FOM: thanks to Simpson

[holmes@catseye.idbsu.edu]

I want to thank Steve Simpson for his challenge to describe the
worldview implicit in NF (though I only described the view implicit
in NFU :-)  It gave me a chance to clarify my thoughts on this subject
a bit more; I'm planning to write an article about it...

There was another good question in one of Steve's posts that I quoted,
about whether there is any categoricity in models of NFU.  That's also
a very good question, and I'm going to try to briefly indicate the
answer, which is No -- in the absence of additional axioms; but there
may well be interesting axioms which will fill the bill.

I'm going to sketch the problem using the fact that models of NFU can
be constructed using nonstandard models of initial segments of the
cumulative hierarchy with external automorphisms moving a level.

There is an internal way of expressing the same idea -- but it is
less accessible to an audience more familiar with ZFC.

A model of NFU can be obtained from a V_alpha (alpha a nonstandard
ordinal, V_alpha the corresponding nonstandard rank) with an
endomorphism J (it's usually derived from an automorphism of the
ambient model of set theory; it is not a set map in the model, of
course!) sending V_alpha to J(V_alpha) = V_(J(alpha)) where J(alpha) <
alpha; the membership relation of the model of NFU is defined thus x
E(NFU) y iff J(x) E y and y E V_(J(alpha)+1).  The idea is that
V_alpha includes the power set V_(J(alpha)+1) of a structure
V_(J(alpha)) that looks just like it; let sets in V_(J(alpha)+1)
represent the corresponding sets in the "real" power set V_(alpha+1)
and let all the elements of V_alpha - V_(J(alpha)+1) be treated as
urelements (note that there are A LOT of urelements).  It is
straightforward to demonstrate that this gives a model of NFU (this
construction is implicit in Jensen's proof; I believe that it was
originally explicitly given by Maurice Boffa).

The point is that one can take any beta < alpha for which J(beta) <
beta and "cut off" the model of NFU at that point (use V_beta as the
universe); so it is impossible for the axioms of NFU or the strong
axioms of infinity usually added to it to specify what the universe
looks like, except by putting lower bounds on its size. (for example,
in NFU + AC + typical infinity axioms one has no leverage on such
questions as whether |V| is a successor cardinal or a limit cardinal).
This is all equally easy to represent strictly in "native" terms of
NFU; but the development would look less familiar.

To make the axioms more categorical, one would need axioms specifying
the characteristics of the "big" structures more precisely.  There are
some reasonable approaches: one would like to have the structure of
the universe resemble the structure of the (proper class) part of the
universe fixed by the automorphism (in terms of the model
construction; there are internal ways of saying the same thing (in
internal terms, I think one could say (given Choice) that a
well-ordering of V should "look like" the (proper class) order on the
strongly cantorian ordinals).  Such considerations are analogous to
reflection principles in ZFC and extensions.  They suggest that |V|
really should be a limit cardinal, to take a simple example.  It is
likely that principles of this kind adjoined to NFU would yield
consistency strength equivalent to extensions of ZFC with large
cardinal hypotheses.  The models of NFU constructed in my forthcoming
JSL paper do have properties of this kind, but I would have to look
carefully at the models in order to express the new axioms sensibly.

There are further quibbles about the urelements, which appear
structureless and therefore somehow arbitrary.  There are ways that
one can enrich NFU so as to assign structure to urelements (even in
internal terms), but I'm not convinced that it is really necessary to
do this for any philosophical reason proper to an NFU-based viewpoint.
The "rigidity" of arithmetic or ZFC is not a universal feature of
mathematical structures; Euclidean space, for example, isn't rigid
(the points of space all look the same).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes