Could you demonstrate and prove the provability and unprovability of G in real arithmetic sentences in T? — Corvus

If T is consistent, then T does not prove both that G is provable in T and G is not provable in G.

We prove that if T is consistent then T does not prove G and T does not prove the negation of G.

How that is all done is quite complicated, especially with the proofs of all the lemmas.

jquill — Lionino

jgill.

You make it sound like he's a sleep medicine.

Norman Wildberger, whose project is to build mathematics without mention of infinity — within the doctrine of finitism — Lionino

Where can one see the project?

Wildberger's video on set theory is atrocious, appalling, obnoxious intellectual dishonesty.

super-pedantic — Brendan Golledge

Curry's paradox has a very technical context. To understand it properly requires being very careful in the formulations.

Who does that? You? Did someone previously define?:

X := (X -> F)
— TonesInDeepFreeze

"X -> F" is supposed to mean, "This sentence is false." "X := (X -> F)" is supposed to mean "This sentence says, 'This sentence is false'." — Brendan Golledge

'X - > F' means "X is false".

'X := X -> F' means that X is the sentence 'X -> F'.

But no one asserted that X is the sentence 'X -> F'. Indeed, X is not the sentence 'X -> F'.

I've seen in multiple sources that Curry's paradox is defined as X := (X -> Y), and some of them then change it to X <-> (X -> Y). — Brendan Golledge

What sources?

If X := X->Y then X <-> (X->Y).
— TonesInDeepFreeze

You yourself said that this is allowed, so I don't know why you are arguing with me about this. — Brendan Golledge

I am not arguing about that.

Again, we have:

If X := X -> Y then X <-> (X -> Y)

but we do not have:

If X <-> (X -> Y) then X := X -> Y

So we don't have:

X := X -> Y iff X <-> (X -> Y)

If I define Y := X + 1 — Brendan Golledge

It depends on the context. For example, it could be in a computer programming language or something. But in this context, we would not write that. (I could explain why, but it's another subject).

then it is impossible to say that Y is false — Brendan Golledge

It's impossible to say Y is false (for given values of X and Y) because, as it seems, you're using 'Y' as variable ranging over numbers not sentences.

the truth table for "This sentence is false" — Brendan Golledge

It's not apparent what such a truth table would be for such self-referring sentence.

If the proof of Curry's paradox is correct, then we get that logic is broken — Brendan Golledge

The logic isn't broken. In English, we can make such utterances. But in the logic, we are not allowed to define a sentence symbol that way. (The following part I'm not well versed enough, so take it with a grain of salt.) But with such things as arithmetization, we can form certain sentences that are "self-referring". In those cases, where Curry's paradox can be performed, we find not that the logic is broken but that the particular theories in which Curry's paradox occurs are inconsistent. (Again, I'm not real clear on that, so take it with a grain of salt.)

I think that in this discussion, we're assuming that there is some context in which we can justify the definition of 'X' and we're reasoning from that assumption. Upon specifying a justifying context, we would then look for the import of the contradiction in that context.

It is blatantly contradictory for x to be both x and not x. — Philosopher19

It's even contradictory just to say that x is not x.

And set theory does not say there is an x that is not x, nor that there is an x that is x and not x.

You ma

It is blatantly contradictory for a set to be both a member of itself and not a member of itself. — Philosopher19

Correct! Indeed that is a crucial point that is used in an important proof I gave you.

Yet you want to persist by saying things like the above. — Philosopher19

That's a lie. Stop lying. I never said anything like that.

Once again:

It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself. — Philosopher19

That is not "once again". Previously you said that "a set cannot be both a member of itself and a member of other than itself". That is different from "a set cannot be both a member of itself and not a member of itself".

I wondered a while ago whether you did not actually mean "a set cannot be both a member of itself and a member of other than itself" but actually meant " "a set cannot be both a member of itself and not a member of itself". But in my reply I addressed the former in such a way that if you hadn't meant it, then you could revise to what you did mean.

It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.

Who would reject this but the contradictory/unreasonable/irrational/absurd/insincere? — Philosopher19

Indeed. (Well, except for dialetheists and paraconsistent-ists.)

