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From: g-rh@cca.CCA.COM (Richard Harter)
Newsgroups: sci.philosophy.tech
Subject: On denumerability (preliminary)
Message-ID: <17406@cca.CCA.COM>
Date: Sat, 4-Jul-87 16:19:48 EDT
Article-I.D.: cca.17406
Posted: Sat Jul  4 16:19:48 1987
Date-Received: Sat, 4-Jul-87 20:45:34 EDT
Reply-To: g-rh@cca.UUCP (Richard Harter)
Distribution: world
Organization: Computer Corp. of America, Cambridge, MA
Lines: 53


	Some time ago I started a discussion in then net.math about
denumerability which I would like to revive.  Let us start with the
Skolem-Lowenheim "paradox" which states that every formal system has
a countable model.  The paradox is that in ZF the continuum is supposed
to be uncountable.  How one deals with this "paradox" is, a matter of
one's tastes in the philosophy of Mathematics.  I propose to deal with
it by rejecting the notion of absolute nondenumberability, and, in the
process, by claiming that Cantor's treatment of countability is
inadequate.  As a preliminary let me define some terms:

First of all I will use "set" uniformly for any aggregate which is
assumed to exist in a particular theory at hand.  I.e. I will use
the word set for both the sets and classes of two sorted theories
(ML, NB, my own H2, etc.).  In reference to any multi-sorted theory
an appropriate modifier will be used (e.g. Zermelo sets for ZF or NB).
If I am discussing a particular theory and a meta theory about it
and the meta theory has sets which are do not exist within the theory
I will use meta-set or something equivalent.

Secondly, I will be speaking of several different usages of the word
"countability", so I will prefix each such usage with a qualifier, e.g.

Cantor-set-countability:  A set is Cantor-set-countable if there is
a function (in the sense of a set of ordered pairs) which is one-to-one
map from the integers to the set.

Cantor-function-countability:  A set is Cantor-function-countable
if there is a logical function (in the sense of a two place wff)
which is a one-to-one map from the integers to the set.

Cantor-couability:  A set is Cantor-countable if is Cantor-set-
countable or Cantor-function-countable, or both.

Subset-countability:  A set is subset-countable if it is a subset
of a Cantor-countable set.

	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.

	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".

	So much for the preliminaries.  More to follow.
-- 

Richard Harter, SMDS Inc. [Disclaimers not permitted by company policy.]
			  [I set company policy.]