smith

First, as I've said ad nauseum, (with exceptions of figures of speech) mathematics doesn't have a noun 'infinity' but rather an adjective 'is infinite'.

You keep using 'infinity' as a noun. Okay, then what is your definition? What object do you think is named by the noun 'infinity'?

And as I've said ad nauseum, infinite sets are necessary for the ordinary system of the mathematics for the sciences. There are alternative systems, for many approaches, but I don't know of an actual ultrafinitist systemization. (Maybe there is one? Someone can adduce one? Quite some time ago I did read through a lot of Lavine's book, but I don't recall how formally satisfying his proposal is.)

We really don't use 3.14159... (the real value of π). Put yourself in an engineer's shoes and answer that question? — Agent Smith

You skipped what I wrote about that. That is your M.O., in true crank style: Skip responding to points that don't support your own position and instead just keep reiterating your position.

Is ∞ like God as Cantor believed? — Agent Smith

Oh come on, why do you keep harping on something that died over a hundred years ago? Mathematicians, at least in their actual formal mathematics, don't relate to the part of Cantor in which he equated an Absolute Infinity with God. That's just not on the table. Forget about it. It's a huge red herring.

it's a simple rule of thumb that if a mathematician wants to propose a new idea, s/he'll use ∞ only if absolutely necessary and that too with much reservation. — Agent Smith

When you make claims like that, I ask you to provide your basis. By what actual evidence, from what actual sources, do you make that claim?

It's patently false anyway. Many mathematicians freely make use of infinite sets. All over in analysis and other other branches. Not to mention set theory itself which is a branch of mathematics. Day one of even a high school class: The real number line. That's an infinite set of points ordered by the less than relation. Day one of freshman calculus: A sequence converging to a limit. That's an infinite sequence. Infinite sets are as basic to ordinary mathematics as wheels are to automobiles.

You're just making claims straight from your smoke blowing orifices, with no regard, no sense of responsibility, for supporting those claims or even thinking for one moment whether you are right about them. Pure crank.

Quine's Methods of Logic — Srap Tasmaner

Quine is always quite the pleasure to read. Church too (for me, the intro chapter in 'Introduction To Mathematical Logic' is the definitive primer). Those are two of my heroes. Smullyan also is a great writer. And I particularly like Boolos. And for textbooks, Enderton is great.

You can stipulate that your account applies to one way of using a word or a phrase, though there may be others — Srap Tasmaner

This is interesting. Sometimes I encounter people who say symbolic logic is dogmatic because it demands that 'if then' must be taken as the material conditional even though English speakers use 'if then' in other senses. And it is true that ordinarily English speakers don't have the material conditional in mind. If you asked 10000 people whether "If London is in Asia then Groucho Marx was an aviator" is true, you'd be lucky if even one person said it is true.

So, it is helpful when an intro textbook makes a disclaimer that use of the material conditional in symbolic logic is not to be construed as a claim that the material conditional captures the many everyday senses of 'if then'. And, as far as I recall, Kalish, Montague and Mar (KMM) does not make that disclaimer.

(There are some comments about looseness on page 10.)

Another thing I wish were different in the book: Truth tables are not mentioned until page 87, after the propostional calculus has all been specified. I think it's much better to explain the truth tables before specifying the proof calculus. That way, the student can see how the proof calculus is truth preserving. Otherwise, the student is first all wrapped up in a bunch of rules while the student doesn't know the motivation for those rules.

I wish there were an intro textbook just as precise as KMM and Mar, but with a more streamlined proof system. The boxes method is intuitive, and helped me a lot as a beginner. But I would like to see a textbook with a more streamlined natural deduction system that uses line accumulations instead. (I think I posted such a system earlier in this thread?)

That said, KMM is still my favorite intro symbolic logic textbook. It certainly set me up with a solid foundation.

Consistency follows from soundness. Proving soundness is not deep. We ordinarily just do induction on the length of derivations.
— TonesInDeepFreeze
Some simplified detail might be fun. — Banno

I hope I say all this right (I'm pretty rusty):

Df. A model M is a model of a set of sentences G iff every member of G is true in M.

Df. A set of sentences G entails a sentence S iff every model of G is a model of {S}.

Df. A system T has the soundness property iff for every set of sentences G and sentence S, if S is derivable from G then G entails S.

Thm. If all the axioms of T are logically true, and T has the soundness property then set of theorems of T is consistent. Proof outline: If the set of theorems of T were inconsistent, then there is sentence P & ~P derivable from the axioms. But the axioms are true in all models, and soundness provides that the axioms entail P & ~P, but that is impossible since P & ~ P is false in every model.

Outline of proving a system has the soundness property: Base step: Prove that all the axioms are logically true. Inductive step: Suppose we are at step n in a derivation and soundness has obtained. For step n+1, show that the rules of inference are truth preserving.

This is found in any textbook in introductory mathematical logic.

