From: lemma_one@my-deja.com
Subject: Re: axioms of modern mathematics
Date: Fri, 05 Jan 2001 05:10:47 GMT
Newsgroups: sci.math
Summary: Common axiom sets for set theory

In article <20010102230631.27553.00000057@ng-fm1.aol.com>,
  ahimog@aol.com (Ahimog) wrote:
> How many axioms are required to prove all of the proven theorems in
mathematics
> a la Bertrand Russell et al.?
>

A few things:

  (1) ZFC has 8 axioms + 2 axiom schemata. It has been proved that ZFC
can't be finitely axiomatized. Every time you take the subset of a set,
you are using one of the subset axioms. Everytime you are given some set
{a, b, c, ...} and replace its elements with sets {A, B, C, ...} you are
using one of the replacement axioms. (The replacement axioms involve so
much formal machinery to apply that you might not even be aware that
"replacing" sets in a set with different sets is all you are doing.)

  (2) VNB (Von Neumann-Bernays) set theory includes a unary relation
symbol V in the language where V(c) <==> c is a set. It is finitely
axiomatizable: you don't need axiom schemata for subset and replacement.

  (3) The 8 axioms + 2 axiom schemata of ZFC aren't logically
independent. You can use the replacement schemata to get rid of the
subset schemata, and you can also get rid of pair-set (by using
powerset+subset) and you can get rid of union-set also. Emptyset,
Extensionality, infinity, power, replacement, and choice are all you
need, I think. And in VNB replacement can be done with classes so that
it is just one axiom. So perhaps 6 is what you were looking for. This
will be true for nearly all of mathematics, but those working in logic
and set theory also use large-cardinal axioms and forms of the continuum
hypthesis, and then there are things like Conway's surreal numbers which
are mathematics, but can't be embedded in set theory, or so I've heard.


Sent via Deja.com
http://www.deja.com/