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The Axiom of Infinity and Transformations j: V → V

Published online by Cambridge University Press:  15 January 2014

Paul Corazza
Paul Corazza*
Affiliation:
Department of Mathematics and Computer Science, Maharishi University of Management, Fairfield, IA, 52557, USAE-mail:, paul [email protected]
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Abstract

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková–Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V → V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V → V known to be equivalent to the Axiom of Infinity.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 1 , March 2010 , pp. 37 - 84
Copyright
Copyright © Association for Symbolic Logic 2010

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