You could fix the interpretation and change the axioms to show that truth depends on the axioms plus the interpretation. — fishfry

I don't know what that means.

The definition is:

sentence S is true in model M if and only if [fill in the definiens here]

and that definiens doesn't mention 'axiom'.

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

There exist sentences that are true or there exist sentences that are false, or both.

"Or both" means: Potentially, there exist as well true as false sentences.

"Or both" is not about an individual sentence. It is about the fact that both existence clauses could be true, i.e. there are true sentences but also false sentences that satisfy the lemma.

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

I meant to say:

∃ phi or ∃ psi, or both exist.

The term "or both" emphasizes that the "or" is not exclusive. The default interpretation in natural language for "or" is actually exclusive.

People can decide for themselves what is too technical or not. — TonesInDeepFreeze

In my experience, it usually is too technical. The consequence is that nobody reads what I just wrote. I could as well not write it at all ...

I don't know what that means. — TonesInDeepFreeze

If your statement S is an axiom, you will understand my point.

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 ((S is true and F is false) and (S is false and F is true))

That is false.

[EDIT CORRECTION: It is not false. I should have said the third disjunct is false, thus otiose as added to the two other disjuncts.]

If you meant something different, involving 'there exists', then you need to write it.

And just to be clear: The theorem is of the form: For all formulas P(x), there exists a sentence S, such that ....

The consequence is that nobody reads what I just wrote. I could as well not write it at all ... — Tarskian

I'm not nobody. You have a problem with quantifiers.

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 theorem pertains to sentences in certain languages for certain theories.

You should generalize over formulas in those languages and not over properties (since there are properties not expressed by formulas).

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 axiomatizations.

It doesn't matter whether S is an axiom or not. The definition doesn't mention 'axiom'. — TonesInDeepFreeze

You should go back to what I originally said that you objected to, and you will see that I am right.

Mathematical truth is always:

Axioms plus an interpretation. — fishfry

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 [fill in the definiens here]

and 'axiom' is not mentioned.

/

'axiom' is a syntactic notion, not semantical.

A system T is comprised of

a language L
a set G of sentences in the language L (the axioms)
a set of inference rules

That induces a set T of theorems.

That is all syntactical.

'true' is a semantical notion.

a sentence S is true in a model M if and only if [fill in definiens here]

It is not the case that 'mathematical truth' means ''axioms plus an interpretation' — TonesInDeepFreeze

If you fix an interpretation and change the axioms, you'll get different truths. This is obvious. Not worth arguing about.

That is a deep misunderstanding.

An interpretation for a language determines the truth or falsehood of each sentence in the language.

Different axiom sets induce different theorems, hence different theories, but for a given interpretation, what axioms are in a given axiomatization has no bearing on that interpretation and no bearing on which sentences are true in that interpretation.

Again:

'sentence S is true in a model' is semantical.

'sentence S is an axiom for a system' is syntactical.

I suspect that what you have in mind may be put this way:

Given a set of axioms, every model in which the axioms are true is a model in which the theorems from the axioms are true.*

That is the case. But it is not a definition. It is a theorem (the soundness theorem). The definition of 'S is true in model M' does not mention axioms. Again: Given an interpretation (a model) M, the truth or falsehood of every sentence in the language is determined irrespective of what axiom sets there are for different systems.

* We also keep in mind that there are axiom sets such that there are sentences such that neither the sentence nor its negation is a theorem from the axioms, so, if the axioms are consistent, then there are models for the axioms in which a given independent sentence S is true and there are models for the axioms in which S is false.

For example, most trivially:

Let 'P' and 'Q' be sentence letters. Let the only axiom be 'P'. There are two models in which 'P' is true:

P is true, Q is true
P is true, Q is false

So the axiom P does not determine the truth or falsehood of Q.

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.

So you didn't write what you meant regarding S and F. — TonesInDeepFreeze

I did. I wrote:

"There are sentences that are like this. There are sentences that are like that. Both could exist."

There's a lot of syntactic noise associated to specifying F.

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.

