Bruno Marchal wrote:
provable(p) does not entail provable(p) and true(p)
This should be astonishing, because we have restricted ourself to correct machine, so obviously
provable(p) entails the truth of p, and thus provable(p) entails "provable(p) and p"; so what ????
What happens is incompleteness; although provable(p) entails true(p), the machine is unable to prove that.
That is the correct machine cannot prove its own correctness. By Tarski (or Kaplan &Montague 1961)
such correctness is not even expressible by the machine (unlike provability and consistency).
But, (and that's what the "meta" shift of level makes it possible); we can define, for each proposition p, a modal connective knowable(p) by "provable(p) and p". Accepting the idea that the first person is the knower, this trick makes it necessary for any correct machine to have a different logic for something which is strictly equivalent for any omniscient outsider. In some sense this explains why there is necessarily a gap between (3-person) communicable proof and (1-person) non-communicable (as such) knowledge.
Why can't the machine just assume that it is correct, until proven otherwise? If its deductions continue to work ( to correspond
to its oberved reality), and it gains an ever growing set of larger and larger and more and more explanatory
theories through induction and abduction, what's wrong with the machine just assuming without deductive evidence (but rather
through a sort of induction about its own meta-level) that it is logically sound and a reliable observer, individuator, conceptualizer
I think the incompleteness issue is a limitation of the meaning of the concept of truth. Just like "speed" and "time" are
concepts of limited range (speed is no use at lightspeed, time is no use (ill-defined) at the big bang) so truth itself, as
a relationship between representative symbols and that which is (possibly) represented, is probably a limited
concept, and the limitation has to do with limits on the information that can be conveyed about one structure
about another structure. Clearly an embedded structure cannot convey all information about both itself and the
rest of reality which is not itself. There is not enough information in the embedded structure to do this.
So we should just live with incompleteness of formal systems of representation, and not worry excessively about
an absolute all-encompassing complete notion of truth. I don't think such a grand notion of truth is a well-formed
This is so important that not only the knower appears to be variant of the prover, but the observables, that is: physics, too.
But that could lead me too far now and I prefer to stop.
Yes, ok. And indeed evolutionnary theory and game theory and even logic are sometimes used to just put that difference under the rug making consciousness a sort of epiphenomenon, which it is not, for incompleteness is inescapable, and introspective machines can only build their realities from it. All this can be felt as highly counter-intuitive, but the logic of self-reference *is* counter-intuitive.
What is one PRACTICAL consequence of a machine only building its reality-representation using incomplete representation?
Only that the machine can never know everything? Well come on, no machine is going to have time or space to know anywhere near
everything anyway, so what's the big fat hairy deal?