Supercompactness and failures of GCH

Sy-David Friedman¹ ; Radek Honzik²

¹ Kurt Gödel Research Center for Mathematical Logic Währinger Strasse 25 1090 Wien, Austria
² Department of Logic Charles University Celetná 20 Praha 1, 116 42, Czech Republic

Fundamenta Mathematicae, Tome 219 (2012) no. 1, pp. 15-36

Résumé

Let $\kappa \lambda$ be regular cardinals. We say that an embedding $j: V\to M$ with critical point $\kappa$ is $\lambda$-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 and $\lambda^{++}$-tall, we preserve $\lambda$-supercompactness of $\kappa$ and kill the GCH at $\lambda$ by directly manipulating the size of $2^\lambda$ (i.e. we do not force the failure of GCH at $\lambda$ as a consequence of having $2^\kappa$ large enough). The direct manipulation of $2^\lambda$, where $\lambda$ can be a successor cardinal, is the first step toward understanding which Easton functions can be realized as the continuum function on regular cardinals while preserving instances of $\lambda$-supercompactness.

Détail

DOI : 10.4064/fm219-1-2

Keywords: kappa lambda regular cardinals say embedding critical point kappa lambda tall lambda kappa closed under kappa sequences nbsp silver showed gch fail measurable cardinal kappa starting kappa being kappa supercompact later woodin improved result starting optimal hypothesis kappa tall measurable cardinal kappa generally suppose kappa lambda regular wishes gch fail lambda kappa being lambda supercompact silvers methods done starting kappa being lambda supercompact note silvers result above special kappa lambda ask there analogue woodins result lambda supercompactness answer question following strong sense starting gch kappa being lambda supercompact lambda tall preserve lambda supercompactness kappa kill gch lambda directly manipulating size lambda force failure gch lambda consequence having kappa large enough direct manipulation lambda where lambda successor cardinal first step toward understanding which easton functions realized continuum function regular cardinals while preserving instances lambda supercompactness