Re: Provable in T?


george wrote:
Rupert wrote:
Here you need to pause to divulge your professional understanding
of the DIFFERENCE between G_T and Con(T).
For civilians, G_T *is* Con(T).

G_T is the diagonalization of the predicate "the diagonalization of the
predicate with Goedel number x is unprovable".

Why you want to allege that this in some important sense might
DIFFER from Con(T) is mysterious,

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, but that is not an obvious fact, it took
someone smart like Goedel to see it. It depends on the
sigma-1-completeness of T.

since ANY statement alleging the
unprovability of ANY other statement is equivalent to the consistency
of T,
since, by definition, "T is inconsistent" means that T proves
everything.
Whether you instantiate the unprovable statement to something simple
or something complicated is NOT the important point, at this
point. The important point is that it does NOT matter what you
instantiate it to.

Fine. But here's what happened: I said G_T will be a theorem of
T+Con(T). You said this is trivial because G_T is Con(T). On some
standard definitions of G_T, and on the definition I was using, that's
not right.

.

Relevant Pages

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  • Re: Provable in T?
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