AWayOut082802 - 1 A WAY OUT Harvey M. Friedman Department...

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1 A WAY OUT Harvey M. Friedman Department of Mathematics Ohio State University August 28, 2002 ABSTRACT. We present a way out of Russell’s paradox for sets in the form of a direct weakening of the usual inconsistent full comprehension axiom scheme, which, with no additional axioms, interprets ZFC. In fact, the resulting axiomatic theory 1) is a subsystem of ZFC + “there exists arbitrarily large subtle cardinals”, and 2) is mutually interpretable with ZFC + the scheme of subtlety. 1. NEWCOMP. Bertrand Russell [Ru1902] showed that the Fregean scheme of full comprehension is inconsistent. Given the intuitive appeal of full comprehension (for sets), this inconsistency is known as Russell’s Paradox (for sets). The modern view is to regard full comprehension (for sets) as misguided, and thereby regard Russell’s Paradox (for sets) as a refutation of a misguided idea. We first give an informal presentation of the axiom scheme investigated in this paper. Informally, the full comprehension axiom scheme in the language L( ) with only the binary relation symbol and no equality, is, in the context of set theory, Every virtual set forms a set. We use the term “virtual set” to mean a recipe that is meant to be a set, but may be a “fake set” in the sense that it does not form a set. The recipes considered here are of the form {x: j }, where j is any formula in L( ). Other authors prefer to use the term “virtual class”, reflecting the idea that {x: j } always forms a class, with the understanding that x ranges over sets. Our terminology reflects the intention to consider only sets, and construct a powerful set existence axiom. We say that {x: j } forms a set if and only if there is a set whose elements are exactly the y such that j . Here y
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2 must not be free in j (and must be different from x). Thus {x: j } forms a set is expressed by ( $ y)( " x)(x y j ). Russell showed that {x: x x} forms a set leads to a contradiction in pure logic. Our way out of Russell’s Paradox is to modify the inconsistent Fregean scheme in this way: Every virtual set forms a set, or _____. We refer to what comes after “or” as the “escape clause”. The escape clause that we use involves only the extension of the virtual set and not its presentation. We are now ready to present the comprehension axiom scheme. NEWCOMP. Every virtual set forms a set, or, outside any given set, has two inequivalent elements, where all elements of the virtual set belonging to the first belong to the second. To avoid any possible ambiguity, we make the following comments (as well as give a formal presentation in section 2). 1. For Newcomp, we use only the language L( ), which does not have equality. 2. Here “inequivalent” means “not having the same elements”. 3. The escape clause asserts that for any set y, there are two unequal sets z,w in the extension of the virtual set, neither in y, such that every element of z in the extension of the virtual set is also an element of w.
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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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AWayOut082802 - 1 A WAY OUT Harvey M. Friedman Department...

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