But there are puzzles I find with infinite sets in certain aspects of mathematical logic. I admit that my framework of understanding hits a wall in those particular aspects.

Then there is the heuristic question: What practical advantage would there be in throwing out the methods of infinitistic calculus for an ultrafinitist approach? Again, how is it any better for me to have to know what degree of accuracy my friend needs when instead I could just say, "pi is the value" and let him use whatever accuracy he needs in its practical application?

And, from a formalizing perspective, we can pretty much anticipate that an ultrafinitist system would have a much more complicated axiomatization.

From an aesthetic point of view: At least for me, it is unbeautiful to pick some particular number, either arbitrarily or on the basis of some physicalist constraints, to be the greatest number. I lean toward being more sympathetic with a system of greatest possible generality, not particularized by whatever physical science determines at any given point in the developments of the sciences.

how can a finite brain grasp infinity — Agent Smith

Depends on what 'grasps' means. If it means mentally seeing an infinite number of concrete or abstract objects all at once, then probably it can't be done.

But if it means grasping that a certain property is held by more than a finite number of objects, such as we describe that situation as there being a set of all and only those objects, then I grasp that easily.

In that regard, I have little difficulty in grasping, for example, that there is the set of all and only the natural numbers. And I am not dissuaded by such mere facts that, for example, there is a bijection between the set of even natural numbers and the set of natural numbers. I don't find that baffling. Especially I don't think I need to reject that fact only because at an earlier stage in my life, as a child, I didn't think about such things so that I might have mistakenly jumped to the conclusion that a proper subset of a set T cannot be in correspondence with T (though that incorrect intuition wouldn't have been stated in those terms when I was a child).

The enquiry can be fruitful. But poorly so if the vines are mangled and torn apart by misinformation and unnecessary confusion.

Of course, Monatague is famous for his work in formalizing natural language. It's an interesting question how much that involvement bears on the introductory textbook 'Logic: Techniques Of Formal Reasoning'. But I would be wary of thinking that the book suggests that natural languages lie down so easily that we can just read off its sentences always unambiguously into formal sentences. (I'm not saying that you're saying that is what the book suggests.)

I would guess that the authors would acknowledge that English (for example) has different senses of the connectives. For example, I would be surprised if the authors held that "if then" is always in English the material conditional.

I agree that the book is very careful indeed in how it states things and formulates things. I always recommend the book.

You think Agent Smith is pretending to be a finitist when he is really an ultrafinitist at heart? — apokrisis

That's addressed to Real Gone Cat, but for myself, I don't think he's pretending that. He doesn't even know what the scope of the rubric 'finitism' includes, and he didn't even know what ultrafinitism is until (maybe) he looked it up an hour or so ago.

Anyway, it's not clear to me what his view is: Does he grant that there is no greatest number, while suggesting that there is a greatest practical number? Or does he hold that there is a greatest number period?

What do you mean by "ultrafinitism to be an accident"?

I think someone is trolling. — Real Gone Cat

I think Agent Smith is sincere that he thinks there should be an alternative to infinitistic mathematics. And he thinks he's contributing to that goal when he broaches certain considerations and (what are to him) juicy tidbits about mathematics.

Where he is not sincere is in not bothering to learn very much about the subject while he serially misrepresents the infinitistic mathematics he objects to.

I sense that where he's actually trolling is in his cutesy, self-effacing, would-be disarming, [parody:] "Aw shucks, I'm just a poor country boy educated in an old one room country school" shtick.

Your notion of “mentioning” is as disingenuous — apokrisis

Granted, I didn't just say that Wilderberger is an ultrafinitist but that it is quite missing the point to refer to him merely as a finitist. In my post just quoted, I didn't intend that my saying that I mentioned that Wilderberger is an ultrafinitist was meant to elide that I also said that it is quite missing the point not to more specfically say he's an ultrafinitist.

That you read into such details that I am dishonest for referring to myself has having mentioned something is nutty.

your definition of “being constructive” — apokrisis

I gave no definition. I just said that there is a sense in which I think posting replies to cranks is constructive. And I as much as happily granted that that is limited.

There was no argument made by you — Metaphysician Undercover

Yet again you repeat your false claim that is refuted by content of the posts.

The traditional early chapters of a logic textbook try to show how the logical constants capture some of what we mean by familiar idioms. (The exception might be Kalish and Montague, because they're not kidding.) — Srap Tasmaner

I'm curious about your take on Kalish, Montague and Mar.

They don't give as many examples and exercises as most books, but they do give some.

What do you mean by "they're not kidding"?

Your quibbles are doubtless correct. But not helpful. — Banno

That was discussed, but I wish also to state for myself:

My comments here have not been mere quibbles. They are important considerations, especially for keeping clear the distinction between syntax and semantics.

The comments would be helpful for anyone who knows about the subject but can use some help with a refresher on these particular points. Or, if one doesn't know enough about the subject, then my comments may suggest learning more about the subject.

