FOM: RE: Cantor's Diagonal Argument

[Insall]

On 28 June 2002,
sometime before 12:28 PM CST,
Richard Arthur wrote:
``What I am about to say will
no doubt bring the Wrath of
Fom down on my head,
but I hope to learn
from the responses.''

No wrath.
Just a suggestion:
Read some finite model theory.
I do not know as much about it as I would like,
but it seems to me that the belief in a
finite universe can be quite
adequately
handled in a particular finite
model of an appropriate size.

If, however, you wish to deal with
``arbitrarily large finite sets'',
then you need a denumerable universe.
The most natural theory to study is then PA,
or some reasonable modification thereof.
PA does not assert the existence of any
infinite sets,
but it is a theory of counting that
can be fairly easily modified into a
formal theory of the combinatorics of arbitrarily
large finite sets.
It is, of course,
undecidable,
and not finitely axiomatizable.
However, PA is a first-order theory,
about which much model theory and abstract logic has been written.


Matt Insall
PS:  I know I should look it up myself,
but I'll ask anyway.
Does anyone know af a conservative,
finitely axiomatizable extension of PA?
I ask this question,
because I find it interesting that Goedel's class-set theory
is a finitely axiomatizable conservative extension of Z(or ZF, depending on
what you mean by ``Class-Set Theory'') set theory.
Perhaps that is exactly what nonstandard analysis really is,
if one approaches it from a Nelsonesque perspective.
Maybe that is why,
in 1988, in Northhampton Massachussetttes,
at a meeting of nonstandard analysts,
Ed Nelson postulated that the cumulative hierarchy may terminate
after some finite number of steps.
I was shocked.  Now, it's just finite model theory.


PPS:  Terminological references available upon request.
Sorry.
I just see this stuff as almost folklorical now,
so that giving a reference is,
to me,
almost like giving a reference for the quadratic formula,
when I need to factor the polynomial x^2.
When I teach a class, it's in the book.
Now, it's all on the web,
and the search engines are amazing.