From: Peter Percival <not.him@nowhere.invalid>
Subject: Re: why do we need axioms?
Date: Sat, 02 Sep 2000 13:58:34 +0100
Newsgroups: sci.math
Summary: [missing]

"David C. Ullrich" wrote:
> 
> On Fri, 01 Sep 2000 19:17:00 +0100, Peter Percival
> <not.him@nowhere.invalid> wrote:
> 
> >Keith Ramsay wrote:
> >>
> >> In article <39A6CCF6.22682D30@nowhere.invalid>, Peter Percival
> >> <not.him@nowhere.invalid> writes:
> >> |There are those (most famously Frege and Russell) who tried to show that
> >> |maths is just logic carried on far enough. There are two problems: (1) a
> >> |very broad definition of "logic" is called for, and (2) the limitative
> >> |theorems of Godel, Turing, etc. show that it still can't be done.  This
> >> |doesn't mean that maths shouldn't be done logically.
> >>
> >> I don't see that reducing mathematics to logic requires having
> >> a complete axiomatization of it.
> >>
> >> Keith Ramsay
> >
> >Well, if every mathematical truth is to be deduced the logic has to be
> >complete doesn't it?  I think I've missed your point, sorry.
> 
>         Exactly what does "the logic is complete" mean? It's a serious
> question -

...

> 
>         The Incompleteness Theorem doesn't
> say that we cannot "reduce mathematics to
> logic", it says we cannot do so with a fixed
> set of axioms...
> 
>         To put it all another way: You say
> that "if every mathematical truth is to be
> deduced the logic has to be complete,
> doesn't it?" One might ask whether
> 
> AxAyAzAw (x*(y*(z*w))) = (x*y)*(z*w)
> 
> is a "mathematical truth". The answer is
> of course not, it depends on what you're
> assuming about the * operator. The
> statement is true in every group, so it
> had better be deducible from the axioms
> of group theory. It is. Otoh it does not
> follow from some other set of axioms.
> 

On the other other hand

if (conjunction of the axioms of group theory) then ( AxAyAzAw
(x*(y*(z*w))) = (x*y)*(z*w) )

is a mathematical truth and it is provable in first order logic.  My
point was (and it may be off-topic) that Frege and Russell wanted one
logical calculus from which all of mathematics could be deduced.  I
believe that Hilbert, too, wanted such a calculus so that by reasoning
about it (i.e. its symbolic presentation as a finitary thing) he could
prove that all mathematics is consistent.

On the other other other hand

if (conjunction of the axioms of PA) then (Paris-Harrington's extension
of the Finite Ramsey Theorem)

is a mathematical truth (because its consequent is true) and it is not
provable in first order logic.  This incompleteness of PA may be
interesting and to a logicist it may be disappointing.  The difference
between group theory and PA is significant isn't it?

I agree with your post, most of which I snipped.

-- 
The From: address in the header is fictional
  I am peter dot percival at cwcom dot net