G (Godel sentence) = the mathematical statement with Godel number xyz is unprovable — TheMadFool
Yet it isn't equivalent to Liar Paradox or perhaps in this case to Russell's paradox. This is something that some writers erroneously think. The self refence doens't fall into a vicious circle (as Russell put it), even if there's a statement / Gödel number referring to a Gödel number.Last but not least, Godel admits that his method of proof has a connection with the Liar Paradox which, to me, should set the alarm bells ringing. — TheMadFool
Things that may be true, but may not be provable:
1. There is a God
2. String theory — jgill
So it avoids retorts of the sort "but we just prove [whatever] in a higher system". — Nagase
Putting these together, we obtain the following: (1) Being provable is definable in PA, but (2) being true is not definable in PA. On the other hand, we have (3) if something is provable, then it is true. Hence, since, by (1) and (2), being provable and being true are different, and, by (3), everything that is provable is true, we have that there must be something that is true without being provable. In other words, PA is incomplete. — Nagase
Also, proof of a proposition is necessary to claim that that proposition is true. — TheMadFool
question might not be so straightforward... — SophistiCat
Does this observation apply exclusively to TheMadFool or does it also range over others? — TheMadFool
Does it matter? Do you hold your intelligence contintgent on the intelligence of people you are arguing with? — SophistiCat
The rules of math were designed by human beings, so don't we design the proof and the truth if we are the designers of the system? What is the proof and truth of how to spell the word, "cat"? Humans didn't design the universe therefore truth and proof of some state of the universe will be difficult to come by however humans did design the means to represent and communicate our experiences of the universe, so any truth or proof would be inherent in the rules we've dictated.For instance, if you substract a positive number from a larger positive number, you get a positive number. There is no proof for this, despite it ensuing from mathematical axioms. No proof, but then again, nobody can find a counter-example for it, either. — god must be atheist
One clear lesson is that claims about truth need to be formulated with extreme precision, not out of knee-jerk pedantry but because in practice correct general claims about truth often turn out to differ so subtly from provably incorrect claims that arguing in impressionistic terms is a hopelessly unreliable method. Unfortunately, much philosophical discussion of truth is still conducted in a programmatic, vague, and technically uninformed spirit whose products inspire little confi dence.
Philosophy can never be reduced to mathematics. But we can often produce mathematical models of fragments of philosophy and, when we can, we should. No doubt the models usually involve wild idealizations. It is still progress if we can agree what consequences an idea has in one very simple case. Many ideas in philosophy do not withstand even that very elementary scrutiny, because the attempt to construct a non-trivial model reveals a hidden structural incoherence in the idea itself. By the same token, an idea that does not collapse in a toy model has at least something going for it. Once we have an unrealistic model, we can start worrying how to construct less unrealistic models.
Godel sentences do not support the colloquial interpretation they receive, that is to say "G is true because G is unprovable, assuming Peano Arithmetic is consistent". This is because
i) Firstly, the hypothesis that PA is consistent is potentially falsifiable, but it can never be verified. Therefore it is irrational for logicians and philosophers to assume or even talk about PA's 'infinitely complete' hypothetical consistency. Instead, they should only talk about PA's consistency within a limited finite scope of derivations.
ii) PA cannot talk about what isn't provable in PA, which is the central conclusion of Godel's first incompleteness theorem, and corresponds to both the Halting Problem and Tarski's undefinability of truth within PA. In general, for any sentence S of PA we have — sime
Let us suppose that everything you say is true. This still does nothing to address two facts: (1) the set of true formulas is not arithmetically definable, but the set of provable formulas is, whence the two must be distinct; — Nagase
(2) truth is not conservative over PA, whence it can't be redundant. I sketched that argument in my first post here precisely so we did not get entangled in fruitless discussions about how we can know that G is true or about the Kirby-Paris theorem. — Nagase
Obviously, that particular argument assumes the soundness of PA, which you have disputed (this is a minority position, but one that I respect, if only because in the case of Nelson it generated some interesting mathematics). But this is not necessary for the argument to go through: one can start with Q and argue that any recursively axiomatized theory that extends Q will fall into the same problem, namely truth will be arithmetically undefinable and theoremhood will be arithmetically definable. Since no one that I know of doubts the soundness of Q (not even Nelson), the argument should go through. — Nagase
By the way, if your ii.c) is correct, then PA is inconsistent. In any case, that is not a valid substitution instance of ii.a): ii.a) says merely that (assuming soundness) PA |- S iff PA |- Prov('S'), not that PA |- S <-> Prov('S') (the latter is a reflection principle and is actually not provable in PA). — Nagase
Unfortunately, from the fact that provability is sufficient for truth, it does not follow that it is necessary for truth (in general, being a sufficient condition is not, well, sufficient for being a necessary condition) — Nagase
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