From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.logic,sci.physics
Subject: Do completed infinite totalities exist? Was: Lowneheim-Skolem theorem
Date: 19 Nov 92 17:53:54 GMT
Followup-To: sci.logic
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <TORKEL.92Nov18193457@bast.sics.se>, torkel@sics.se (Torkel Franzen) writes:
> In article <361@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:
> 
>   >This would be a valid argument if uncountable had an absolute definition.
>   >I think uncountable is only meaningful relative to some formal system.
> 
>   Yes. However, this is a peculiar philosophical dogma which on the face of
> it has nothing to recommend it, and in particular, has nothing to do with
> ordinary mathematics.

Whether it has anything to recommend it depends on your philosophical
inclinations about the infinite. What it has to do with ordinary mathematics
is an interresting question. If one rejects the notion that there exist
completed infinite totalities, what does one make of the rich and
beautiful mathematics that seemingly depends on this notion? 

I think of real numbers as being meaningful if they represent properties
of a Turing Machine that may be of interest to beings in a finite but
potentially infinite universe. I think this is a natural way of thinking
about things once one rejects completed infinite totalities. I will not
be surprised if you and most readers disagree.

It is not clear that there are any real numbers that mathematicians would
generally agree are meaningful that cannot be defined in this way. A real
number that encodes the truth or falsity of the continuum hypothesis is
an example of a real number that many mathematicians would not consider
meaningful.

Is an example of what I have in mind consider the following question
about biological evolution in a potentially infinite universe: will
any species evolve into an infinitely long chain of descendant
species? Assuming a species may be thought of as a Turing machine,
(I can hear the protests now) we can restate the problem as follows:
Nondeterministically simulate this TM and interpret each of its outputs
as the Godel number of another TM. The equivalent question is does
there exist an infinite chain of TM's each of which was output by its
parent? This question which we might call that of well foundedness for
Turing Machines takes us through the hyperarithmetical hierarchy of
reals and the notion can be generalized beyond this.

You protest that to define this property requires quantification over
the reals and thus the notion that the reals are a completed infinite
totality. (You may have other protests as well.) I disagree. I think
the notion of an arbitrary path in a recursively enumerable tree does
not require that we accept philosophically that the reals are a completed
infinite totality. I think we can adopt the same formal approach to
mathematics at this level but use a different philosophical justification.
I believe that thinking about mathematics in this way can help us to
understand what is mathematically meaningful.

More importantly it can change the focus of mathematical research. We
can study `experimentally' the question of what TM's are well founded
and we can use this research to refine and enhance are knowledge and
intuition about mathematical objects. I think we can learn a lot more
about the notion of a real number from such research than we can
from proving theorems related to the continuum hypothesis.

Follow ups are directed to sci.logic.

Paul Budnik
Newsgroups: sci.logic