[FOM] Cantor's Theorem

[Matt Insall]

Neil,
     Your story of Mother and the Naif is very nicely done,
but Mother could have given the Naif an empty set at the
beginning.  In this case, at no point in learning set theory
does the Naif have the illusion that no sets exist.  Better
yet, Mother may mereely say ``there is a set'', at the very
beginning, and let the naif play around in the set theoretic
garden to find the empty set or some other set.  In this way,
we have set theories of different ilks at each stage when Mother
introduces a new axiom, and the existant models become more
and more restrictive with each new axiom fed to the Naif by
Mother.
     You wrote also:
<<By "in no model of set theory is Cantor's Theorem false" I mean
"in no model of the set-theoretic axioms used to prove Cantor's Theorem is
Cantor's Theorem false.">>

As I see it, no specific models of the languaga of set theory are used to
prove
set-theoretic statements.  One proves results from the axioms, using modus
ponens,
etc.  Now, I am platonic enough to believe that models of consistent
theories
actually exist.  I also have not seen evidence that would convince me that
ZF
is inconsistent.  Thus, I may have in mind some particular model of the
languaga
of set theory when I formulate axioms for a set theory I would like to
communicate,
like Mother to the Naif, to my students or collaborators.  But the method of
proof
that is typically required of me is reducible to classical first-order
predicate
calculus with equality, in the language of ZF.  Handing to someone a model
of set
theory seems to be out of the question, in most cases, difficult at best in
others.
(Even the existence of Solovay's model of ZF, in which the subsets of the
reals
are all measurable, finds itself contingent upon the consistency of a large
cardinal
axiom, so that we need to first have a model of a theory for which there is
provably
no proof of the consistency of the theory, in order to construct Solovay's
model.)






You wrote:

<<Thus your worry about not being able to tell, when listing the axioms of
ZF, whether you have now specified all of set theory is irrelevant.
(Listing the axioms of ZF will take you an infinitely long time, by the
way, if you really do mean *axioms*, rather than "axioms or
axiom-schemes".)>>

Instead of axioms or schemes, let us discuss models.  I presume that the
models you would like to consider have in them some sets, so the empty
universe is out.  (If not - if you want to allow all universes to be
models of set theory, then you have made the choice that set theory is
merely a language.  Fine.  But when you allow this, you cannot reasonably
claim that "in no model of set theory is Cantor's Theorem false", for
you pointed out yourself what is needed to prove Cantor's Theorem:
``a separation axiom using a formula that contains parameters but no
quantifiers.''  In models that fail to satisfy this very weak condition,
it may be the case that Cantor's Theorem fails.)


Thus, instead of concerning myself with listings of axioms that are to
specify what we mean by ``set theory'', I ask for a consensus about
which models of the language of set theory are to be considered models
of set theory.  (Personally, I see this as no different from asking for
a consensus about which sets of axioms should be considered to be minimal
sest of axioms for ``set theory''.  But I am playing your game today.)

If we cannot have a consensus, perhaps we should ask instead for a vote.
In our democratic society, this may be the most reasonable way to do things.
But it seems to me that the current situation, in which the author picks a
stance without making clear what that stance is except by stating
conclusions
that are not provable from the empty set of axioms, and then the readers and
editors and referees attack as if some other stance is the one which should
be taken, is arbitrary and unjust to the author.  Had I been the author of
your
claim, namely that "in no model of set theory is Cantor's Theorem false", I
believe I would have preferred to say "in no model of ZF is Cantor's Theorem
false",
because that is something I know to be the case.  But in this forum, we have
seen time and again that others may not accept various axioms of ZF, so I
may
have said instead "in no model of Z is Cantor's Theorem false".  My question
was
one with the spirit of reverse mathematics:  If I want to claim that "in no
model of set theory is Cantor's Theorem false", then what must I mean by
``set
theory''?