Re: Godel did not destroy the Hilbert Frege Russell programme

On Mar 10, 3:04 pm, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:

"In the second edition Whitehead and Russell took the step of using the
simplified theory of types dropping the axiom of reducibility and not
worrying to much about the semantical difficulties"


As I said, this one is simply false. But if it were true, it would in
no way vindicate your point. The logic of Goedel's system P is the
simple theory of types. The simple theory of types is essentially
equivalent to the ramified theory of types with the axiom of
reducibility, in a sense which I defined precisely in a previous post.
If this one were true, it would strengthen the case for saying that
Goedel's system P is essentially the same as PM.

But as I said, it's just wrong. You can't rely on secondary sources.
You've got to read the book. And you can't expect anyone to take you
seriously when you refuse to read the book and continue to respond to
people who have read the book by endlessly referring them to the same
secondary sources. I've explained to you in detail many, many times in
what sense it's true and in what sense it's false that Russell
"repudiated" the axiom of reducibility in the second edition. Why
don't you listen? You might learn something.

Anyway, the issue is trivial. If you don't like the axiom of
reducibility that's fine. Tell us a theory you do like.

For the thousandth time, these are the important points:

(1) You claim that the axiom of reducibility was abandoned in the
second edition of PM. This is not true. I've given you detailed
information about what actually is in the second edition of PM, and
what the truth and lack of truth is in this assertion. You refuse to
listen. I obviously know the book and you obviously don't. Instead of
taking advantage of the opportunity to learn something, you farcically
continue to refuse to read the actual book and refer me endlessly to
the same secondary sources, at least one of which is just plain wrong.
P is essentially the same as PM with the axiom of infinity, in a sense
which I made precise in a previous post. The title of Goedel's paper
is perfectly apt.

(2) While your contributions to the above debate are farcical, the
issue is trivial in any case. The question of what Russell did and did
not do in the second edition is a question of historical interest
only. It is not in any way relevant to the assessment of the merits or
interest of Goedel's proof.

(3) You have repeatedly been given at least two major reasons why the
"validity" of the axiom of reducibility has no bearing on the merits
or interest of Goedel's proof, namely that the proof goes through in a
weak fragment of finitary number theory which even the most extreme
mathematical skeptics accept, and that it applies to a very large
class of systems, finitary, predicative, and impredicative, namely
every system in which a consistent recursively enumerable extension of
Robinson Arithmetic can be interpreted. So your claim that Goedel's
proof only applies to P is just plain wrong. He examines the special
case of P in order to show how his construction works in one
particular case. The theorem would still be of significant interest if
it only applied to impredicative systems, especially since
predicativists are a minority among mathematicians these days, but in
fact it applies to every system which is a serious candidate for
serving as a foundation for mathematics. If you don't like the axiom
of reducibility, you should tell us a theory you do like, as I keep
asking you to, and we can tell you whether or not Goedel's proof
applies and if so, why.

(4) You have never engaged in any way with Goedel's actual reasoning.
You can't find a problem with a proof just by quoting the definitions
it uses. The definitions are in principle eliminable. That can't be
where the mistake is. You've got to find fault with an actual step in
the argument.
.

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