TonesInDeepFreeze

His invention of the predicate calculus is great intellectual wisdom.

When people say the three laws of thought are A=A ... identity A v ~A .... excluded middle ~(A & ~A) ... non-contradiction they are using 'A' for two ...

And your emoticon doubly seals it! Who could ever defeat an emoticon?

Well that truly settles the question!

Empty generalization and bluster. I quite understand that human thinking, including about mathematics, involves intuition. Indeed I'm interested in th...

It is a formal definition. But it still captures the ordinary sense of "To claim a contradiction is to claim a statement and its negation." For exampl...

You are entirely ignorant of what contradiction is in mathematics. Moreover, even if contradiction were, in some sense, couched semantically, then no ...

The statement that there exists sets that are infinite is not a logical impossibility. The idea of allowing 'there exist potentially infinite sets' bu...

There is no magic. Very much to the contrary. At a bare minimum, it is algorithmically verifiable whether a given formal expression is well formed and...

Look at how the subject of identity is handled in formal logic, philosophical logic, mathematical logic, set theory, and mathematics. Those clear up a...

From your personal stash. Side effects include confusion and disorientation. Explains a lot.

You regularly and willfully spread ignorant disinformation about mathematics throughout many threads, over the course of at least several months, even...

You should not be disappointed. You should be encouraged. I gave you an outstanding example: when one does not have sufficient knowledge then one shou...

I know so little about physics or cosmology that I can't answer that.

No, set theory does not say that there is a proper subset of a set such that the proper subset is the set. Set theory does say that there are sets suc...

'Describe the universe' is a notion not defined by you. Clearly that is false. Infinite sets are basic for calculus. Let that number be M. Then M+1. P...

That's not a good place to start. Infinitary logic is a special topic in mathematical logic that is not so much directed to the basic notions of infin...

The mathematics that in particular delves into infinity is set theory. There are no known contradictions in axiomatic set theory.

There is a fixed point lemma in the proof of incompleteness: G <-> prov(#G) But I don't know what fixed point is involved in the sequence of theories ...

There have been controversial puzzles in epistemology for centuries. I don't see any crashing down of the world related to this. What do you think wil...

How so?

PPS. I'm not sure, but I think that not just showing that the union is a recursive axiomatization might require simultaneous recursion, I think simult...

PS. We could take the union of all those theories. (And maybe not hard to show that it is recursively axiomatized? We might need to use simultaneous r...

I'd like to add that the books I recommended are beautifully written, precise, and pedagogically excellent. One can see that each of them was written ...

That's true! I cannot understand why more people don't lie awake at night about it! It requires a global response. We need the World Bank, the World H...

I haven't looked at that one. I did like using the Schaums topology intro, but only as a supplement to other textbooks.

For example, even at page 9, just a few pages in, he writes (Zorns' lemma): "Say that a collection of sets C is a chain iff for any elements x and y o...

You can study Enderton's mathematical logic book first. However, the book is intended for upper division students who are already studying abstract ma...

'valid' has a technical meaning. I wonder whether you are using 'valid' in some other sense. Anyway, a formula is a contradiction if and only if it is...

Correct. But, as far as I can tell, your purported definition is not justified by any instance of a definition by recursion theorem. That is, yours is...

Absolutely I can recommend the very best textbooks I have found after looking at and reading many of them: First. Learn symbolic logic. How formulas a...

Let's say for context that we are interested in a particular model M at some point in discussion. So temporarily we'll take 'true' and 'false' to stan...

For this discussion, in order to be as clear as possible, I suggest sticking with my technical distinction between an argument and a proof, even thoug...

Regarding the mathematical handling of Russell's paradox as opposed to dealing with the problem informally, Russell gives a non-mathematical analogue:...

Also, don't forget that a paraconsistent logic, depending on its formulation, can have both LNC and non-explosiveness.

A system for use with a multi-valued semantics can be paraconsistent or not. However, as far as I know, a paraconsistent system can't have a classical...

What does 'AP' stand for? And what article online (if it's online) do you refer to in relation to Tarski?

You did in so many words. If you object to the paraphrase, then substitute the actual words you used. Appears to you, whose perception is poor. Moreov...

I don't know how you ever came up with the strawman that I don't take the statement as paradoxical in its everyday language context. Indeed, very much...

Ordinary language changes in the course of millions of individual choices toward variation but also sometimes in decisive strokes. If you wish to argu...

Your condescension is belied by comparing our familiarity with the subject. And you just skipped what I wrote about this. It depends on the author whe...

And I don't propose any specific changes to the explication of the paradox per mathematical logic. On the other hand, no matter what you propose or do...

It's subject to revision in the sense that anyone can propose different approaches. Meanwhile, in terms of its ordinary mathematical context, it has p...

I have never proposed any argument that it is not paradoxical in ordinary language.

Again, you're not seeing the point among your unnecessarily split hairs. Sometimes informally we use 'sentence' and 'statement' synonymously. Whether ...

Yes, mathematical logic offers the freedom for anyone to present alternative formulations, definitions, methods, and paradigms. That's a good thing. I...

I said that we can evaluate it by formal methods. I didn't say that we must evaluate it only by formal methods.

Oh come on, of course we admit that natural language utterances don't have a single definitive unequivocal context. But given some reasonable understa...

Sure we do. "Provo is in Utah" bears truth. "Provo is not in Utah" bears falsehood. See the Introduction in Alonzo Church's 'An Introduction To Mathem...

First we need to be very clear in our terminology: Formally, an argument is merely an ordered pair <G P> where G is a set of statements and P is a sta...