TonesInDeepFreeze

A more general subject: entailment. It occurs to me that perhaps a good way to express the advantage of the rigorous model theoretic notion of entailm...

In mathematical logic, 'valid' is used differently from the way you use it. Here's a quick breakdown of the terminology for ordinary first order logic...

In mathematics (and often crossing into philosophy) there is the subject of mathematical logic in which the notions of truth and provability are given...

I am not familiar with Wittgenstein's views on Moore's paradox.

"This square is not a square" is seen as a self-contradiction on its face, and its truth value is falsehood, and there is no contradiction in saying i...

But it's not gibberish. It's syntactical and it talks about the property of truth as pertaining or not to a given sentence, which is a well understood...

Yes, I didn't write that correctly. What I meant: Classical logic with added mathematical axioms works just fine for a vast amount of the mathematics ...

Rather than get bogged down in whatever vagaries there might be in the Epimenides paradox, I would suggest the clearer, simpler, mathematically "trans...

Of course, if we wish to have theorems that are contradictions but without explosion, then classical logic doesn't work, and if one wishes to have con...

Yes, so?

There are systems with all three: LNC, contradictions, and non-explosiveness. You can look it up yourself; you can educate yourself about this subject...

With a paraconsitent logic, one can have both LNC and non-explosiveness. In such a logic, we may have LNC as a theorem (or theorem schema) and also ha...

In the sense you mention a 'truth predicate', we actually say a 'truth function'. On the other hand, as to truth predicates, (Tarksi) for an adequatel...

(1) I don't know your meaning of 'homological' applied to relationships between a mathematical theory and empirical observation. (2) There are two sen...

So there you have my reply. What was your point in asking your question?

Agent Smith is ignorant of how it actually works in formal mathematics.

Yes, and Agent Smith ignores the most obvious choice.

For there to be a culprit there needs to be a misdeed. It's a bizarre view that the culprit is not the one irresponsibly spreading misinformation but ...

That sounds like dialetheism. Paraconsistent logic is characterized by the absence of the explosion principle. What Graham Priest text do you refer to...

I gave you copious explanation why you are wrong about that. You are blatantly wrong about it. I cannot fathom what reward you find in posting blatant...

I'm going back over concepts in notes I already gave you. As I mentioned for other examples, your question depends on whether we are working with pred...

It is not difficult. excluded middle: P or not-P non-contradiction: not(P and not-P) bivalence: (P or not-P) and not(P and not-P) So bivalence is just...

You propose that there are closed well formed formulas that are meaningless (have no interpretation or the valuation function also has meaningless in ...

Also, aside from providing semantical interpretation, and myriad other result in model theory, we use models for consistency proofs, relative consiste...

Also, I overlooked that you said "not adopting". So I added more response accordingly.

That is incorrect. No matter about models, if you have inconsistent axioms, then you derive Russell's paradox. Then, it is merely an additional note, ...

Per the valuation function for truth in models (the Tarski definition by recursion on formulas), every sentence is either true in the model or false i...

You know the derivation of Russell's paradox, right? Assume EbAy(yeb <-> ~yey) "There exists a set b such that for all sets y, y is a member of b if a...

See post above that I added to. EbAy(yeb <-> ~yey) is false in not just some models but it is false in every model.

The schema of unrestricted comprehension specifies an infinite number of axioms: If F is a formula and b is a variable that does not occur free in F, ...

It would be false in some models if it were formalized as a first order sentence, or, for a schema, it would have false instances if the schema were f...

Sure, where the set theory is not formalized with axioms, we can at least point out that the pre-formal principles it uses are non-logical, at least i...

Cantor didn't have axioms. But of course he did use non-logical principles even if not formalized as axioms. Except for the pure predicate calculus it...

'throw in', 'bag', and 'fill' (in your context) are not mathematical terms, so I can't give you a mathematical answer to your question. However, the m...

Those are your personal, impressionistic locutions. Real analysis doesn't have such terminology. The real continuum is constructed in formal axiomatic...

That's wrong. This is correct: Hx for "there is an x such that x is a Scotland Yard detective named 'Sherlock Holmes'". ~ExHx for "there does not exis...

The proof shows that there is no enumeration of the set of real numbers. As formalized, the proof uses only first order logic from the axioms of Z set...

That's wrong. This is correct: Mx for "x is maximally great" ExMx for "there is an x such that x is maximally great", which is to say "a maximally gre...

If I'm not mistaken, there is work in combining formal paraconsistent logic with formal fuzzy logic. But fuzzy logic itself is not a formalization of ...

Each axiom of group theory is a non-logical axiom.

Good. You are at your most eloquent with emojis.

Explaining that ZFC does not have unrestricted comprehension that yields the existence of the Russell set is not silly game playing.

I said exactly what is wrong with what you said just a few posts ago! And I explained also in posts yesterday. And from months ago I've explained why ...

Historically, infinitesimals were not given a rigorous treatment. However eventually non-standard analysis was devised in which infinitesimals are con...

As your posting history suggests, your thousand apologies will be followed by a thousand more of your egregiously misinformational posts. One only has...

The diagonal argument does not require reductio ad absurdum. And 'negative self-reference' needs a definition. The diagonal argument is constructive a...

Let's select that quote in particular. It is pure misinformation. ZFC, which is the common set theory for mathematics, is formulated so that it does n...

I explained why you are incorrect. You are terribly mixed up and you don't know what you're talking about. And you add additional confusions and misin...

First, I see a purported definition of U that mentions U in the definiens, which is circular. Second, I see T mentioned in the definiens of a definiti...

I read the post. Then I went back to the first place that, as far as I can tell, he doesn't make sense. His theory U is not defined; his proposed defi...