A definable failure of the Singular Cardinal Hypothesis

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11210%2F12%3A10127068" target="_blank" >RIV/00216208:11210/12:10127068 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s11856-012-0044-x" target="_blank" >http://dx.doi.org/10.1007/s11856-012-0044-x</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11856-012-0044-x" target="_blank" >10.1007/s11856-012-0044-x</a>

Result language
angličtina
Original language name
A definable failure of the Singular Cardinal Hypothesis
Original language description
We show first that it is consistent that $kappa$ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of $H(kappa^+)$. Then with further forcing we show that it is consistent that GCH fails at $aleph_omega$, $aleph_omega$ strong limit, while there is a lightface definable wellorder of $H(aleph_{omega+1})$ (''definable failure'' of the singular cardinal hypothesis at $aleph_omega)$. The large cardinal hypothesis used is the existence of a $kappa^{++}$-strong cardinal, where $kappa$ is $kappa^{++}$-strong if there is an embedding $j:V to M$ with critical point $kappa$ such that $H(kappa^{++}) sub M$. By work of M.~Gitik and W.~J.~Mitchell cite{GITIKo2}, cite{MITcoreI}, our large cardinal assumption is almost optimal. The techniques of proof include the ''tuning-fork'' method of cite{FRIEDMANperfect} and cite{FRDOBtree}, a generalisation to large cardinals of the stationary-coding of cite{FRFprojective} and a new ''definable-co
Czech name
—
Czech description
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Type
J<sub>x</sub> - 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

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