FOM: categorical cardinals (in two senses)

Stephen G Simpson simpson at
Wed Apr 12 21:52:25 EDT 2000

In my posting of 1 Mar 2000, I discussed the ZFC-style foundational
picture given by the cumulative hierarchy.  I said that this picture
tends to inspire confidence, because it is ``demonstrably categorical
in a certain sense''.  I asked whether the alternative foundational
pictures embodied by NF and topos theory have any analogous
``categoricity properties''.

Later in the same thread, Eric Palmgren (18 Mar 2000) gave some
references to intuitionistic set theory, etc, and saying that IZF
provides a ``categorical framework of at least the interpretational
strength of set theory''.

To eliminate any possible confusion, let me point out that Palmgren
and I were using the word ``categorical'' in two completely different
senses, and we were making completely different points.

In my remarks, I was referring to categoricity of axiomatic theories,
meaning that a theory has only one model up to isomorphism, as when
one says that Hilbert's second order axiomatization of geometry is
categorical.  My point was that it is well known and provable in ZFC
that, for all ordinals alpha, the structure (V_alpha, epsilon|V_alpha)
can be characterized uniquely up to isomorphism, as the unique
structure of height alpha satisfying certain obvious second order
axioms.  Furthermore, this structure is *rigid*, i.e., it has no
automorphisms other than the identity.  This kind of categoricity and
rigidity give us the feeling that we are dealing with something very

On the other hand, Palmgren in his remarks was evidently referring,
not to categoricity of axiomatic theories, but rather to *category
theory*, i.e., morphisms, functors, toposes, etc.  One of Palmgren's
points was that there is hope for interpreting set theory and large
cardinal axioms into topos theory, because of the close relationship
between topos theory and intuitionistic set theory.

I would like to follow up a little bit on Palmgren's point.  I don't
really see how large cardinals can be fruitfully studied in a topos
setting.  Can someone enlighten me?

Of course it is known that the axioms of topos theory can be
supplemented by a list of additional axioms to make it similar to (a
reasonable fragment of) classical ZFC set theory.  So in this sense
Palmgren's point is obvious.  But one of the virtues of topos theory
is supposed to be its generality.  Thus it seems reasonable to ask
what can one say about large cardinals in the general topos framework?

For example, one of the well known large cardinal properties is weak
compactness.  If X is a set, let [X]^2 denote the set of 2-element
subsets of X.  The cardinality of X is said to be weakly compact if
and only if X is uncountable and, for every partition of [X]^2 into
two parts, A and B, there exists a set Y included in X of the same
cardinality as X, such that either [Y]^2 is included in A, or [Y]^2 is
included in B.  

So, the set-theoretic formulation of weakly compact cardinals is
obviously very elegant.  Is there a corresponding topos formulation?
Is it comparably elegant?  Is it comparably fruitful?

-- Steve

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