Bruno Marchal wrote:
> Now all (sufficiently rich) theories/machine can prove their own
> Godel's theorem. PA can prove that if PA is consistent then PA cannot
> prove its consitency. A somehow weak (compared to ZF) theory like PA
> can even prove the corresponding theorem for the richer ZF: PA can
> prove that if ZF is consistent then ZF can prove its own consistency.
Of course you meant "..then ZF cannot prove its own consistency."
> So, in general a machine can find its own godelian sentences, and can
> even infer their truth in some abductive way from very minimal
> inference inductive abilities, or from assumptions.
> No sound (or just consistent) machine can ever prove its own godelian
> sentences, in particular no machine can prove its own consistency, but
> then machine can bet on them or "know" them serendipitously). This is
> comparable with consciousness. Indeed it is easy to manufacture thought
> experiements illustrating that no conscious being can prove it is
> conscious, except that "consciousness" is more truth related, so that
> machine cannot even define their own consciousness (by Tarski
> undefinability of truth theorem).
But this is within an axiomatic system - whose reliability already depends on
knowing the truth of the axioms. ISTM that concepts of consciousness,
knowledge, and truth that are relative to formal axiomatic systems are already
to weak to provide fundamental explanations.
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