The wholeness axiom and Laver sequences
type: Item
instance of: scholarly article
title: The wholeness axiom and Laver sequences
zbMATH Open document ID: 0999.03048
DOI: 10.1016/S0168-0072(99)00052-4
author: Paul Corazza
published in: Annals of Pure and Applied Logic
publication date: 2002-12-03T00:00:00Z
review text: The ``Wholeness Axiom'' WA discussed in the paper is formalized in the language \(\{\in,{\mathbf j}\}\) where \({\mathbf j}\) is a unary function symbol. It consists of the followng axioms: (1) \(\exists x({\mathbf j}(x)\neq x)\), (2) For every formula \(\varphi(x_1,\dots, x_m)\) in the language \(\{\in\}\), \[ \forall x_1,\dots, x_m(\varphi(x_1,\dots, x_m)\leftrightarrow \varphi({\mathbf j}(x_1),\dots,{\mathbf j}(x_m))), \] (3) Each instance of the usual Separation scheme involving a formula \(\psi\) in the language \(\{\in,{\mathbf j}\}\). These axioms assert that \({\mathbf j}\) is a nontrivial elementary embedding from the universe into itself and that Separation holds for formulas with occurrences of \({\mathbf j}\). It is shown that \(\text{ZFC}+ \text{WA}\) has consistency strength strictly between \(I_3\) and the existence of a cardinal that is super-\(n\) huge for every \(n\). (By a result of \textit{K. Kunen} [J. Symb. Log. 36, 407-413 (1971; Zbl 0272.02087)], the stronger theory obtained by asking that Replacement holds for formulas with occurrences of \({\mathbf j}\) is inconsistent.) \(\text{ZFC}+ \text{WA}\) is used as a starting point for studying generalizations of Laver sequences (a Laver sequence at an infinite cardinal \(\kappa\) is a function \(f:\kappa\to V_\kappa\) with the property that for every set \(x\) an every \(\lambda\geq \max(\kappa,|\text{TC}(x)|)\) there is a normal ultrafilter \(U\) on \(P_\kappa(\lambda)\) such that \(x= i_U(f)(\kappa)\), where \(i_U\) is the canonical embedding determined by \(U\)). For example, it is shown that if WA holds and \(\kappa\) is the critical point of the WA-embedding \(j: V\to V\), then there is a Laver sequence \(f\) at \(\kappa\) with the following property: there exist \(\lambda\), \(U\) such that \(\lambda>\kappa\) and \(U\) is a normal ultrafilter on \(P_\kappa(\lambda)\), \(|\text{TC}(i_U(f)(\kappa))|\leq \lambda\) and if \(D\) is the normal ultrafilter on \(\kappa\) derived from \(i_U\), then, …
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E65
zbMATH DE Number: 1546877
zbMATH Keywords: wholeness axiom
zbMATH Keywords: elementary embedding
zbMATH Keywords: Laver sequences
zbMATH Keywords: large cardinals
Wikidata QID: Q114012777
reviewed by: Pierre Matet
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1546877