Supercompactness and Failures of GCH

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11210%2F12%3A10127077" target="_blank" >RIV/00216208:11210/12:10127077 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.4064/fm219-1-2" target="_blank" >http://dx.doi.org/10.4064/fm219-1-2</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4064/fm219-1-2" target="_blank" >10.4064/fm219-1-2</a>

Result language

angličtina

Original language name

Supercompactness and Failures of GCH

Original language description

Let $kappa < lambda$ be regular cardinals. We say that an embedding $j: Vto M$ with critical point $kappa$ is emph{$lambda$-tall} if $lambda<j(kappa)$ and $M$ is closed under $kappa$-sequences in $V$. Silver showed that GCH can fail at a measurable cardinal $kappa$, starting with $kappa$ being $kappa^{++}$-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a $kappa^{++}$-tall measurable cardinal $kappa$. Now more generally, suppose that $kappa le lambda$ are regular and one wishes the GCH to fail at $lambda$ with $kappa$ being $lambda$-supercompact. Silver's methods show that this can be done starting with $kappa$ being $lambda^{++}$-supercompact (note that Silver's result above is the special case when $kappa = lambda$). One can ask if there is an analogue of Woodin's result for $lambda$-supercompactness. We answer this question in the following strong sense: starting with the GCH and $kappa$ being $lambda$-supercompact

Czech name

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Czech description

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Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

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Project

<a href="/en/project/GP201%2F09%2FP115" target="_blank" >GP201/09/P115: Logical and set-theoretical properties of the continuum function</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2012

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Fundamenta Mathematicae

ISSN

0016-2736

e-ISSN

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Volume of the periodical

219

Issue of the periodical within the volume

Listopad

Country of publishing house

PL - POLAND

Number of pages

21

Pages from-to

15-36

UT code for WoS article

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EID of the result in the Scopus database

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