Re: Reals without infinity

From: Chas Brown (cbrown_at_cbrownsystems.com)
Date: 10/30/04 

Date: 30 Oct 2004 14:11:57 -0700

David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<7b17o09ced48dvcqauvvg6705c21vreoe1@4ax.com>...
> On 30 Oct 2004 00:46:02 -0700, cbrown@cbrownsystems.com (Chas Brown)
> wrote:
>
> >While responding to Eray, I found myself wondering, if we abandon the
> >axiom of infinity...
> >
> >I think I can see how to construct something isomorphic to the ring Z
> >in ZF,
>
> I doubt that.
>

You're no doubt correct (and the curtness of your reply makes me think
I'm either missing something extremely basic or extremely abstract);
but I still can't see what's wrong with this approach (feel free to
jump in at the first glaring error):

We define "something like the naturals" via the usual successor
function; so a set has the property "is a natural" iff it is the
result of a finite sequence of applications of the successor function
to the empty set.

This is not the same as asserting that the _set_ of all naturals
exists; it just asserts that I can tell you whether any (finite) set S
is, or is not, "a natural".

Attach a "sign" via cross product with the finite set (Z_2), and call
any set which can be so constructed "an integer". It seems clear to me
that we can always determine, in a finite number of steps, whether or
not a given finite set S satisfies the property "integer".

We can then construct, for any two sets satisfying the above
definition of "integer", a new, finite set which also provably has the
property "integer" for each of multiplication and addition. Given any
set S with the property "integer", I can prove the existence of a set
"-S" with property "integer" which is its additive inverse.

To show an isomorphism: in ZF, we have the axiom of infinity, and it's
obvious that each element with the property "integer" is a member of
Z. Conversely each member of Z in ZF is clearly the result of a finite
number of applications of the successor function (and a sign); and so
has the property "integer". The operations multiplication and addition
are preserved in both directions.

What's wrong with this approach (aside from the fact that I don't
actually have any problems with the axiom of infinity, and this
alternative seems like a tremendous pain in the neck)?

It seems somewhat like the idea of "proper classes"; I specify the
"class" of integers by specifying a property, without asserting that
the resulting collection of "all integers" actually exists as a set.

Cheers - Chas "Ignorance is not bliss" Brown

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