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From: webber@brandx.rutgers.edu (Webber)
Newsgroups: sci.philosophy.tech
Subject: Re: On denumerability (preliminary)
Message-ID: <284@brandx.rutgers.edu>
Date: Sat, 4-Jul-87 22:49:03 EDT
Article-I.D.: brandx.284
Posted: Sat Jul  4 22:49:03 1987
Date-Received: Sun, 5-Jul-87 03:04:30 EDT
References: <17406@cca.CCA.COM>
Organization: Rutgers Univ., New Brunswick, N.J.
Lines: 53
Summary: is LST a problem for the concept of size or the concept of logic?

In article <17406@cca.CCA.COM>, g-rh@cca.CCA.COM (Richard Harter) writes:
> 
> 	Some time ago I started a discussion in then net.math about
> denumerability which I would like to revive.  Let us start with the

Hmmm.  Do you have a copy of that discussion that you could send me?
If there is something interesting at the tail end of this discussion,
I personally would rather know about it now before sufferring through
all the messages about whether logic is philosophy or math and what the
definitions of all the basic terms of logic are (just selfish, I guess).

> ...	My interpretation of the Skolem-Lowenheim paradox is as
> follows:  The specification of the Universe (of Logic and Mathematics)
> is Cantor-countable.  The Universe and the Continuum are subset-
> countability; however the Continuum is not Cantor-countable.

Since this is about the significance of a piece of math, it clearly will
have some philosophic aspects.

> 	In consequence the entire Cantor programme of transfinite
> cardinals is, I believe, ill-conceived.  Cantor cardinals do measure
> in some sense, a richness of structure.  They do not measure "size".
> The notion of "large cardinals" rests on a fundamental confusion
> between different notions of "size".

Two good references here are: J. N. Crossley et al, What is Mathematical
Logic?, Oxford University Press, 1972 (for an overview of what is up) and
C. C. Chang and H. Jerome Keisler, Model Theory, North-Holland, 2nd Edition,
1978 (for getting one's hands wet -- particularly the open problems in
the back).  Also of interest is an intro article written by Abraham Robinson
entitled ``Model Theory'', which appears in his collected works.

The L\:{o}wenheim-Skolem-Tarski Theorem in this instance boils down to a
statement that First Order Predicate Calculus cannot be used to describe
systems that cannot be modelled by a subset of the integers.  Thus, when
you attempt to use FOPC to describe things like real numbers, you end up
with a theory that has a `nonstandard' model.  

Regardless of what you think the `size' of the set of real numbers is,
the LST theorem is going to claim that there is a one-to-one mapping
between a subset of the integers and any FOPC formulation of real numbers.

Thus, this does not seem like a reasonable place to start an analysis of
the concept of `set size' from.  On the other hand, it does strongly bring
into question the signficance of FOPC to `naive set theory'.

By this I do not mean to say that there is nothing wrong with the current
mathematical formulation of `set size'.  Only I see no indication that one
really has to go as far as LST before one starts to run into problems with
it.  However, admittedly I have not yet seen the rest of your proposal
and look forward to being surprised.

--- BOB (webber@aramis.rutgers.edu ; rutgers!aramis.rutgers.edu!webber)