I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.
— TonesInDeepFreeze

I didn't say all others are ignorant. I just said there are people who are like this. I did not specify who. — Philosopher19

You are very confused. Yes, you didn't say all others are ignorant. And I didn't say that you said that all others are ignorant. Rather, as now you mentions again, you said that some people have regarded all others as ignorant.

You didn't specify anyone in particular. Good. Because there is no one who has even hinted at a suggestion that all others are ignorant. You take the sneaky road of impugning but leaving it open-ended who you are impugning though it is obvious who you mean. And my point stands: You are ignorant on the subject. That doesn't entail that others are ignorant on it.

Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?
— Philosopher19

Where is my response? Is it me who ignores you or you who ignores me? — Philosopher19

My response is right where it was when I gave it.

And I also responded to your previous tu quoque, and you ignore that too.

I gave you a refutation. You started with insults — Philosopher19

On some crucial points, you didn't even recognize them, let alone refute them. And when you did attempt to refute points, you failed, as your supposed refutations were false and confused.

You started with insults — Philosopher19

That's a lie. I started with plain, cold information. And I did that for several posts. Eventually, it became clear that you are immune to rational discussion, and so I factually pointed out that you are confused, ignorant of the subject and in bad faith.

emotional or biased — Philosopher19

As to bias, I have read a pretty good amount of the literature of this field with informed and responsible debates regarding classical mathematics. I am fascinated by and greatly enjoy informed and responsible critiques of classical mathematics. As to emotion, exasperation with cranks is natural.

You have not yet answered:

Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it? — Philosopher19

I did answer it. Specifically and exactly.

And don't say to me something like "some set theories allow for this or that". — Philosopher19

The relative consistency of those theories indicates that it is not contradictory that a set is a member of itself and also a member of other sets.

There is no need to dance around anything. — Philosopher19

There's nothing terpsichorean about my reply. I gave you an exact refutation. The fact that you are ignorant of the context of set theory and alternative set theories is not my fault.

I don't think I'm the one that has been showing the disrespect — Philosopher19

I don't care to say you are "disrespectful", but you are irrational and in bad faith when you skip refutations and explanations given you and instead just keep repeating your false and confused claims.

I don't think I entered the discussion closed-minded or dogmatic. — Philosopher19

You are closed minded to the fact that you are close-minded and dogmatic. And you still won't face that your hyper-opinionating on a subject you know nothing about. If really were the fair minded person you claim to be, then you would get a book and find about the subject rather than posting misinformation and confusions about it.

"expert" in the field — Philosopher19

Just for the record, I don't claim to be an expert in anything other than jazz, and even in that field I'm deficient in important ways.

They want to hold on to their paradoxical or contradictory theory — Philosopher19

There it is again! You say 'contradictory', again ignoring all the explanation given you about that.

incomplete — Philosopher19

There is it is again! You say 'incomplete', again ignoring all the explanation given you about that.

act as though they are the knowledgeable ones whilst all others are ignorant — Philosopher19

I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.

What good is an expert in multishapism geometry that deals with the study of shapes such as round triangles and circular pentagons? — Philosopher19

Nope. Set theory doesn't do that.

The cardinality of N = the cardinality of P iff there is a bijection between N and P.

There is a bijection between N and P.

Therefore, the cardinality of N = the cardinality of P.

Meanwhile, there is no apparent meaning in "from the point of view".

Yes, P is a proper subset of N. Indeed the point is that it is a property of infinite sets that there are bijections between them and certain proper subsets of themselves.

The fallacy is in saying "half" in this context. For infinite sets, there is no division operation such that there is 1/2 the cardinality of an infinite set.

That a set cannot be both a member of itself and a member of other than itself is the equivalent of saying that a shape cannot be both a square and a triangle — Philosopher19

He did it again! He completely skipped recognizing the refutation given him.

We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.
— TonesInDeepFreeze

Evidently, there's no point in continuing this discussion. — Philosopher19

As I said, the discussion will go in circles given that you skip answers given you and instead just repeat your refuted claims.

If you believe your mathematics is free from contradictions or paradoxes — Philosopher19

It's not my mathematics. I don't have allegiance to it. I find value in it, find wisdom in it, recognize that it axiomatizes reasoning used in the sciences, and enjoy it. But I don't claim that there might not be better approaches - philosophically, intuitively, and practically.

