Re: Provable in T?


george wrote:
Rupert wrote:
You were saying the proof of G_T in T+Con(T) is a one-line proof. The
way I define G_T, that's not the case. It is true that G_T is
equivalent to Con(T) in T,

In WHICH T??
There are a lot of DIFFERENT T's to which Godel's
1st incompleteness theorem applies. I still am not
sure you even know what a T *is*.


My statement is true for any recursively enumerable extension of
Sigma-1-Induction Arithmetic. Believe you me, there is nothing you know
that I don't know.

but that is not an obvious fact, it took
someone smart like Goedel to see it.

Well, having seen it, he proved the SECOND
incompleteness theorem, after which it is abundantly
clear that Con(T) will serve perfectly well in the role
of the thing diagonally constructed in the first one.

It depends on the
sigma-1-completeness of T.

T~Con(T) is NOT sigma-1-complete,

It is. It is just not sigma-1-sound.

but in T~Con(T)+Con(T~Con(T)),
the proof of Con(T~Con(T))
is STILL "it's an axiom".

.