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  1. Paul Corazza (2010). The Axiom of Infinity and Transformations J: V→V. Bulletin of Symbolic Logic 16 (1):37-84.
    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? (...)
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  2. Paul Corazza (2008). Lifting Elementary Embeddings:→. Archive for Mathematical Logic 46 (2):61-72.
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  3. Paul Corazza (2007). Forcing with Non-Wellfounded Models. Australasian Journal of Philosophy 5:20-57.
    We develop the machinery for performing forcing over an arbitrary model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is transitive. However, for establishing properties of a given model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling standard steps of presentation in Kunen and Jech.
     
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  4. Paul Corazza (2007). Lifting Elementary Embeddings J: V Λ → V Λ. [REVIEW] Archive for Mathematical Logic 46 (2):61-72.
    We describe a fairly general procedure for preserving I3 embeddings j: V λ → V λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I3 embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in (...)
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  5. Paul Corazza (2006). The Spectrum of Elementary Embeddings J: V→ V. Annals of Pure and Applied Logic 139 (1):327-399.
    In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j:V→V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V→V relative to models of ZFC. We do this by working in the extended language , using as axioms all the usual axioms of ZFC , along with an axiom schema that asserts that j is a nontrivial elementary embedding. Without (...)
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  6. Paul Corazza (2000). Consistency of V = HOD with the Wholeness Axiom. Archive for Mathematical Logic 39 (3):219-226.
    The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language $\{\in,j\}$ , and that asserts the existence of a nontrivial elementary embedding $j:V\to V$ . The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for j-formulas. We show that the theory ZFC + V = HOD + WA is consistent relative to the existence of an $I_1$ embedding. This answers a question about the (...)
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  7. Paul Corazza (2000). The Wholeness Axiom and Laver Sequences. Annals of Pure and Applied Logic 105 (1-3):157-260.
    In this paper we introduce the Wholeness Axiom , which asserts that there is a nontrivial elementary embedding from V to itself. We formalize the axiom in the language {∈, j } , adding to the usual axioms of ZFC all instances of Separation, but no instance of Replacement, for j -formulas, as well as axioms that ensure that j is a nontrivial elementary embedding from the universe to itself. We show that WA has consistency strength strictly between I 3 (...)
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  8. Paul Corazza (1999). Laver Sequences for Extendible and Super-Almost-Huge Cardinals. Journal of Symbolic Logic 64 (3):963-983.
    Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses, Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for (...)
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  9. Paul Corazza (1992). Ramsey Sets, the Ramsey Ideal, and Other Classes Over R. Journal of Symbolic Logic 57 (4):1441 - 1468.
    We improve results of Marczewski, Frankiewicz, Brown, and others comparing the σ-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis of σ-ideals to include the completely Ramsey (...)
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