I don't claim to perfect certainty that set theory is consistent. But it seems to me to be an extremely good bet that it is. (1) No contradiction has been found in it under incredibly intense and indefatigable scrutiny for about 125 years. (2) We can see specifically how it was devised to avoid Russell's paradox. (3) The concept of sets as a hierarchy itself suggests an intuitive approach that is consistent.

Again, you use the word 'incompleteness', thus ignoring the information that was given you about incompleteness.

I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise). — Philosopher19

You haven't proposed an alternative framework, let alone in axiomatic form. Articulate the principles by which you propose to derive mathematics adequate for the sciences, or, better yet, put it in axioms; then we can put it to the test.

Set theory gets the job done of axiomatizing the mathematics for the sciences. By analogy: Set theory is an airplane that flies. If one thinks it's not a good airplane, then one is welcome to show us a better one.

How do you prove then N is different size to P? — Corvus

We don't. He proved that they are the same size.

I have an ability to understand concepts without even knowing of them — Vaskane

Please forgive the cliche, but it is especially apt: Above is Dunning-Kruger on steroids.

The question is not well formed. It is not apparent what "infinitely infinite infinitely infinite infinitely infinite infinitely infinite infinitely… (etc.) infinities" means.

But here are exact statements that might answer what the poster is wondering about:

Any set of all the real numbers in any non-empty interval is infinite.

Any set of all the real numbers in one non-empty interval is equinumerous with the set of all the real numbers in any other non-empty interval.

There is no greatest cardinality.

For any infinite set of cardinalities, there are greater infinite sets of cardinalities.

So what semantic are mathematicians using when they use the world/label "infinite"? — Philosopher19

We don't say "using semantic".

Rather, we just state the definitions.

I stated the definitions in my first post in this thread:

https://thephilosophyforum.com/discussion/comment/878326

You know, its a funny thing, but when I don't know much about a subject, I pay attention to people who do know something about it. And especially I don't slather the Internet with stubbornly false and confused claims about it.

We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.

So it is again regarding 'contradiction'.

Go back and read what I wrote about "contradiction".

Something cannot be both a member of itself and a member of other than itself at the same time. — Philosopher19

In certain alternative set theories, there are sets that both members of themselves and of other sets.

In ordinary set theory, no set is a member of itself.

By the way, we don't need to use temporal phrases such as "at the same time". Set theory does not mention temporality.

Then the rest of your z's and v's is irrelevant if it is supposed to refute the proofs I gave. Moreover, if you knew anything about this subject or even mathematical discourse you'd see that your prose about it is ungrounded, impenetrable double-talk.

To refute a purported proof, you need to show a step in the proof that is not permitted by the inference rules (which in this case are those of ordinary predicate logic).

And you separately quoted me saying "The axiom schema of separation". What was the point of that? Did you look up what the axiom schema of separation is and you think your remarks relate to it in some way?

I'm not sure what you mean by "So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference." — Philosopher19

It is clear. You don't know what it means, because you are virtually completely ignorant of the subject matter.

I'll spell it out even more:

In set theory, to prove there exists a set having a certain property, we must do so from the axioms and rules of inference alone.

In this instance, the property in question is "has as members all the natural numbers"

Without the axiom of infinity, we cannot prove that there is a set with the property "has as members all the natural numbers". But with the axiom of infinity we can prove that there is a set with the property "has as members all the natural numbers".

You argued that from "after each natural number there is a next natural number and there is no greatest natural number" we cannot infer "there is a set of all the natural numbers". And you are correct about that!

So I pointed out that indeed set theory does not make that unjustified inference, but rather, set theory has an axiom from which we CAN infer that there is a set of all the natural numbers. And THAT inference, from the axiom, does NOT use the unjustified inference from "after each natural number there is a next natural number and there is no greatest natural number" to "there is a set of all the natural numbers".

/

You know virtually nothing about set theory. You should present whatever concept of infinity you like, but you shouldn't be presenting it as a refutation of a subject you are ignorant about.

