From: torkel@sics.se (Torkel Franzen)
Subject: Re: Do completed infinite totalities exist? Was: Lowneheim-Skolem theorem
Organization: Swedish Institute of Computer Science, Kista
	<369@mtnmath.UUCP> <TORKEL.92Nov21195445@bast.sics.se>
	<1992Nov22.074740.13322@smds.com>
Date: Sun, 22 Nov 1992 10:55:49 GMT


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.

  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.

  To make this a bit more concrete, consider the example of two ordered
sets: the rational numbers and the real numbers, with their usual orderings.
These orderings are elementarily equivalent - that is, any first order
formula using only "<" and "=" is true of one ordering if and only if it
is true of the other. In mathematics we learn that these orderings are
nevertheless not isomorphic, since the order of the real numbers is
complete, but that of the rational numbers is not.

  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. How is the formal system to be used in
the explanation? Which formal system is to be used? What are the variations
in the concept resulting from various choices of formal system? These
questions never arise in any ordinary mathematical context. In practical
terms, neither the explanation or understanding of this distinction, nor the
use made of it, has any such formal system as a parameter. On the other hand,
such concepts as "formally undecidable" or "definable" do have formal systems
as explicit and implicit parameters, and are understood, explained, and used
on that understanding.

  This is not to reject the philosophical ideas that usually underlie
such statements as "uncountability is relative to ...", namely various
forms of anti-realism or formalism wrt set theory. What I am saying is only
that it is a mistake to believe that any such ideas obliterate ordinary
mathematical distinctions between absolute and relative concepts, or
necessarily shed any light on how mathematics is or can in fact be understood
in practice.