Mikael Johansson wrote:
<....my problem with the God III is that he is supposed
to be able to break against axiomatic laws
without stepping out of that axiomatic system.>
But the consistency of the axioms seems to be undecidable.
And that opens the door to Type III (or not?).
There are axiomatisable theories that are not recursive.
If we accept the Church Thesis, even the First Order Calculus
is actually undecidable. That is to say: we cannot determine
algorithmically whether a logical formula is (logically) valid.
Btw, if a mathematical statement is neither
provable nor refutable, should it be considered
neither true nor false?
Somebody wrote: let us suppose that I have a safe,
but nobody knows the combination to. If I tell that the safe
contains a gold coin (and it really does contain the coin)
then I'm telling the truth, whether or not anyone can prove it.
If it does not contain the gold coin, then I'm not telling the truth.
Of course: it is possible to describe statements that are
either definitely true or definitely false (but whose truth-value
is not knowable). It is also possible to describe statements
that are not either definitely true or definitely false (but in between).
Any way R.P. Feynman wrote: "God is always associated with those
things you do not understand."
- s.
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