You should generalize over formulas in those languages and not over properties (since there are properties not expressed by formulas). — TonesInDeepFreeze

Well, it is about properties that have formulas in PA. That was also implied.

Mentioning all of that, including the above, will make the entire explanation impenetrable.

I just refer to a link that contains all these details but I can pretty much guarantee that few people will ever read it.

My rendition is not suitable for a mathematical forum, but I had hoped that it would be for a philosophical one.

I wrote:

"There are sentences that are like this. There are sentences that are like that. Both could exist." — Tarskian

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. Moreover, even if one did figure out that you meant something different from what you wrote, then it is still appropriate to point out that what you wrote is incorrect as it stands no matter that in your mind you meant something different.

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

No, a person doesn't know that you're not making the generalization that you stated. Instead of saying "logic statements" (and what is a "logic statement"?) it would have been correct to say "sentences in the languages of said theories" or something like that. Or say, "For theories of a certain kind". People are not supposed to guess that you don't mean what you say.

it is about properties that have formulas in PA. — Tarskian

Then you need to say that. People are not required to not take you to mean what you say when you say "all properties".

Actually, all you needed to say is, "all predicates in the language" rather than "all properties" which is not only mathematically wrong but philosophically wrong.

Mentioning all of that, including the above, will make the entire explanation impenetrable. — Tarskian

Actually, it's inpenetrable when you don't specify what you mean but instead put the burden on the reader to suss out what you might mean.

I just refer to a link that contains all these details — Tarskian

Another way of putting it on the reader to then wade through an article to figure out what you mean in a post, when you don't at least say what particular passages in the article you have in mind, or at least say that something like "my thesis depends on the bulk of this article that needs to be understood first". And linking to Wikipedia about mathematics is rank. Wikipedia articles about mathematics are too often incorrect, inaccurate, poorly organized or poorly edited. It's often to the subject matter what fast food is to nutrition.

My rendition is not suitable for a mathematical forum, but I had hoped that it would be for a philosophical one. — Tarskian

Your writing about mathematics is so often incorrect and ill-formed That it is in a philosophy forum and not a mathematics forum doesn't alter that it is so often incorrect and ill-formed.

You leave out crucial points because you think they are too technical. But then people who don't know that there are such crucially needed points are liable to be misled by your bad oversimplifications.

What you call "noise" is actually needed clarity of the signal. What you think is your signal is the sound of a blown woofer.

@Tarskian

Back to this matter:

Whether there are uncountably many truths or whether there are unexpressed truths depends on what is meant by 'a truth'.

In context of mathematical logic, I would take a truth to be a certain kind of sentence relative to a given model. So, for a countable language, there are only denumerably many truths (i.e. true sentences).

Or one might also say that truths are states-of-affairs, such as a certain tuple being in a certain relation is a truth, even if there is no sentence that states that fact about that certain tuple and certain relation. In that sense, yes, for an infinite universe, there are uncountably many truths.

@Tarskian

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.)

[EDIT CORRECTION: I think it is incorrect that (Z\I)+~I is bi-interpretable with PA. This is correct: If every set is finite, then the axiom schema of replacement obtains and (Z\I)+~I = (ZF\I)+~I. But I don't think that works; I was thinking that the negation of the axiom infinity implies that every set is finite. But I think that itself requires the axiom schema of replacement.]

Wikipedia articles about mathematics are too often incorrect, inaccurate, poorly organized or poorly edited — TonesInDeepFreeze

There are over 20,000 articles about math on Wikipedia. My own experience has been that accuracy improves with advanced topics, and I have found that as an introduction to a topic Wikipedia is very good. But I know little of foundations.

"[...] as an introduction to a topic Wikipedia is very good."

I'll fix that:

as an introduction to a topic Wikipedia is very good lousy.

W*kipedia on physics also has some pretty bad articles. Horrible on history. Propagandist on politics.

"[...] as an introduction to a topic Wikipedia is very good."
I'll fix that: as an introduction to a topic Wikipedia is very good lousy. — TonesInDeepFreeze

Two opposing opinions. Here is a discussion on Quora.

