From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.logic
Subject: Re: Do completed infinite totalities exist? Was: Lowneheim-Skolem theorem
Date: 22 Nov 92 18:23:51 GMT
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <TORKEL.92Nov22115549@echnaton.sics.se>, torkel@sics.se (Torkel Franzen) writes:
> 
> In article <1992Nov22.074740.13322@smds.com> rh@smds.com (Richard Harter) 
> writes:
> 
>    >Perhaps the distinction is quite simple, but I fail to see it.  In
>    >particular I fail to see why there isn't an implicit formal system
>    >in the concept of "uncountable".  Unless one accepts the notion of
>    >the absolute uncountable, uncountability is always relative to a
>    >system and a model for the system, is it not?
> 
>   I'm not sure what you mean by "the notion of the absolute uncountable".
> What I am saying is just this, that when we learn or explain to others
> the concept of uncountable sets, there is no formal system as a parameter
> in that explanation, and the large majority of people who use this
> concept neither know nor need or care to know anything about formal systems.

I alluded to this in a previous posting. It is a convention that
all formal systems in which reals are definable allow Cantor style
diagonalization of maps from reals to integers. Because of this convention
you can speak of uncountability without a formal system as a parameter.
This is not necessarily a good convention. I would argue from my philosophical
position that it would be preferable to see from within the system that
everything definable in the system is countable. This would force one to
use a type hierarchy that precludes the diagonalization of some maps from
integers to reals.

I disagree that most people who use this concept do not need to know or care
about formal systems. It may be a pragmatic reality given the way mathematics
functions today. I do not think it makes for good mathematics in the long
run. I think the belief that one is studying some objective mathematical
structure, when this is not the case, leads to stagnation in mathematics.
The best example of this is Cantor's insanity from trying to prove the
continuum hypothesis.
 
>   This is quite independent of whether or not there is an iota of
> "absolute truth" in mathematics. Let us stipulate that the mathematics we
> are talking about is a piece of fantasy and that sets have no existence
> outside our (essentially arbitrary) imagination. It remains that we will do
> gross violence to our ordinary understanding of mathematics if we try to
> attach some formal system as parameter to the concept of "uncountable set",
> and it's far from clear that it can be done at all.

Sometime gross violence is called for. I do not believe that the mathematics
we are discussing is a fantasy.  That is why I consider this debate important. 
I am not a formalist. I think there is objective truth to all mathematical
propositions that would interest the inhabitants of a finite but
potentially infinite universe. I think that this definition encompasses
most of contemporary mathematics. It leaves out questions like the continuum
hypothesis and what cardinals exist.

> ... 
>   Now suppose we actually try to take seriously the idea that "complete" is
> a concept that must be understood relative to a formal system. Then the
> explanation of the difference is on the face of it meaningless as long as
> no formal system has been produced.
> ...

It is not the concept of uncountable or complete that is relative to a
formal system. It is the question of what sets (if any) have these properties
that is relative.

Paul Budnik