Supercompactness within the projective hierarchy
type: Item
instance of: scholarly article
title: Supercompactness within the projective hierarchy
zbMATH Open document ID: 0989.03054
DOI: 10.2307/2695035
author: Howard Becker
author: Steve Jackson
publication date: 2002-07-14T00:00:00Z
Mathematics Subject Classification ID: 03E60
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 1658166
zbMATH Keywords: projective ordinals
zbMATH Keywords: supercompactness
zbMATH Keywords: axiom of determinacy
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2695035
OpenAlex ID: W2033824611
cites work: On the determinacy of games on ordinals
cites work: Determinacy implies that \(\aleph_ 2\) is supercompact
cites work: Descriptive set theory
cites work: 𝐴𝐷 and the very fine structure of 𝐿(ℝ)
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published in: The Journal of Symbolic Logic
review text: In the (choice-free) ZF-context, a cardinal \(\kappa\) is called {\(\lambda\)-supercompact} if there is a fine, normal, \(\kappa\)-complete ultrafilter on \(\mathcal{P}_\kappa(\lambda)\). (Without the Axiom of Choice, this is not necessarily equivalent to the standard definition via embeddings, in particular, \(\lambda\)-supercompact cardinals can fail to be inaccessible and can actually be quite small.) NEWLINENEWLINENEWLINEA general feature of the theory under the Axiom of Determinacy is that the projective ordinals NEWLINE\[NEWLINE\pmb {\delta}^1_n:=\sup\{\alpha;\;\alpha\text{ is the length of a }\pmb{\Delta}^1_n\text{ prewellordering}\}NEWLINE\]NEWLINE have large cardinal properties. In this paper (and in this review), the base theory is \(\text{ZF}+\text{AD}\). NEWLINENEWLINENEWLINEResults of \textit{L. A. Harrington} and \textit{A. S. Kechris} [Ann. Math. Logic 20, 109-154 (1981; Zbl 0489.03018)] combined with results of \textit{D. A. Martin} and \textit{J. R. Steel} [Cabal Semin. 79-81, Proc. Caltech-UCLA Logic Semin. 1979-81, Lect. Notes Math. 1019, 86-96 (1983; Zbl 0529.03027)] yield that \(\pmb{\delta}^1_1 = \aleph_1\) is \((\pmb{\delta}^2_1)^{\mathbf{L}(\mathbb{R})}\)-supercompact and \textit{H. Becker} showed [Isr. J. Math. 40, 229-234 (1981; Zbl 0521.03036)] that \(\pmb{\delta}^1_2 = \aleph_2\) is \((\pmb{\delta}^2_1)^{\mathbf{L}(\mathbb{R})}\)-supercompact as well. NEWLINENEWLINENEWLINEIn the paper under review, the authors show that all projective ordinals are \(\lambda\)-supercompact for \(\lambda < \aleph_{\omega_1}\). NEWLINENEWLINENEWLINEThe proof consists of two parts: (a) Becker defined a notion of \(\lambda\)-regularity and showed that under certain assumptions \(\lambda\)-regular cardinals are \(\lambda\)-supercompact (Theorem 2.1). (b) Jackson proved that the projective ordinals have some degree of regularity (Theorem 4.1). NEWLINENEWLINENEWLINEBecker and Jackson combine these steps to the above result which has been superseded by a theorem o…
reviewed by: Benedikt Löwe
description: scientific article; zbMATH DE number 1658166