Re: Godel did not destroy the Hilbert Frege Russell programme

On Mar 11, 3:38 pm, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:
you say

Let's go back up a few posts. A little while ago I wrote this:

i say
the point was
you say you agree with the quote so

 so what are these capitals in the quote

 In the Introduction to the second edition of Principia, Russell
REPUDIATED Reducibility as 'clearly not the sort of axiom with which
we can rest content'...Russells own system WITH OUT  reducibility was
rendered
  incapable of achieving its own purpose"

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That doesn't refute anything I've said.

Anyway, never mind all that. I've just thought of an interesting
point.

Suppose we let PM* be the system discussed in the *introduction* to
the second edition (not the main text). That's the ramified theory of
types with the axiom of extensionality (and we'll throw in the axiom
of infinity as well), but not the axiom of reducibility. Now, not all
the primitive recursive functions are representable in this system.
The elementary recursive functions are, but that may not have been
known in 1931 (and I myself have never read a proof of this fact). To
clarify whether it was known at the time, we should check out the
second appendix to the second edition of PM, where Russell discusses
which instances of induction can be justified in this system. We
should also look at the more recent work showing that EFA can be
interpreted in this system.

So suppose you formulated your objection as follows: "Let PM* be the
system discussed in the introduction to the second edition. Goedel did
not show in 1931 how to give a finitary proof that if PM* is omega-
consistent, it is incomplete."

Then you might actually have a point. How about that? :)

I'll look into the matter further for you.
.

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