So prima facie there doesn't seem to be a problem with saying that a sentence is true iff it is provable – it's just that a second language is what proves the sentence. This would seem to fit with Tarski's claim that a metalanguage is required to make sense of an object language's truth. — Michael
I think it gets a bit dicey; if you can assume anything that doesn't trivialise proof in a system as part of the system as an axiom, and a system proves all its axioms, this trivialises truth. I think such a statement "true iff provable" has to be thought of in terms of "P is true iff there exists system C which proves P", but so long as you can conjure a C which has P as an axiom, P is true since it is by assumption provable in C. — fdrake
I think such a statement "true iff provable" has to be thought of in terms of "P is true iff there exists system C which proves P", but so long as you can conjure a C which has P as an axiom, P is true since it is by assumption provable in C. — fdrake
It might be that "truth as provability" trivialises truth, but do we have any reason to believe that truth isn't trivial? — Michael
(which is why second order arithmetic doesn't trivialise the theorem but this second system does). — Michael
Trivial truth makes 1+1=3. — fdrake
Trivial truth also makes "truth is trivial is false" true.
If we had a formal system that used the Peano axioms to define the numbers and addition but also axioms that entailed that 1 + 1 = 3 then we'd have an inconsistent formal system, and so could dismiss it on those grounds. — Michael
If you wanna dismiss a formal system for being inconsistent - ultimately the reason you'll do that is because it trivialises truth. — fdrake
Isn't true but unprovable a contradiction? — TheMadFool
Imagine a game of chess with a rule that says that pawns can only move forward 1 square and a rule that says that pawns can only move forward 2 squares. I'd dismiss it as unplayable, not because I think that it "trivialises the truth" of the rules. — Michael
A true contradiction is one which is true and false. You're not gonna get the same notion of contradiction out if you jettison truth. AFAIK paraconsistent logics don't jettison truth, they modify how it behaves in a manner that limits explosion/trivialism. — fdrake
the reason I might dismiss a system that allows me to prove p ∧ ¬p or that has the single axiom p isn't "ultimately ... because it trivialises truth" but because it isn't usable. — Michael
As another example, how about "there is a number greater than zero and smaller than every real number". Is it true or false? Does it even make sense to say that it's either true or false? Or is it more correct to say that it's provable if using the hyperreals and that its inverse is provable if not? — Michael
Refer to my reply to ↪sime for a clearer sketch of why truth is not provability, and which has nothing to do with G or whatever. And, again, you're evading my point that, if you want to reduce A to B, the mere fact that B is sufficient to A is not enough: you must also show that B is necessary for A. But this is just what Gödel's theorems deny.
Your juridical reasoning may be convincing in the intuitive level, but, as Williamson has said, in these matters it is not enough to argue impressionistically. As counter-intuitive as it might be, it is demonstrable that truth (on most reasonable axiomatizations of the notion, anyway) is not redundant and is not the same as provability. — Nagase
You've made a good point. Godel's theorems depend on proof not being necessary for truth but the catch here is Godel, like all mathematicians and logicians I suppose, seems to have felt the need/necessity to prove his incompleteness theorems. Why didn't Godel simply state his theorems without proof? — TheMadFool
1. Proof is necessary for truth [Godel assumes and thus proves his incompleteness theorems]
2. The incompleteness theorems proves that proof is unnecessary for truth
3. Proof is unnecessary for truth (from 2)
1 and 3 contradict each other, no? This is a meta-cognitive statement regarding Godel's thought processes. — TheMadFool
You seem to be confusing knowledge with truth. Obviously, to establish a proposition as true, I need to, well, establish as such. And, in mathematics, to establish a proposition as truth---i.e. to know it---is precisely to prove it. But there can be true propositions that are unknowable, and hence that we cannot establish, and hence that we cannot prove. That there are such propositions is established by Gödel's theorems. Again, note that, in the form I have presented, the theorem does not rely on pointing to one such proposition and saying "This proposition is true, but we can't know it as true". Rather, it proceeds from general properties about truth and provability to show that these concepts must come apart. — Nagase
There is no contradiction. One can hold that proof is necessary to establish truth, yet hold that it is not necessary for truth (cf. my point about unknowable truths). And, in the form I have presented, his theorem does not require him to establish the truth of any unknowable proposition. — Nagase
What are you about, Madfool? You're on notice - been told repeatedly - that the proof in question is a meta-mathematical proof, and that the unprovability is a consequence of the particular system. Failure to grasp this means you don't understand, and apparently don't want to understand, anything Godel.Secondly, the incompleteness theorem itself speaks of a statement that's true but unprovably so — TheMadFool
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