TonesInDeepFreeze

You said something similar to that. But later you said something very different. It's not the reader's job to suppose you don't mean what you write. M...

Also, look at it this way: Given a set of axioms G and a different set of axioms H, it may be the case that the class of models for G (thus for all th...

That is a deep misunderstanding. An interpretation for a language determines the truth or falsehood of each sentence in the language. Different axiom ...

It is not the case that 'mathematical truth' means ''axioms plus an interpretation'. The definition is: sentence S is true in model M if and only if a...

It doesn't matter whether S is an axiom or not. The definition doesn't mention 'axiom'. By the way, every sentence is an axiom of uncountably many axi...

So you didn't write what you meant regarding S and F. And still no recognition of these: You should not say 'logic sentences' in general, since the th...

You wrote: "(S is true and F is false) or (S is false and F is true) or both." Which is: (S is true and F is false) or (S is false and F is true) or (...

I don't know what that means. The definition is: sentence S is true in model M if and only if and that definiens doesn't mention 'axiom'.

I have made the point that whether there are uncountably many truths or whether there are unexpressed truths depends on what is meant by 'a truth'. In...

You still resist recognizing these points: You should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languag...

First, I made a mistake as I misread your disjunction for conjunction. I made edit notes for that in my posts now. You don't know what every person is...

(1) You skipped that I pointed out that: (S is true and F is false) and (S is false and F is true) is never the case. (2) "S ? ¬F(r(#S)" is not the sa...

(1) Your quoted characterization did not have the specifications you are giving now. Your quoted characterization was a broad generalization about pro...

('r' for 'the numeral for' and '#' for 'the Godel number of') Let C be this theorem: For certain theories T, for every formula F(x) there is a sentenc...

You wrote: "For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has it, or both."...

No, I meant what I wrote, I showed you a property of sentences that every sentence has. And what you wrote doesn't even make sense. # S is a number no...

? ? ¬ F(°#(?)) is not: "For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has i...

Qualification: Presburger arithmetic is usually stated with a finite axiomatization. But it also can be finitely axiomatized. On the other hand, the o...

One point though: Godel-numbering is in the meta-theory, but we want to know why we need multiplication in the object theory. But, if I'm not mistaken...

See a proof Godel-Rosser. Diagonalization is available in any case. But we need multiplication for Godel numbering. We also need exponentiation, but G...

Where did Carnap write that?

That depends on what things are truths. If a truth is a true sentence, then there are exactly as many truths as there are true sentences, which is to ...

(1) I would avoid the word 'valid' there, since it could be misunderstood in the more ordinary sense of 'valid' meaning 'true in every model'. What yo...

We know it is so because having both addition and multiplication entails incompleteness, so, since Presburger arithmetic is complete, it can't define ...

Robinson arithmetic is incomplete.

The truth of a sentence is per interpretation, not per axioms. Some sentences are true in all models. Some sentences are true in no models. Some sente...

For starters, the system represents all the primitive and recursive functions, and is incomplete.

For perspective, keep in mind that Skolem arithmetic and Presburger arithmetic are not fully analagous, since Skolem arithmetic has more detailed axio...

Yes, there is a mathematical definition of 'true in a model'.

No, a fragment of set theory with also the negation of the axiom of infinity is bi-interpretable with PA. I pointed that out previously.

The law of identity, the indiscernibilty of identicals, and the identity of indiscernibles are different. With a semantics for '=' such that '=' is in...

You claimed that I don't distinguish between material implication and everyday discourse. But I had explicitly said, about three times, that I do reco...

One is a statement in the meta-language and the other in the object language. They are different levels of statement.

You lie again. You lie in the face of my having said the exact opposite. And you again make coherent discussion impossible. I said that I study differ...

It's not a translation of the sentences discussed. That point was recognized by @"Lionios" who originally claimed it to be a translation.

Banno may speak for himself, but I don't know what difference in reference you mean by spelling 'false' without caps and with all caps. Nothing is "re...

There are particular apples and we can generalize about them. There is no apple that is not a particular apple. But we do say things like "If x is an ...

'non-particular' is your word. It's up to you to say what you mean by it. There are particular contradictions and we can generalize about them. One su...

Again: We derive that a set of premises G implies a sentence ~P by using any of the members of G as lines, then entering P on a line, then deriving a ...

There's no metabasis (change). Again, to show a derivation that a set of premises G proves a sentence ~P, we may use any of the members of G as premis...

Thank you for recognizing my point. / For any A, we have these possibilities: (1) There is a contradiction Q such that A implies Q. (2) There is a con...

The system S could be inconsistent, in which case, if "S is consistent" is expressible in the language of S, then S proves "S is consistent" even thou...

What does "absolute sense" mean? Godel-Rosser is that system of a certain kind don't prove their own consistency. That doesn't entail that there are n...

RAA is an inference rule. A sentence Phi is truth functional if and only if the truth or falsity of the sentence is a function of the assignment of tr...

I would not accept that reading, for the reasons I mentioned several posts ago. Most briefly: Yes, if P implies Q & ~Q, then P implies a contradiction...

What I say is that "P implies a contradiction" is not a translation of "P -> (Q & ~Q).

To prove a negation, we must have a rule to use to do that. And any alternative (that adheres to soundness and provides for the completeness of the ca...

The inference rules don't opine as to falsity. Rather, syntactically, when a contradiction results from a conditional premise, then the rules allows e...

Modus tollens is an inference rule or axiom, depending on the system. That's syntactical. The notion of falsity is semantical. Syntax and semantics wo...

To add to my remarks about the number of posts and length of posts. A reply to a short post or to a small portion of a post can be long because of the...