[FOM] Recursion Theory and Goedel's theorems
Charles Silver
silver_1 at mindspring.com
Thu Aug 2 15:55:02 EDT 2007
I want to make a small point, which from my point of view is
significant. Take this statement of Gödel's theorem: IF PA is
consistent, then there is a sentence G that is not provable, nor is ~G.
To me, all allegations that G can be "seen" to be true (without
qualification) neglect the fact that PA must first be "seen" to be
consistent. The theorem itself doesn't refer to truth or falsity at
all (by Gödel's intention). So, one has to "see" or "interpret" G to
be true. As mentioned, though, this is impossible without first
"seeing" or "interpreting" PA to be consistent.
(This small point may be unimportant in the present context, but
unalloyed statements that Gödel's G is true allow people like Lucas
and Penrose to make all sorts of [what seem to me to be] erroneous
claims. [See Putnam and Feferman])
Charlie Silver
> The Pi^0_1 nature of Godel statements is particularly significant,
> as it can be for any other such statement. Because if they are
> found to be
> undecidable (in some basic theory, such as Q), then they are
> AUTOMATICALLY
> true in the natural numbers. This is enormously significant.
>
> ->More important: Goedel's proof includes a procedure which
> ->given an r.e. true theory, provides a true sentence
> ->(which we know to be true if we know that the theory is true!)
> ->which is undecidable in that theory. For me this is a
> ->positive part of the first incompleteness theorem which
> ->is no less important than the negative part.
> ->Does recursion theory provide such a procedure too?
>
> Well, yes, isn't it? It is within recursion theory that "creative
> sets"
> are fully defined and examined; these are precisely the sets of the
> type
> you describe.
>
> -- Bill Taylor
>
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