Yes, the theorem itself, as you quoted it, does not mention truth. But from the theorem, we do go on to remark that the undecided sentence is true.
And the statement is neither true or false in the system on an even more fundamental basis than that it is undecided by the system: — TonesInDeepFreeze
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. — TonesInDeepFreeze
When we're talking about plain arithmetical truths, I don't know why we would have to go down the road of wondering about realism. I mean, non-realists still recognize the truth of arithmetical statements. — TonesInDeepFreeze
Mathematical theorems are true or false; their truth or falsity is absolute and independent of our knowledge of them.
Pure mathematics... seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way. — G.H. Hardy
But if we adhere to strict "truth=provability" principle, then the sentence is not true even in the metatheory, if it assigns truth to sentences subject to their provability in the object theory. — SophistiCat
Anti-realists recognize arithmetical statements as true relative to particular mathematical theories — SophistiCat
which are as fictitious as any other such theories — SophistiCat
It depends on the definition of 'mathematically proven'. — TonesInDeepFreeze
what you [Pfhorrest] said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable. — Janus
It is the case that there is an infinite escalation of theories, each proving arithmetical truthts not provable in the lower theories.
Would this fact render all such proofs non-exhaustive and/ or trivial — Janus
The theories are not exhaustive, indeed. But I don't see why that would make the proofs trivial. — TonesInDeepFreeze
for incompleteness, it's not just a matter of having to assume things to prove things. — TonesInDeepFreeze
Pure mathematics... seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way. — G.H. Hardy
It's a matter of quantifier order:
Godel: For any system S of a certain kind, there exist statements undecided by S.
False: There exist statements F such that for any system S of a certain kind, F is undecided — TonesInDeepFreeze
Thanks. I have learned from this thread to avoid discussion of this topic in future. — Wayfarer
What dogmatism do you think you have witnessed? — TonesInDeepFreeze
Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.
(1) Sentences are not true in a language. They are true or false in a model for a language.
(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.
(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.
(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system. — TonesInDeepFreeze
I hate all those Vienna Circle types. That’s one of the reasons I’m going to keep out of these discussions. — Wayfarer
Now, let's look at a certain kind of arithmetical sentence. These are sentences in the language of arithmetic all of whose quantifiers are bounded. — TonesInDeepFreeze
The declared aim of the Vienna Circle was to make philosophy either subservient to or somehow akin to the natural sciences. As Ray Monk says in his superb biography Ludwig Wittgenstein: The Duty of Genius (1990), “the anti-metaphysical stance that united them [was] the basis for a kind of manifesto which was published under the title The Scientific View of the World: The Vienna Circle.” Yet as Wittgenstein himself protested again and again in the Tractatus, the propositions of natural science “have nothing to do with philosophy” (6.53); “Philosophy is not one of the natural sciences” (4.111); “It is not problems of natural science which have to be solved” (6.4312); “even if all possible scientific questions be answered, the problems of life have still not been touched at all” (6.52); “There is indeed the inexpressible. This shows itself; it is the mystical” (6.522). None of these sayings could possibly be interpreted as the views of a man who had renounced metaphysics. The Logical Positivists of the Vienna Circle had got Wittgenstein wrong, and in so doing had discredited themselves. — Wittgenstein, Tolstoy and the Folly of the Logical Positivists
And one may have one's own reasons for eschewing a conversation — TonesInDeepFreeze
Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. — TonesInDeepFreeze
When you reject such, and insist on the other, it's dogmaticism. — Metaphysician Undercover
Let me explain it clearly then, since you seem to be having trouble understanding. — Metaphysician Undercover
argues a philosophy — Metaphysician Undercover
Many principles employed in modern mathematics, axioms, have not been empirically proven. — Metaphysician Undercover
labeling those who doubt these unproven principles as cranks — Metaphysician Undercover
That's condescension coming from a person who can least afford it. — TonesInDeepFreeze
Many principles employed in modern mathematics, axioms, have not been empirically proven. — Metaphysician Undercover
There is a Grand Canyon of difference between, on one hand, doubts and reasoned critique and, on the other hand, attacks from willful ignorance, frothing confusion, and sophomoric dogmatism. — TonesInDeepFreeze
I haven't argued a philosophy. — TonesInDeepFreeze
empirical validation isn't relevant. — Wayfarer
You are arguing a philosophy of truth. — Metaphysician Undercover
empirical validation isn't relevant.
— Wayfarer
That's exactly what makes arguing for mathematics as the purveyor of truth, dogmatism. — Metaphysician Undercover
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