Consistency of V = HOD with the wholeness axiom | Zendy

Paul Corazza | Zendy

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Consistency of V = HOD with the wholeness axiom

Author(s) -

Paul Corazza

Publication year - 2000

Publication title -

archive for mathematical logic

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.618

H-Index - 32

eISSN - 1432-0665

pISSN - 0933-5846

DOI - 10.1007/s001530050144

Subject(s) - mathematics , axiom , axiom of choice , embedding , transitive relation , mathematical proof , schema (genetic algorithms) , discrete mathematics , zermelo–fraenkel set theory , combinatorics , class (philosophy) , set (abstract data type) , set theory , computer science , geometry , artificial intelligence , machine learning , programming language

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 existence of Laver sequences for regular classes of set embeddings: Assuming there is an \(I_1\)-embedding, there is a transitive model of ZFC +WA + “there is a regular class of embeddings that admits no Laver sequence.”

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