Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable" — Wikipedia
reflexivity — Wayfarer
If L is true then L is false = L is false or L is false = L is false — TheMadFool
My disagreement with Pfhorrest seemed to perhaps hinge on his use of the term "definitely true". He hasn't responded to say whether he would claim that "There's never a statement in any given language that is both true according to the rules of that language and also not provable in that language, because to be true according to the rules of a language just is to be provable in that language." (The statement he made leaving out the word "definite"). — Janus
we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part. — Pfhorrest
It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that — Pfhorrest
I too will plead, as at this time, especially as I am rusty in the subject, I wouldn't be able to marshal enough knowledge to explicate the details of the mathematics mentioned there. — TonesInDeepFreeze
It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system. — TonesInDeepFreeze
Leaving out the "definitely"s completely changes the meaning, so no, I wouldn't claim that modified sentence.
We cannot know for sure ("definitely", or "certainly") that some proposition is true, without in the process having proven it, so we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part.
It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that, because in somehow saying for sure that something was true, we just would be proving it. — Pfhorrest
that lower-level system has no idea whether or not it's true (because it's unprovable) — Pfhorrest
I might be able to offer an opinion as to whether Godel has proved that there are true propositions within mathematics that cannot be mathematically proven. — Janus
Is there any proof that such a "formal system that incorporates methods going beyond ZFC." will or even could be found? — Janus
what you [Pfhorrest] said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable. — Janus
Would this fact render all such proofs non-exhaustive and/ or trivial — Janus
It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold. — TonesInDeepFreeze
What gets to me, and maybe you can clarify, is how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven” — Pfhorrest
I seem to recall reading somewhere that Gödel was a mathematical Platonist. Are you suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language? — Janus
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F. — Raatikainen 2015
There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability. — TonesInDeepFreeze
At the end of Sextus’ discussion in PH II, he clearly signals, as one would expect, that he suspends judgment on whether there are criteria of truth:
You must realize that it is not our intention to assert that standards of truth are unreal (that would be dogmatic); rather, since the Dogmatists seem plausibly to have established that there is a standard of truth, we have set up plausible-seeming arguments in opposition to them, affirming neither that they are true nor that they are more plausible than those on the contrary side, but concluding to suspension of judgement because of the apparently equal plausibility of these arguments and those produced by the Dogmatists. (PH II 79; cf. M VII 444) — Stanford Encyclopedia of Philosophy
According to Chisholm, there are only three responses to the Problem of the Criterion: particularism, methodism, and skepticism. The particularist assumes an answer to (1) and then uses that to answer (2), whereas the methodist assumes an answer to (2) and then uses that to answer (1). The skeptic claims that you cannot answer (1) without first having an answer to (2) and you cannot answer (2) without first having an answer to (1), and so you cannot answer either. Chisholm claims that, unfortunately, regardless of which of these responses to the Problem of the Criterion we adopt we are forced to beg the question. It will be worth examining each of the responses to the Problem of the Criterion that Chisholm considers and how each begs the question against the others. — Internet Encyclopedia of Philosophy
how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven” it. — Pfhorrest
Something that is assumed to be so, but can’t be proven to be so? Isn’t that what the issue is about? — Wayfarer
Godel's about whether there are things that are true but aren't provable. — Pfhorrest
Godel only shows something about the relationship between a formal system and statements in it: that some systems can't prove some things they're capable of talking about either way, even though we can know, through in a proof made in a higher-level system, that those things are true. — Pfhorrest
I’m afraid to say that you’re [Pfhorrest] splitting hairs. — Wayfarer
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
— Raatikainen 2015
Here the statement is not said to be either true or false [...] in the context of system F the Gödel statement is neither true nor false, since it can be neither proved nor disproved in that system. — SophistiCat
There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. There's a subtle difference: A theorem of the object-theory is true in any model of the theory, so in that sense one would say that the object-theory does determine the truth of certain sentences. But the object-theory does not itself have a theorem that the sentence is true in models of the theory (or else, the object-theory would be inconsistent per Tarski's theorem). — TonesInDeepFreeze
realist/Platonist — SophistiCat
It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system. — TonesInDeepFreeze
we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold. — TonesInDeepFreeze
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