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From: Marc Alcobé García on 1 Apr 2010 13:52
On 30 mar, 19:01, zuhair <zaljo...(a)gmail.com> wrote:
> Suppose we have a set theory T, which cannot define
> the notion of "natural number", would that make T
> escape the incompleteness theorems of Godel's.
>
> Zuhair

I know of a partial result due to H. M. Löb quoted in Boolos' "The
logic of provability" Chapter 2 (page 16). The reference given there
for that result is "Solution of a problem of Leon Henkin" Journal of
Symbolic Logic, 20 (1955), 115-18.

The result in question is this one:

If Z is a theory in which a few simple facts about natural numbers can
be proved (namely the first six axioms of PA and "every non-zero
number is the sucessor of some number", and there is a formula B(x)
such that for all S, T sentences of PA:

1. If Z proves S then Z proves B([S]), where [S] is the numeral that
stands for the Gödel number of S (Boolos' uses Quine's corner
notation, but I don't know how to type it).
2. Z proves that B([S->T]) -> (B([S]) -> B([T]))
3. Z proves that B([S]) -> B([B([S])])

then Z cannot prove its own consistency. These three numbered
conditions over Z are called derivability conditions.

Usually a theory that proves things about natural numbers is one for
which in some extension by definitions (these are always conservative)
one defines 0, s(x), + and ·, and proves some fragment of PA (usually
varying only in how much induction you take into account).

I have not read all contributions, so I hope not having repeated
anything already said. Sorry if I have and I hope this helps.
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