Re: Internet, Goldbach conjecture, cranks and anticranks.

Denis Feldmann wrote:
Aatu Koskensilta a écrit :
It is certainly true that ZF can't be proved inconsistent in ZF - all of the axioms of ZF are true,

How do you know? (in fact, how could you know?)

I accept the basic set theoretical principles - finding them obviously true under the conception of the hierarchy of sets underlying modern set theory -, and from them the soundness of the formal theory ZFC trivially follows.

Of course, anyone is free to doubt the basic principles of set theory, but then there is no particular reason for him to care about ZFC or to be interested in proofs of ordinary mathematical statements that use these principles. ZFC is not special in this regard, as Rudolph notes - we could with equal justification claim that no one can know that the axioms of PA are true, or even that no one can know anything at all in mathematics. It just happens that as an empirical fact most mathematicians accept the basic set theoretical principles, in the sense that they freely employ them in proofs of e.g. simple arithmetical statements that can't possibly be regarded "meaningless", or however one might wish to dodge the question of the truth (or falsity) of the principles.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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