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Symmetry as a criterion for comprehension
by
M. Randall Holmes
Boise State University
For any permutation f of the universe, define j(f) as the map which sends a set A to f[A] (the elementwise image of A under f). If extensionality holds, j(f) will also be a permutation. We define a permutation f of the universe as ``setlike" if jn(f)(A) is a set for any natural number n and set A (this will be a trivial concept if Replacement holds (in the form which says that a class the same size as a set is a set). We say that a class is ``n-symmetric" iff it is fixed under jn(f) for all setlike permutations f. We say that a class is ``symmetric" if it is n-symmetric for some n. It is easy to see that any parameter-free stratified instance of the comprehension scheme defines a set which is n-symmetric. We will discuss the converse program of ``symmetric comprehension": we explore the development of set theory from an axiom which asserts that a class is a set if and only if it is symmetric. There are some technical difficulties in formalizing such a theory (and there is more than one possible way to do it). Such theories extend Quine's New Foundations. Choice is false in such theories not only because it is false in NF (a surprising theorem of Specker) but because symmetric comprehension implies that choice is obviously false. Set theory with symmetric comprehension is not just NF: it proves theorems which cannot be proved in NF and it has a different kind of motivation. We will also discuss reasons to think that symmetry is a philosophically reasonable criterion for comprehension. We do not claim any consistency result!
Date received: March 7, 2003
Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakf-06.