[FOM] Cantor Theorem

Matt Insall montez at fidnet.com
Mon Feb 10 19:26:31 EST 2003

>From Stephen Newberry, we get:

We know already from the Loewenheim-Skolem Theorem that models exist in
which the Cantor Theorem is false. Hence it is necessarily contingent.

Neil Tennant replied:

But in no model of set theory is Cantor's Theorem false.


I would have to agree that in no model of ZF is Cantor's result false.
But is it universally accepted that any model of set theory is a model
of ZF?  I doubt it.  (Witness many arguments about this and related issues
in this very forum.)  If we begin listing the axioms of ZF, at which point
do we conclude ``Now that is an axiomatization of a set theory.''?  I have
no problem with saying we need all of ZF, but if there is not to be found
a consensus on this issue, can we eliminate some axioms until there is
a consensus, or would an empty set of axioms be as acceptable as ZF for
a ``set theory'', only because it is a set of axioms ``in the language
of set theory''?

Thus, my question to Stephen Newberry is ``models of what?''.  He says
``models exist''.  But we need no Lowenheim-Skolem application for this
conclusion, if any set of axioms in the language of set theory is an
axiomatization of a theory of sets.  For, although I have not tried to
construct a model of the negation of Cantor's Theorem, I am fairly certain
that if C denotes Cantor's Theorem, then {~C} is a consistent set of axioms
in the language at hand.  I am not very interested in such set theories
that have no axioms, or so few axioms as to be of no use to mathematics.
(But if you think you can convince me that I should find them interesting,
or that they are useful, please try.)

My question to Neil is between the lines in my above comments.  Basically,
if Stephen is unreasonably ambiguous by saying ``models exist'', then
Neil is not clarifying his foundational assumptions in saying ``in no model
of set theory'', for again, without a more explicit statement about what
he means by ``set theory'', I have difficulty determining the truth value of
his claim, precisely because I am uncertain what he means by it.

>From a cold, but sometimes not so cold, Rolla.

Matt Insall

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