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

On Fri, 02 Nov 2007 00:16:31 +0100, G. Frege <nomail@invalid> wrote:


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

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 relevant 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.)"


Though '~p -> (p -> q)' would do as well; and the derivation of the
latter is extremely simple (in the system described in the book
augmented with the definition: S1 -> S2 =df ~S1 v S2, where S1 and S2
are wffs).

Theorem:

~p -> (p -> q)

Proof:

(1) p -> ( p v q) Ax. 1
(2) ~p -> (~p v q) Subst. 1 [p/~p]
(3) ~p -> (p -> q) Df. 2


F.

--

E-mail: info<at>simple-line<dot>de
.

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