It makes a real difference. By saying 'infinity' as a noun and then that there are different sizes of infinity is to picture an object that has different sizes. There is no such object in mathematics.
— TonesInDeepFreeze

I don't think I'm picturing an object. I think I'm just focused on the semantic of Infinity. — Philosopher19

Whether described as 'picturing an object' or 'positing that there is such an object' my point is that set theory does not mention, describe or posit any such object, so saying 'Infinity' as a noun as you do is misleading as it does suggest that one should take set theory as suggesting that such an object can be countenanced, considered or pictured, etc.

Good faith in posting a critique of mathematics would entail at least knowing something about it.
— TonesInDeepFreeze

I think it is from all that I have seen and heard [...] — Philosopher19

What are your sources? What specific texts in set theory or mathematics do you think have said the things you claim set theory to say?

Whether all that I have seen or heard is enough, is another matter. You don't think I have. I think I have. — Philosopher19

It's the heart of the matter of why your are ignorantly misrepresenting set theory.

You think you've read enough set theory to understand its axiomatic treatment of infinite sets? What specifically have you read, let alone studied sufficiently to competently discuss it?

If so, then you understand that a line of an interval of 2 represent twice the length, as the line of an interval of 1. And thus you're perhaps an even lower wisdom score than 8 after I already pointed out several times that there's an error in communication and even held myself accountable for that error, that you're too dumb to understand a line has length/area/size whatever the fuck you wanna call it, after I clearly stated a communication error upon the context ... I mean fuck dude, you're like Marine when he sees red. — Vaskane

As that was added in edit, I missed it.

Whatever you think of me, or whatever error you think there was in communication, I accurately responded to your posts as they were written.

You claimed that the size of the set of numbers between 1 and 2 is less than the size of the set of numbers between 1 and 3. If that's not what you meant, then it's not my fault. Then you deflected to the fact that the distance between 1 and 2 is less than the distance between 1 and 3, which is true, but it does not bear on the fact that the size of the sets is the same. At the time of posting I saw no post in which you "held yourself accountable" for that error.

And I explained in perfect detail about length, but instead of recognizing that, you incorrectly suggest that I don't understand length and you resort to juvenility such as "too dumb".

[set theory says] Nothing can be the set of all things (which logically implies Existence is not the set of all existents) — Philosopher19

Mathematics doesn't mention "all existents" or "set of all things".

The heart of your attack on infinitistic mathematics is your own mistaken fabrication of what you think the mathematics is. In other words, you're putting up a huge strawman.

if each cardinal is STRICTLY larger than the one before it, I suppose they do indeed have different sizes. — Vaskane

By definition, a successor cardinal is strictly greater than its predecessor.

(By the way, aside from successor cardinals, there are cardinals other than 0 and aleph_0 that are not successor cardinals, and they are greater than any previous cardinal.)

Anyway, without even getting into successor cardinals and limit cardinals (cardinals that are not 0 and not successor cardinals), it is easy to prove that for any set, there is a set of greater cardinality.

G is a sentence in the language of arithmetic.

There are many ways to couch incompleteness proofs. Here is one outline:

Let T be a recursively axiomatized, consistent theory that is "sufficient for a certain amount of arithmetic".

We adduce a sentence G that is is true (to be more precise, it is true in the standard model for the language of arithmetic) if and only if G is not provable in T.

Then we prove that G is not provable in T. So G is a true sentence that is not provable in T. Moreover we show also that ~G is not provable in T. So T is incomplete.

'S is infinite' is equivalent with 'S has infinitely many members'.

Or as you say:

if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them. — Bob Ross

'the set of natural numbers is infinite' is equivalent with 'there is an infinite amount of natural numbers'.

Proving that the set of natural numbers is infinite is the same as proving that there are infinitely many natural numbers.

In a textbook in set theory, you would see how a theorem of the form:

S is infinite

is actually proven.

Right, those working in the various branches don't usually concern themselves with the foundations. So use of infinite sets is ubiquitous without concern for the foundational axiomatization concerning them. Proverbially, infinite sets are the water mathematics swims in. The fish doesn't have to know anything about water, but it still needs that water to swim in.

