Bulletin of Symbolic Logic

The Axiom of Infinity and transformations j: V→V

Paul Corazza

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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.

Article information

Source
Bull. Symbolic Logic, Volume 16, Issue 1 (2010), 37-84.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.bsl/1264433797

Digital Object Identifier
doi:10.2178/bsl/1264433797

Mathematical Reviews number (MathSciNet)
MR2656117

Zentralblatt MATH identifier
1196.03069

Subjects
Primary: 03E55: Large cardinals
Secondary: 03E40: Other aspects of forcing and Boolean-valued models

Keywords
Axiom of Infinity WA Wholeness Axiom large cardinal exact functor critical point Lawvere universal element

Citation

Corazza, Paul. The Axiom of Infinity and transformations j: V→V. Bull. Symbolic Logic 16 (2010), no. 1, 37--84. doi:10.2178/bsl/1264433797. https://projecteuclid.org/euclid.bsl/1264433797


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