Re: A missing definition in "Gödel's Proof" by Nagel & Newman (open letter)
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Thu, 01 Nov 2007 06:42:09 -0600
On Wed, 31 Oct 2007 20:20:29 +0100, G. Frege <nomail@invalid> wrote:
On Wed, 31 Oct 2007 15:55:11 -0000, translogi <wilemien@xxxxxxxxxxxxxx>
wrote:
Thanx for your support. Actually, I already got a slap in the face bySend an private email to Douglas Hofstadter
Without this definition the exposition of the system (described in the
book) is simply incomplete (for example, without it not even p -> p
can be derived); and hence imho it should be added to the text.
Actually, a footnote by the Ed. would do.
Hope he will respond
Hofstadter. His reply to my email was as follows:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Hello, and thanks for your note. I appreciate
your interest in "Gödel's Proof" and your
thoughts. There are always things that one
person or other would like to see "fixed", but at
this point the text is "fixed" (in another sense
of the word) and unlikely to be changed. But
thanks for the suggestion. -- Douglas Hofstadter.
You should really get a grip - that's not a slap
in the face, it's a perfectly polite reply. He
even thanks you.
Anyway: Can you tell us exactly what the axioms
and rules of the system in the book are?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Thus I decided "to go public". My point (which Hofstadter kindly decided
to ignore) is that several distinguished claims by Nagel & Newman (made
in their book) are _false_ without explicitly stating the missing
definition _as part of the system described_.
This is especially unfortunate because they spend a whole chapter
(chapter V) to _rigorously prove_ that the system in question is
/consistent/. This proof relies on the (alleged) "theorem" 'p ->
(~p -> q)'. But this "theorem" is _not_ derivable in the system as
described in the book.*)
Actually, they write/claim:
"Now, it happens that 'p -> (~p -> q)' (in words_ 'if p, then if not-p,
then q') is a theorem in the calculus. (We shall accept this as a fact,
without exhibiting the derivation.)"
Well, actually, it's NOT a fact, since 'p -> (~p -> q)' cannot be
derived in the system /as described/.
If THIS is not something that SHOULD be "fixed" (at least by mentioning
the missing definition in a footnote) what then?!
F.
______________________
*) In chapter V Hilbert & Ackermannn's variant of Russell & Whitehead's
system for propositional logic (in PM) is introduced. Even the formation
rules for wffs are given. The ONLY thing that is missing is the
_crucial_ information that 'A -> B' means '~A v B' (where A, B are
wffs).
************************
David C. Ullrich
.
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