As I argue, the first day of class when we are told "We have the natural numbers and we have the real numbers and the real number line", boom, we are presented with infinite sets, even if the instructor doesn't happen to mention, "And don't forget, those are infinite sets".

As to age, the mathematics is pretty durable, thus also is the wisdom of those who learn it.

we could determine S is infinite either by stipulation—e.g., if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them. — Bob Ross

That is circular.

And we don't just stipulate that the set of natural numbers is infinite. We prove it.

If S1 is a set with size 2 elements ad infinitum and S2 is a set with size 1 of elements ad infinitum, then S1 > S2 (and I don’t need to count them). — Bob Ross

It's nothing like that.

I'm not sure, but I think bread and butter analysis might touch on the cardinality of the power set of the set of reals (?), but I don't have enough information to dispute that even higher cardinals don't come up much.

But infinite sets are regarded as objects. The set of real numbers is a set theoretic object. Boom, from page 1 we are dealing with an infinite object. The real plane is the Cartesian product of the set of reals with the set of reals. Takes an two objects (or at least one object twice) to make a Cartesian product.

What is the domain of the function f where, for all natural numbers n, we have f(n) = 2*n? f is a function, and every function has a domain, and the domain of f is the set of natural numbers, which is an infinite set.

Even when we say "let n go from 0 to inf", that really is just to say that the domain of the function is the set of natural numbers.

I don't see the point in saying that mathematics such as analysis doesn't use infinite sets, when plainly, at the very outset, to even start in the subject, we see that we are using infinite sets.

As for transfinite math, it rarely if ever comes up in classical analysis. — jgill

Depends on what is meant by 'transfinite math'. 'transfinite' is just another word for 'infinite', and, of course, analysis uses infinite sets. Moreover, there are mathematicians who work (and not in obscurity) with higher cardinals vis-a-vis analysis, though that work might not be prominent in the bread and butter mathematics you have in mind.

As a connoisseur of cranks and sophists, I beg to differ. This thread is run of the mill in that regard. And there are routinely far more risible ignorance and confusion posted.

You can use a truth table to prove NOT X <-> (X -> F).
(X -> Y) <-> (X -> F) in the case where Y is false, so this applies to Curry's paradox as well as "this sentence is false". — Brendan Golledge

Yes:

|- ~X <-> (X -> F)

If Y is false then (X -> Y) <-> (X -> F) is true.

That's not Curry's paradox.

Then you take your definition X := (X ->F) and substitute NOT X for the second part. — Brendan Golledge

Who does that? You? Did someone previously define?:

X := (X -> F)

Maybe a winter pastime for some of us. — Mark Nyquist

It is all-seasonal and perennial I assure you. People spouting hyper-opinionated uninformed and confused misinformation about this subject goes on constantly and forever on the Internet.

Still Cantor proves Cardinals do indeed have varying sizes. — Vaskane

Of course we prove that there are cardinals of different size. We know the proof well.

That doesn't even the least bit suggest that there is a mathematical object called 'Infinity' that has different sizes.

Yo, like I said, I came here making a comic relief joke, to which it tumbled into the argument on the size of infinities, and lo and behold I have an actual theorem that shows sets of cardinals being larger than the last, that backs me up and it's a theorem that has had impact upon reality my friend: Theory of Computation: Cantor's diagonal argument, used to prove the existence of different sizes of infinity, inspired Alan Turing's work on the undecidability of the halting problem, a foundational result in the theory of computation and computer science. — Vaskane

What?

(1) Whatever jokes you made, you also made the incorrect claim about the size of the set of numbers in the intervals. You still haven't recognized the the thorough explanations why your claim is incorrect.

(2) Yes, theorems:

If C is a cardinal then there is a cardinal greater than C.

If C is a cardinal then there is an infinite set of cardinals greater than C.

If C is an infinite set of infinite cardinals, then there is an infinite set of cardinals greater than C and such that it has infinitely many members that each is greater than every member of C.

There is no set k such that every cardinal is a member of k.

We already know that. It's not something that somehow vindicates your incorrect claim about intervals or anything else you might have said.

(3) Yes, we already know that Turing used a diagonal argument and that the diagonal technique was made prominent by Cantor. Moreover, we know that infinitistic set theory applies to the theory of computability. I don't how in the world you think think any of this some how "backs you up" in terms of any controversy there's been with you.

