OK, but you said there's a difference between truth and provability.
Exactly. That's Godel's point! So if those statements are true, they are NOT true because they are provable.
In formal logic that property is called satisfiability. There exists an equivalence relation between (1+1=2) and ⊤ such that (1+1=2) and ⊤ belong to the same equivalence class.promethean75 wrote: ↑Mon Nov 18, 2024 3:40 am "What makes an evaluation "true"?"
OMG, what makes it true is what meets the agreed-on criteria established between two or more conventional language users.
Religions are what mankind has formed from wo/men having experiences of the divine within various cultures and then, rather crudely formed said religions.Dr Faustus wrote: ↑Sun Oct 27, 2024 5:50 pm What is religion ?
The question seems to be trivial. When we talk about religions, every one knows what it refers to : Christianity, Islam, Judaism, etc.
Why?
Soundness of PA is provable from ZFC:
The soundness of a theory is provable in a metatheory of which the soundness is provable in a meta-metatheory until at some meta-level the soundness can only be assumed.ChatGPT: Does model theory assume the soundness of Peano Arithmetic?
In model theory, the soundness of Peano Arithmetic (PA) is typically not assumed as part of the foundational framework. Instead, the soundness of PA (i.e., the property that if a statement is provable in PA, then it is true in the standard model of natural numbers N) is often proven within a stronger meta-theory. This meta-theory is usually something like Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), which is strong enough to handle such meta-theoretical results.
ChatGPT: Does model theory assume the soundness of ZFC?
To establish the soundness of ZFC, we need to rely on a stronger meta-theory, such as second-order arithmetic, classical higher-order logic, or other robust formal systems that are capable of proving the consistency and soundness of ZFC. For example, within ZFC itself, one cannot prove the soundness of ZFC, due to Gödel's second incompleteness theorem (ZFC cannot prove its own consistency).
ChatGPT: Does model theory assume the soundness of second-order arithmetic?
Meta-theoretical frameworks such as Zermelo-Fraenkel set theory (ZFC) or higher-order set theory (including systems like Second-Order Set Theory) can be used to prove the soundness of second-order arithmetic, provided they are assumed to be consistent.
ChatGPT: Does model theory assume the soundness of Second-Order Set Theory?
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That's not true. You are equivocating equisoundness with soundness.
I know. So if soundness is merely assumed why are you claiming that Provable(X) = >True(X) ?
All you get is infinite (meta)regress.
I guess my theory predicted this, huh?
No, the problem is caused by Carnap's diagonal lemma. For each predicate, there always exists a true sentence for which the predicate is false or a false sentence for which the predicate is true. Hence, truth cannot be defined as a predicate.
Yes, Tarski's convention T has the same problem. All the problems that the use of a metalanguage solves in the object language simply reappear in the metalanguage.
Skepdick So 1+1=2 if true; and 1+1=2 is NOT true because it's provable - why is it true?
I know.
Tarski's workaround for the undefinability of the truth as a predicate in the object language is to define this predicate in the metalanguage.
This does not solve, however, the problem of defining a truth predicate that operates on sentences in the metalanguage itself. This can only be achieved in the meta-meta-language.Google: AI Overview
In Tarski's theory of truth, the "truth predicate" - a term used to denote whether a sentence is true or not - is considered a predicate that exists solely within the "metalanguage," meaning it is used to talk about the sentences of the "object language" and cannot be expressed within the object language itself; this is crucial to avoid paradoxes like the liar's paradox.
Key points about Tarski's truth predicate:
Object language vs. metalanguage:
The object language is the language being analyzed (e.g., a formal logic system), while the metalanguage is the language used to talk about the object language.
Avoiding paradoxes:
By placing the truth predicate in the metalanguage, Tarski prevents self-referential statements that could lead to contradictions like "This sentence is false".
Convention T:
The core principle of Tarski's theory is often referred to as "Convention T," which essentially states that a sentence in the object language is true if and only if the metalanguage predicate "is true" applies to that sentence.
I know. Hence my question. About the problem-predicate..