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1 FROM RUSSELL’S PARADOX TO HIGHER SET THEORY by Harvey M. Friedman Ohio State University Philosophy Colloquium October 10, 1997 ABSTRACT: Russell’s way out of his paradox via the impre- dicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END. The famous Russell’s paradox for sets arises out of the intuitively appealing naive principle of full comprehension written ( $ x)( " y)(y x j ), where j is a logical formula involving " , $ , , , and variables in which x is not free. The variables are thought of as ranging over sets. Here x,y are any distinct variables. In particular, the simple special case ( $ x)( " y)(y x y y) generates the inconsistency by fixing such an x and uni- versally instantiating y by x, thus obtaining x x x x. Russell’s Paradox also makes perfectly good sense in other contexts besides set theory – e.g., in a theory of predicates. Here the naive comprehension axiom scheme takes the notationally similar but distinct form ( $ P)( " Q)(P(Q) j ),
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2 where j is a logical formula involving " , $ , ,(), and variables, where P is not free in j . This time the variables are thought of as ranging over predicates. Here P,Q are any distinct variables. Why do I draw a distinction between the set version of Russell’s Paradox and the predicate version? One big difference between the way sets and predicates are thought of is in terms of extensionality. We have pre-ferred not to use equality as primitive, and so extension-ality for sets takes on the form ( " z)(z x z y) ( " z)(x z y z). Extensionality is in accordance with the usual way of thinking about sets. However, the corresponding statement about predicates, ( " R)(P(R) Q(R)) ( " R)(R(P) R(Q)), does not seem appropriate. For instance, one may distinguish different predicates P,Q such that P,Q fail at every argument, where P,Q can be distinguished by another predicate R. Then we have ( " R)(P(R) Q(R)) and yet ( " R)(R(P) R(Q)). Yet another important distinction between certain notions of predication and sets will be relevant later in the discussion. Frequently people still use the notation x y even if x,y are predicates and we are asserting that y holds at x. We will follow this convention. Then the distinction between predicates and sets may come at the point at which we decide to include or exclude EXT. Now after his discovery of Russell’s Paradox, Russell embarked on a series of developments to repair the damage. It is customary to identify two proposals of Russell to get out from under his Paradox.
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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