Quora — jgill

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.

(1) The layout of the threads is quite illogical and very impractical. The illogical organization style of answers and comments makes discussions incoherent. Like the rest of the site, the design is not to facilitate reading but to add to click counts and ad views. A site is not to be faulted for having ads, but the entire design of Quora is egregiously manipulative. (2) Posters are allowed to delete their posts, which is okay, but deletion of one's posts includes deletion of replies to the deleted posts. Thereby, a poster can wipe out all your replies. (3) Mathematics and logic discussions at Quora are inundated with prolific, persistent, chronic, serial cranks who slater the threads with misinformation, disinformation, confusion and ignorance. It's a disgusting cesspool. It's the dark web of discussion.

The only thing worse is "AI", which can always be relied upon for absurd misinformation and computer generated lies, all under the imprimatur of computing.

Quora — TonesInDeepFreeze

Better than stackexchange nowadays, and it is not as if Quora got better with the years.

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

The original article that establishes and proves the bi-interpretability:

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

I have already linked to this original publication in a previous comment. The abstract says the following:

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.

The official name for the set theory that it is about, is ZF-inf.

As I already have written previously in a previous comment, if is about ZF with the axiom of infinity removed and denied.

I don't know why you believe that the term ZF-inf would be wrong because that is exactly the term that the original authors use, i.e. Richard Kaye and Tin Lok Wong.

As far as I know, Wikipedia does not mention this publication anywhere in connection with bi-interpretability. There are a few placeholder pages on the subject but they look very much like a draft at this point.

Two opposing opinions. Here is a discussion on Quora. — jgill

I agree with the following comment:

In other words, Wikipedia articles tend to be written in technical jargon that is impenetrable to non-initiates.

I disagree with the following comment:

Many of those original articles should be scrapped entirely and rewritten by a knowledgable scholar.

That is only going to make the problem worse.

In my opinion, it is preferable to mention the scholarly publications in the footnotes. That will allow anybody who is interested in the exact technical details to investigate them there.

That is a deep misunderstanding. — TonesInDeepFreeze

Better deep than shallow.

I discussed this with myself and determined I'm right.

I prefer not to argue the point, if you'll forgive me.

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

I'll read and consider his when I get a chance. Thanks for posting it.

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

Another discussion: Are mathematical articles on wikipedia reliable?

StackExchange also has a bad discussion design. And often some confused discussions, But at least as far as math and logic, I have found it to be far better than Quora, which is the pits.

with the infinity axiom negated (ZF−inf)

Ah, that is not a notation I would have thought means "the axiom of infinity negated". I would have thought it means "ZF without the axiom of infinity". The notation with which I am familiar indicates (1) the axiom of infinity is dropped and (2) the negation of the axiom of infinity is adopted. But since the ZF-Inf is also used for that, of course, with that use, ZF-Inf is bi-interpretable with PA. And as I said, so is Z-Inf. [EDIT: cross out previous sentence.]

better deep than shallow. — fishfry

Better deep in knowledge and shallow in misunderstanding. Better deep in love and shallow in hate.

I discussed this with myself and determined I'm right. — fishfry

Then you're discussing with the wrong person.

I prefer not to argue the point, if you'll forgive me. — fishfry

You may prefer whatever you want; there's no need for forgiveness for preferring whatever you like; meanwhile, I prefer to show how you are wrong in saying that 'true' is not defined entirely with the notion of interpretations and not the notion of axioms.

There used to be something called scholarpedia http://www.scholarpedia.org/article/Main_Page
There still is, but it is dead

ZF-Inf is bi-interpretable with PA. And as I said, so is Z-Inf. — TonesInDeepFreeze

[EDIT CORRECTION: I think it is incorrect that (Z\I)+~I is bi-interpretable with PA. If every set is finite, then the axiom schema of replacement obtains, so (Z\I)+~I = (ZF\I)+~I. I was thinking that the negation of the axiom infinity implies that every set is finite. But I think that itself requires the axiom schema of replacement.]