TonesInDeepFreeze

I don't know that. The axiom of infinity says there is an inductive set and, with other axioms, entails that there is an infinite set. Set theory does...

No. I did not begrudge you hyperbole. Rather, (1) I explained why previously it was not unreasonable for me not to infer that you were writing hyperbo...

I don't know what hyperbole you have in mind. Maybe 'nobody'. Because you seemed adamant with all-caps, and, as I recall, three variations of 'no', I ...

I don't seek to be assuaged. You don't need to assent to 'plenty' on my account. Rather, one can assent to it merely on the grounds that it is obvious...

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.

I neither denied it nor affirmed it. Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case". Today when I read "Nobody ...

That's your view. My point is not nor has been to convince you otherwise. Rather, my point is that no matter that it may be your view, it is not true ...

A progression of views (not necessarily your own): (1) "Hilbert said that mathematics is only a meaningless game of manipulating symbols." False. Hilb...

Yes, which makes it even more curious what one would mean by saying the axioms of ZFC are false, while proposing a theory that is equivalent to ZFC PL...

I would expect that there is a wide range of interest in foundational axioms among mathematicians - from no interest to intense interest. But even amo...

Many mathematicians and philosophers of mathematics regard certain axioms and theorems to be true not just relative to models. It might even be the do...

It might be fair to say that for Hilbert the syntax of logic does not include content. But Hilbert did not consider content irrelevant for mathematics...

Thanks very much. I appreciate it.

I didn't complain. I merely added the information. And I used 'connective' in line with the notion of a connected relation. I don't begrudge you striv...

No, I got your point that your posts are meant only as an overview. But that doesn't entail that I can't mention clarifications and some more exact fo...

Just to be clear, in set theory, the existence of a set that has all the natural numbers as members is not proven by taking a limit or a union. Rather...

Within any limit ordinal, there is no last successor. For example, w is a limit ordinal, and there is no member of w that is the last successor.

My point was not that understanding definitions is sufficient, rather that understanding definitions is necessary.

'successor' for ordinals is simply this, by definition: successor of x = x u {x}. Defintions!

I think fishfry addressed that. epsilon_0 is a limit ordinal, not a successor ordinal. epsilon_0 is the union of the set of ordinals of the form w^x, ...

w = N. No matter what order. That does not contradict that also w is the order-type of <w standard-ordering-on_w> = the order-type of <N standard-orde...

You are skipping the definitions: w = the set of natural numbers w+1 = w u {w} Mathematics doesn't have a separate definition of 'number' in general. ...

Whatever many people may think, such books are key to understanding. But of course, a combination of books and teachers is best. In any case, in mathe...

For clarity, I prefer to use 'permutation' in its exact mathematical meaning. A permutation is a bijection from a set onto itself. In that regard, a p...

We should mention that limits (aka 'sups') in regard to ordinals are unions that are not successors. We need to have a good understanding of both bina...

That is an entertaining book, but one might need to take it with a grain of salt regarding certain technical matters (I don't recall the particular ma...

Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem. From the definition of 'order-type', every order-type is an ordinal...

I didn't say one needs to be a specialist. Having an adequate grasp of the basic terminology doesn't require that one be a specialist. And one can get...

Here is some of the terminology (not necessarily in logical order) that one must have a very clear understanding of in order to have a clear understan...

No. The set of all permutations of S is the set of all bijections from S onto S. The set of all well orderings of S is something different. For a natu...

'first' in the sense that there is no member that precedes it in the ordering. Usually we say 'least' or 'minimal'. In ordinary mathematics, other tha...

Further, in greater generality, a well ordering R of a set S is relation such that both (1) R is a subset of the set of ordered pairs of members of S,...

A well ordering of set S provides that every non-empty subset of S has a first element. And S is a subset of S, so if S is non-empty, then S has a fir...

Re: The question was asked by tim wood: "What is an infinite ordinal?" As direct an answer I can provide: S is an ordinal if and only if all three: (1...

Of course, semantics for intuitionistic systems are different from semantics for classical systems. But the question of equivalence is that of derivab...

Explicitly constructive mathematics goes back at least a hundred years, and with roots in the 19th century too. It has great importance toward underst...

That is not the reason. The reason is that LEM does not imply AC, whether with intuitionistic or classical logic. I looked at that article briefly. I ...

That is not correct. It is the case that Z (even without the law of excluded middle (LEM)) and the axiom of choice (AC) together imply LEM. But it is ...

Sure, if "captures the essence" means grossly mischaracterizes with ignorant confusions.

If you think there is anything wrong in my rebuttal, then you should be able to point to it exactly.

I asked you what version of Godel's proof have you read in a paper or book. That is, what writing did you base your previous post on?

I pretty much figured that you didn't know what you were writing when you said that there is a 1-1 correspondence. I pointed out that you have failed ...

Yes, it's possible he might get a chuckle at your hapless ignorance. It wasn't just that Kronecker criticized the work. But it does seems reasonable t...

That is is nothing like Godel's proof. On so many levels it is nonsensical. What actual version of a Godel's proof have you read in a paper or book?

Incorrect. 2 = {0 1} and has cardinality 2. K = {K} and has cardinality 1.

If you link to where you first posted it, then I'll link to where I answered it. You still have not addressed my rebuttal.

I don't know how he reads in the original German, but the above is not how the set theory that came from Cantor works. We don't define "infinity" as a...

Since it's a restatement, I don't need to address it again, since I've replied to your "proofs" already, in quite detail. And you have not gotten back...

That entire passage is merely a report of notions and terminology of mathematical logic. It's nowhere even close to a philosophical statement. Except ...

Your boorish condescension is stupid. I never said that the set of orderings of a set is not inherent to the set. I said over and over and over that s...