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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Lightface $ \Sigma^1_2$-indescribable cardinals


Author: David Schrittesser
Journal: Proc. Amer. Math. Soc. 135 (2007), 1213-1222
MSC (2000): Primary 03E35, 03E55, 03E65
Published electronically: October 13, 2006
MathSciNet review: 2262928
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Abstract: $ \Sigma^1_3$-absoluteness for $ {ccc}$ forcing means that for any $ {ccc}$ forcing $ P$, $ {H_{\omega_1}}^V \prec_{\Sigma_2}{H_{\omega_1}}^{V^P}$. ``$ \omega_1$ inaccessible to reals'' means that for any real $ r$, $ {\omega_1}^{L[r]}<\omega_1$. To measure the exact consistency strength of `` $ \Sigma^1_3$-absoluteness for $ {ccc}$ forcing and $ \omega_1$ is inaccessible to reals'', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) $ \Sigma^1_2$-indescribable cardinal; $ \kappa$ has this property exactly if it is inaccessible and $ H_\kappa \prec_{\Sigma_2} H_{\kappa^+}$.


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Additional Information

David Schrittesser
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Währinger Straße 25, A-1090 Wien, Austria
Email: david@logic.univie.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08571-6
PII: S 0002-9939(06)08571-6
Keywords: Forcing axioms, indescribable cardinals
Received by editor(s): April 15, 2005
Received by editor(s) in revised form: October 9, 2005, and November 16, 2005
Published electronically: October 13, 2006
Additional Notes: During the preparation of this article, the author was supported by FWF-Project 16334. The author also would like to thank everyone at the Centre de Recerca Matemàtica, Barcelona, for their support. Lastly, many thanks to the referee for her (or his) work and patience. This article is also available from arXiv.org.
Communicated by: Julia Knight
Article copyright: © Copyright 2006 by the author