TonesInDeepFreeze

The basic subjects of the original post deserve to be stated clearly: (1) Godel-Rosser is a conditional. The antecedent is: T is a formal, consistent ...

(1) Formalism comes in variations, many of which are not the view that mathematics is only a symbol game. Indeed, Hilbert himself stressed that mathem...

On a philosophy forum, one of its most prolific posters cannot fathom the use-mention distinction. "To the Lounge with this rubbish" indeed!

The law of identity is a philosophical principle. It is adopted in mathematics. Ax x=x is math. / Using '=', 'equals', and 'is identical with' interch...

There was discussion about whether incompleteness pertains to systems with infinitely many types. It does. Indeed Godel's original proof was about suc...

Regarding placement of threads: Some of the moderation of this forum is quite irrational.

If '=' in set theory is to mean 'is the same as', it is not the case that the treatment of identity in set theory can dispense semantics. Again, usual...

This is telling: The poster challenged by asking where in a certain Wikipedia article it says that 'equals' means 'the same'. I pointed out: The artic...

Meanwhile, I'm still interested in hearing what one would claim to be "the" order of the set of all and only the bandmates in The Beatles. That is jus...

I didn't say anything about 'constitutive'. And it is exactly my point that use of terminologies in different fields are often not compatible with one...

Here are some of the details: Theorem: There is no formula T(x) such that for every sentence S, T(g(S)) is true if and only if S is true. Proof: Towar...

I've posted explanation previously in this forum. But it seems it needs to be resaid: Tarski's undefinability theorem is that, in the relevant context...

When we are studying formal languages, formal semantics and formal theories, we would need to know how "This sentence is not true" would be formalized...

I didn't say that it is not the case that undecidability is fully met by self-contradictory expressions. I didn't say that because I don't know what "...

The posts have come full circle, at least three times today. If any new points arise, I'll consider addressing them.

I cannot provide for progress in a conversation by repeating that I cannot provide for progress in a conversation by repeating refutations and explana...

No important point has been ignored . It's the other way around. I pointed out that the footnote pertains to informal heuristic analogy and is not par...

I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reass...

Again, whatever "the axiom of extensionality indicates identity means": (1) If we use identity theory at the base of set theory, then the axiom of ext...

Again, as has been mentioned very many times on this forum, the use of the symbol '=' and the words 'equal' and 'identical' in mathematics are by stip...

Mathematical logic does not assign "fault". Fault though would be vital to assign if one were a judge in a traffic accident case. The Godel sentence i...

I didn't quote. The proof itself does not mention 'epistemological antinomy'. Godel's footnote pertains to analogies of the proof, the proof itself do...

Godel never said any such nonsense that if a system proves a contradiction then the system is incomplete. Indeed, if a system proves a contradiction t...

No self-contradiction is provable in a consistent theory, irrespective of incompleteness.

These are stipulative definitions. Anyone may use different definitions. To accommodate someone who insists that we don't use a technically defined te...

One may consult introductory textbooks in mathematics to see how we can prove undefinability from incompleteness or prove incompleteness from undefina...

If we define 'true' as 'provable', then of course all bets are off regarding these theorems as they are stated. And if in baseball we define 'hit' as ...

There is no proof of G in F. That's the point. Too miss that point is to utterly not know what the theorem is about. "Why" is not a technical term, mo...

With identity theory, '=' is primitive and not defined, and the axiom of extensionality merely provides a sufficient basis for equality that is not in...

As to manipulation of symbols, the incompleteness theorem can be be done in mere primitive recursive arithmetic, so the assumptions and means of reaso...

Regarding Tarski's undefinablity theorem, Tarski proved that in certain systems, there does not even exist such a sentence. Not only did Tarski not us...

Again, as has been explained several times in this forum: G asserts that G is not provable in system P. But P does not prove G, and P does not prove t...

The incompleteness theorem requires no notion or terminology 'True(L, x)' where L is a set of axioms or system. Rather, using the above style of notat...

"Did you lie?" doesn't have a truth value, because it is not a declarative sentence. Indeed, interrogatory sentences do not appear as lines in proofs.

Contrary to a claim made in this thread (and made by the same poster several other times in this forum), it is not the case the Godel sentence require...

I don't prefer Wikipedia as a reference on such matters, but it was asked where in the Wikipedia article on the 'Axiom of extensionality' is it said t...

I don't think they're stupid. Rather, I find that there is complacency and sloppiness in the writing of certain articles, sometimes to the extent that...

Before the reply to my post, I deleted "To see that, you just need to read the article that you yourself say is "clear and accurate"", as I thought it...

It's not a question of what was relevant to your point. I cited faults in the article, whether or not those faults bear on your point. Tarski's proof ...

Yet I showed exactly what is amiss in the Wikipedia article recently cited.

"Is there a proof of T?" is a question. But a proof of T does not have questions in it.

One can couch things as questions. But the proofs themselves do not have questions in them.

All steps in proofs are statements, not questions.

Whatever the case may be with your characterization of the subject, at least we know that disallowing sets to be members of themselves does not avoid ...

Yes, going back to your p and q would be going back full circle yet again. To break the circle requires that you give serious consideration to the fac...

What is incorrect is the assumption that there is a barber who shaves all and only those who do not shave themselves. And we don't even need any set t...

No one knows what you mean by such locutions as "x is a member of itself only in its own set". You have not defined what it might mean. Someone might ...

I did not contradict myself. And, again, as I just explained, disallowing sets from being members of themselves does not avoid inconsistency. Again, a...

Usage may vary. One prominent definition of 'theory' is that a theory is a set of sentences closed under derivability. Then, any set of axioms determi...

S might not be countable. But, yes, we do have that either S in S or S not in S. Either S in S or S not in S. Suppose S in S. Then S not in S. So S no...