It was apokrisis who first mentioned Wilderberger in this context.

Then Agent Smith replied that Wilderberger is a finitist.

Then I mentioned that, more particularly, he's an ultrafinitist.

And for my doing that, apokrisis is ludicrously reading into my comment.

If you were not calling me back to defend my statement, then what were you doing [...] — Metaphysician Undercover

Stating my own argument about it.

Whether you reply to my statement of my argument about your contradiction is entirely your own prerogative.

And, since you seem now to be leaning toward a claim that I called you back to defend your argument, I withdraw my comment that I might have taken you too literally.

So you made a pointless point — apokrisis

It had a good point. This thread is about a greatest number. Wilderberger is an ultrafinitist. By pointing that out, we see that he's especially relevant to this thread.

as if you were adding some significant and necessary correction to the discussion. — apokrisis

I made no such pretense that it was necessary. You keep essentially putting words in my mouth. It's beneath you.

you troll — apokrisis

I decidedly do not troll. I do not post to provoke reactions, and especially not for gratuitous reasons. I do not post to confuse or gaslight people.

By merely commenting that we gain by a more specific designation, I am not at all trolling.

Meanwhile, it seems to me that I my postings have rankled you to the degree that you've lost your balance so that you read into my posts the worst interpretation even though that interpretation is false, and your imbalance is causing you to now strawman me as a pattern.

you have to make it seem untrue — apokrisis

I did not suggest at all that it was untrue or even that it seems to be untrue. I addressed that in a post a few back. Please stop claiming that I've said or suggested things I have not.

You never even considered what I actually wrote. — Metaphysician Undercover

You are just repeating that falsehood, ignoring that there are actual posts in which I discussed your argument in detail, quoting it and analyzing it explicitly.

i never said you "asked", or "suggested" that I reply — Metaphysician Undercover

You wrote:

you called me back to defend what I wrote — Metaphysician Undercover

But I did not call on you, or call you back, to defend what you wrote not did I call on you, or call you back, to do anything at all.

But maybe what you meant is that merely you felt, upon your own sense, that you should defend what you wrote. Not that, as you literally wrote that I called on you to do that. In that case, fair enough. If I should not have taken you literally, then you may consider my previous comments about that now scratched.

/

for the love of all who participate in this forum, shut the fuck up! — Metaphysician Undercover

Talks about audacity!

You don't speak for all who participate.

How is the statement that Wildberger may be a finitist rendered untrue by him being also some subset of that set? — apokrisis

I didn't say it was untrue. I said the generality of him being a finitist misses the most salient point about him that he's an ultrafinitist. And I didn't even say the poster is personally remiss in that regard. I merely pointed out that we can address the reference to Wilderberger more specifically than generally.

Please do not continue to misconstrue my plain words to the point of essentially strawmanning me.

I’m sure that is a basic error that needs pointing out to stop the internet descending into crackpottery and ignorance. — apokrisis

Now you have become egregious. I didn't claim that the poster's comment was crankery. I merely added that his comment admits of additional specificity. That I have crossed with the poster in other posts and said that he posts crankery does not entail that every reply to him includes a suggestion that the particular post is crankery.

Please stop essentially putting words in my mouth.

What was the salient point in the discussion that demanded a need for the further distinction? — apokrisis

It wasn't demanded, but it is helpful.

The topic of this thread is whether there is a greatest number for practical purposes. That goes to utrafinitism. So it's quite pertinent to mention that alluding to Wilderberger in this discussion suggests highlighting that he's not just a finitist but an ultrafinitist.

only referring to it as confused, ignorant, dishonest, rambling, obfuscations. — Metaphysician Undercover

And now you're flat out lying about my posting.

I gave specific analysis, quoting you, and arranging my argument about it in conspicuously clear formatting.

You took two months to reply to my post.

There is no expiration on a poster's prerogative to reply.

After two months, you called me back to defend what I wrote. — Metaphysician Undercover

That is false. I did not ask nor suggest that you reply. I stated no preference whether you replied or not. Indeed, it would be just as well if you hadn't replied, since your replies are invariably even more obfuscation and sophistry.

You're putting words in my mouth again.

ultrafinitist
— TonesInDeepFreeze

What's that? — Agent Smith

You're joking, right?

It's pretty much at the basis of this thread you started.

And you ask me without even bothering to type it into a search?

You're trolling.

So am I. That's why I usually find emoticons inadequate and ugly.

Sometimes they're useful, but yours directed to me strike me as mere insouciance.

If memory serves, [Widlerberger is] a finitist. — Agent Smith

That is quite missing the point. Wilderberger, as I glean, is an ultrafinitist.

Finitism has a broad range. Hilbert was a finitist, yet Hilbert famously celebrated infinitistic mathematics. And that is not a contradiction, as one would see upon learning the specific sense of Hilbert's finitism.

So what is salient about Wilderberger is not just that he's a finitist but that he's an ultrafinitist.