From: wjcastre@magnus.acs.ohio-state.edu (W.Jose Castrellon G.)
Subject: Re: Axioms of set theory, infinity and R. Rucker
Organization: The Ohio State University,Math.Dept.(studnt)
Date: Sat, 7 Nov 1992 00:14:59 GMT

In article <1992Nov6.182447.25955@infodev.cam.ac.uk> gjm11@cus.cam.ac.uk (G.J. McCaughan) writes:
>In article <1992Nov6.133138.16642@prl.philips.nl> schiller@prl.philips.nl (schiller c) writes:

I'll comment to both the proposer and responder, to save bandwidth

>>
>>
>>In the definition of a set, one axiom is the existence
>>of infinity. It is one of the usual Zermelo-Fraenkel
>>axioms.
>>
>>Reading the book "infinity and the mind" by Rudy Rucker
>>(by the way, it is delighting), 

I haven't read it yet, but let me mention the very nice book _Introduction to
Modern Set Theory_  by Judith Roitman (1990). It contains lots of interesting
material and is very readable (it is intended for and undergraduate course).

>>                                one learns that
>>there are many different types of infinities which
>>exist, of different "size".
>>
>>Which of these is the infinity specified in the
>>axioms of set theory ? Is it important to decide this
>>question ? Does this have any effect on set theory ?

In some sense it does have an effect: with the Axiom of Infinity one can
prove that there are many sizes of infinite sets, and that given anyone
(or a set -not a class-) there is always one bigger than that (them). Now
as is pointed out below, it won't matter which one of these one postulates
to exist: one will get the same set theory. However one could add as an
axiom, in place of the Axiom of Infinity, the statement that there is an
inaccessible infinity: one that is bigger than that of the natural numbers,
and bigger than the size of power set (set of subsets) of any set smaller 
than it, and also bigger than the size of the union of a set (that has 
smaller number of members than the inaccessible) of sets smaller than it.
In this case one gets a stronger set theory that can prove new theorems,
that were previously independent; there are also other axioms which imply
the existence of sets of size bigger than some inaccessibles, and so on.
You can read about these in the late chapters of Judith Roitman's book.

>
>The usual axiom of infinity guarantees a countably infinite set; that is,
>one the same size as the set of natural numbers.
>
>With the axiom of choice, every infinite set contains a countable set, so
>an axiom saying "There is an infinite set" without being so specific about
>just what sort of infinite set there was would be OK. Without the axiom of
>choice, there is a difference; and it is useful to have a guarantee that
>there is a set that can function as a set of natural numbers, for instance.

Actually without using the Axiom of Choice, it is possible to prove the exis-
tence of the set of natural numbers, from the other axioms of set theory with
"There is an infinite set" in place of the Axiom of Infinity.

>
>With the axiom of choice, the natural numbers are as small as an infinite
>set can be. Without it, that's still almost true but it's not always possible
>to compare the sizes of infinite sets.
>
>I hope this helps.
>