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Nam Nguyen wrote:
Provability couldn't be regressed forever; it has to stop with axioms.
Similarly, we don't have infinite knowledge of all truths; we have to
start from certain presumed knowledge/truth. So yes, it's a trivial
observation: too trivial to even say it, imho. As for what "absolute"
is supposed to be, it's supposed to be *not* relative.
How informative.
For instance,
the truth of G(PA) is relative, hence is not absolute. Similarly, so is
the truth of x=x.
How is the truth of the Gödel sentence of Peano arithmetic any more
relative than any other arithmetical truth of similar complexity?
But a lot of mathematics is also *not* about arithmetic; so Virgil's "Axioms are
not required to be true" is true - and shouldn't be protested with a
purely subjective statement as "It's a tedious and pointless task
to derive conclusions from some random bunch of axioms unless the
axioms are taken to be true "
Certainly in mathematics there are axioms that are not "required to be
true" - it makes no sense to say that the group axioms are true or
false, for example. But even outside arithmetic people happily accept
e.g. that every set has a powerset. Simple arithmetical statements are
illustrative in another respect, though: while people may claim that set
theoretical statements are "meaningless" or that their truth means
provability in ZFC, everyone - apart from such hard-headed anti-realists
as Wittgenstein - accepts that in mathematics arithmetical claims such
as "every even integer greater than two is the sum of two primes" are
meaningful and that their truth does not amount to provability in this
or that theory. It can then be observed that highly infitistic
assumptions - the existence of an inaccessible cardinal, the existence
of V_omega+omega - have arithmetical consequences and thus one can't
simply disregard the question of their truth - which, after all, in
practice is the only guarantee we have that their arithmetical
implications are true.
--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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