Re: A missing definition in "Gödel's Proof" by Nagel & Newman (open letter)

On Thu, 01 Nov 2007 06:42:09 -0600, David C. Ullrich
<ullrich@xxxxxxxxxxxxxxxx> wrote:


Can you tell us exactly what the axioms and rules
of the system in the book are?

Sure.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Axioms:

Ax. 1 (p v p) -> p
Ax. 2 p -> (p v q)
Ax. 3 (p v q) -> (q v p)
Ax. 4 (p -> q) -> ((r v p) -> (r v q))

Rules of derivation:

- Substitution
- From S1 and S1 -> S2 derive S2. (MP)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

My point 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 (p. 50):

"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/.

Hence imho the missing definition SHOULD be added to the text (or at
least mentioned in a foodnote).


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_ definition: A -> B is short for ~A v B.

.

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