Cantor treats a number sequence that goes on forever as being infinite. But something going on forever does not make it infinite (if my counting to infinity goes on forever, that neither makes my counting infinite, nor does it mean I will eventually reach infinity). It also makes no sense to say something like "assume that your counting to infinity is completed such that you have counted the set of all natural numbers and have successfully proven that there are an infinite number of natural numbers" and then label this as {1,2,3,4,...} — Philosopher19

I answered that exactly already.

You truly are not in good faith.

You make claims about a subject of which you are ignorant. Then when it is explained to you exactly what your confusion is, you ignore that explanation and instead just go on to make the confused claim again.

So I'll give you the explanation yet again so you can ignore it again:

We do NOT claim that from "after each natural number there is a next number" and "there is no greatest natural number" that we can infer that there is a set of all the natural numbers. Indeed such an inference IS a non sequitur. And every mathematician and logician knows it is a non sequitur. So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference.

Also, it is good to study Cantor for historical context, to appreciate his intellectual power, and to gain insights into the concepts. But Cantor has been supplanted for 125 year or whatever by axiomatic set theory. If you are sincere in wanting to fairly critique the mathematics then you would get a book on set theory and read it.

Cantor’s Theorem and the Unending Hierarchy of Infinities: still hasn't been disproven;
"For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem,we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity")." — Vaskane

Yes, it's the colloquial part that is so often abused by people who know virtually nothing about the subject. Especially among beginners in the subject, if we refrain from that unfortunate usage, then (1) We avoid having set theory look ridiculous as if it claims that there is an object that has different cardinalities. (2) We adhere to the way the actual mathematics is couched, which is that is the property 'is infinite' but not an object that is 'infinity'.

So those who post uninformed, intellectually bigoted and confused lashing out against set theory would not have the slippery wedge of presenting mathematics as if it is itself absurd. People who are unfamiliar with the mathematics ordinarily don't think of infinitude the way mathematicians do. In everyday life and conversation, and even academically in certain contexts, it would strike as extremely odd to hear that there are "greater and greater infinities". But if it is said instead, "In mathematics, 'finite' and 'infinite' are properties of sets, and there are infinite sets, and there are greater and greater sizes, such as there are more real numbers than natural numbers" then it may strike one as much more sensible and mathematics is not made to look ridiculous as if it has an object that is infinity but that it comes in different sizes.

By the way, Cantor did not work axiomatically. The results of set theory are on much firmer ground now as we work axiomatically now.

if someone came to me and said they have demonstrated how infinity comes in various sizes, I would say to them that that's nonsense to me. — Philosopher19

And it is fine that the concept is nonsense to you. That is not at issue. But that something strikes you as nonsense doesn't thereby render it nonsense.

You don't know anything about the mathematics, so you are in no position to try to convince other people that it is nonsense. The only thing you could fairly say is, "I don't know anything about the mathematics, so I can't fairly opine on it, but my own concept of infinity does not allow that there are sets of different sizes of infinity".

But then you keep harping on this "semantic" issue you have. If I hadn't said in this thread, I'll say what I've said in other threads:

(1) Words have different meanings and senses in different areas of study. In biology the word 'cell' means one thing and in criminology it means another. Even when the senses are closely related and in closely related or even overlapping fields, the senses can be different. When mathematicians talk about 'infinite' they don't thereby declare that that mathematical sense applies to all other areas including philosophy, cosmology or theology. However, yes, the mathematical sense may be applied in other areas, but it's up to the author to make clear what sense they mean. At the very least, mathematicians ought to be allowed to talk about 'infinite' in the mathematical sense without someone who knows virtually nothing about the mathematics arrogating to convince people that it is "nonsense" and even to claim to "prove" that it is nonsense with unstated standards of "proof" quite different from mathematical proof.

(2) Even if you don't have the least bit of reasonability to grant (1), then we could say, "Fine, from now on consider every book and article and post in mathematics as if the word 'infinite' were replaced with 'zinfinite'. The formal mathematics would be just the same except the word you arrogate to your own meaning would not be used and you could rest easy that mathematics does not impinge on you own concepts.

