You could fix the interpretation and change the axioms to show that truth depends on the axioms plus the interpretation. — fishfry
Disjunction is inclusive, but it is never the case that both of these are true: "P is true and Q is false" and "P is false and Q is true". — TonesInDeepFreeze
In general, a disjunction 'phi or psi' might not allow 'phi and psi', depending on the content in phi and the content in psi. — TonesInDeepFreeze
∃ phi or ∃ psi, or both exist.
People can decide for themselves what is too technical or not. — TonesInDeepFreeze
I don't know what that means. — TonesInDeepFreeze
The consequence is that nobody reads what I just wrote. I could as well not write it at all ... — Tarskian
It doesn't matter whether S is an axiom or not. The definition doesn't mention 'axiom'. — TonesInDeepFreeze
Mathematical truth is always:
Axioms plus an interpretation. — fishfry
It is not the case that 'mathematical truth' means ''axioms plus an interpretation' — TonesInDeepFreeze
So you didn't write what you meant regarding S and F. — TonesInDeepFreeze
You should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languages for certain theories. — TonesInDeepFreeze
You should generalize over formulas in those languages and not over properties (since there are properties not expressed by formulas). — TonesInDeepFreeze
I wrote:
"There are sentences that are like this. There are sentences that are like that. Both could exist." — Tarskian
You should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languages for certain theories.
— TonesInDeepFreeze
It's related to PA or similar. That is always implied. — Tarskian
it is about properties that have formulas in PA. — Tarskian
Mentioning all of that, including the above, will make the entire explanation impenetrable. — Tarskian
I just refer to a link that contains all these details — Tarskian
My rendition is not suitable for a mathematical forum, but I had hoped that it would be for a philosophical one. — Tarskian
Wikipedia articles about mathematics are too often incorrect, inaccurate, poorly organized or poorly edited — TonesInDeepFreeze
Quora — jgill
Quora — TonesInDeepFreeze
And your claim about ZF\I is incorrect. ZF\I is not bi-interpretable with PA. Rather, it is (ZF\I)+~I that is bi-interpretable with PA. (Actually, we can simplify to (Z\I)+~I, which is bi-interpretable with (ZF\I)+~I and bi-interpretable with PA.) — TonesInDeepFreeze
On interpretations of arithmetic and set theory
Richard Kaye and Tin Lok Wong
School of Mathematics, University of Birmingham, Birmingham, B15 2TT, U.K.
Notre Dame Journal of Formal Logic Volume 48, Number 4 (2007), 497-510. doi:10.1305/ndjfl/1193667707
This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way.
Two opposing opinions. Here is a discussion on Quora. — jgill
In other words, Wikipedia articles tend to be written in technical jargon that is impenetrable to non-initiates.
Many of those original articles should be scrapped entirely and rewritten by a knowledgable scholar.
That is a deep misunderstanding. — TonesInDeepFreeze
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 the theorems from G) is different from the class of models for H (thus for all theorems of H). So let's say S is a member of G or a theorem from G, and S is inconsistent with H. Then, yes, of course, S is true in every model of G and S is false in every model of H.
But, given an arbitrary model M, whether S is true in M is determined only by M. — TonesInDeepFreeze
Even worse than Wikipedia, which much too often is, at best, slop. Quora is close to the absolute lowest grade of discussion. It is a gutter of misinformation, disinformation, confusion and ignorance. Quora is just disgusting — TonesInDeepFreeze
with the infinity axiom negated (ZF−inf)
better deep than shallow. — fishfry
I discussed this with myself and determined I'm right. — fishfry
I prefer not to argue the point, if you'll forgive me. — fishfry
ZF-Inf is bi-interpretable with PA. And as I said, so is Z-Inf. — TonesInDeepFreeze
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