(3) Again, formal contradiction is any statement P asserted along with the assertion of the negation of P. There is no known contradiction in set theory. A DIFFERENT matter is that in certain senses set theory is not compatible with your own philosophy, and again that a certain word is used in mathematics differently than you use it.

To my understanding, mainstream maths claims:
There are infinites of various sizes (or at least infinite sets of various sizes, but that amounts to the same thing)
The set of all sets is contradictory — Philosopher19

(1)

There is no object called 'Infinity' in the sense you have been using it.

Here is a way to say what you want to say:

In mathematics, there are sets that are infinite but that have different cardinality from one another.

Better yet:

If x is infinite then there is a y that is infinite and y has greater cardinality than x.

(2)

The statement that there exists a set z such that every set is a member of z is inconsistent with the axioms of set theory; and set theory proves the negation of the statement that there is a set z such that very set is a member of z.

Better yet:

'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y).

And we prove that easily:

First, we prove:

There is no x such that for all y, y is a member of x iff y is not a member of y. Proof:

Suppose, toward a contradiction, that there is a such and x. Then x is a member of x and x is not a member of x. Contradiction. So there is no such that x for all y, y is a member of x iff y is not a member of y.

Next, Suppose, toward a contradiction, that there is a z such that for all y, y is a member of z.

Then from separation we have an x whose members are all and only those y that are both a member of z and not a member of y.

But since every y is a member of z, we have x whose members are all only those y that are not a member of y. That contradicts that there is no x whose members are all only those y that are not a member of y. So there is no z such that for all y, y is a member of z.

Do you agree that lines have length? — Vaskane

Whether you intend it or not, that is a trollish question.

Of course, lines have length. That is implied by my talking about distance.

I already explained to you that there is a difference between the distance of an interval and the cardinality of the infinite set of numbers in the interval. But instead of recognizing that information, you hit me with a mindlessly posed quiz.

So I suppose I need to lay it out for you in even greater detail:

The real line is the set of numbers ordered by the usual less-than relation on the set of real numbers.

The intervals [1 2] and [1 3] are segments of that real line.

For any pair of numbers there is the distance between them, which is the absolute value of their difference. The distance between 1 and 2 is 1, and the distance between 1 and 3 is 2.

But the intervals are a set of numbers, not just their max and min. The interval [1 2] is the set of real numbers that are greater than or equal to 1 and less than or equal to 2. The interval [1 3] is the set of real numbers that are greater than or equal to 1 and less than or equal to 3. Both of those sets have the same cardinality, i.e. both those intervals have the same cardinality.

Your non sequitur is to infer that intervals having different distances implies that they have different sizes. (And I say that because you still have not said that you didn't mean what you said when you said the context here is mathematics.)

Since this keeps getting lost with you, for the third time:

Intervals may have different distances (lengths, if you prefer) but the same cardinality.

I think it's nonsense to say Infinity comes in various sizes. — Philosopher19

He did it again! He still persists in mischaracterizing mathematics as claiming that there is an "Infinity" [capitalized, no less] that has different sizes. That is after the mistake of that has been pointed out at least a dozen times, and as he protests that he is posting in good faith.

Then we have people saying that there are contexts other than set theory so one should be tolerant not to demand that set theory is the only context we may consider. Quite so. But the context of the original poster is not just a proposal for another concept but a claim that the mathematical set theoretic concept "is nonsense". And that is intolerance. To know nothing about the mathematics behind the concept of cardinality but instead just call it nonsense. That is egregious intolerance.

That's a matter of perspective really, I was a Navy Cryptologist, that you only view numbers as a row and now both rows and columns, again revolves back to your objective perspective. — Vaskane

I said nothing about rows and columns.

First, you incorrectly objected to taking size as cardinality when you said yourself that the context is mathematics.

Second, you conflated distance with cardinality.

Third, you claimed it's about area, though area is not involved.

Fourth, you were a Navy cryptologist and something about me regarding rows and columns, though I said nothing about rows and columns.

Will there be a fifth attempt to evade the plain fact below?:

"the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3" is incorrect.

I would think it would be much less wear and tear on your credibility to instead just think about the correction given you and then recognize it.