Supercompactness and failures of GCH

• Autorzy: Sy-David Friedman , Radek Honzik
• Języki publikacji: EN

Abstrakty

EN

Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V.
Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver's methods show that this can be done starting with κ being λ⁺⁺-supercompact (note that Silver's result above is the special case when κ = λ).
One can ask if there is an analogue of Woodin's result for λ-supercompactness. We answer this question in the following strong sense: starting with the GCH and κ being λ-supercompact and λ⁺⁺-tall, we preserve λ-supercompactness of κ and kill the GCH at λ by directly manipulating the size of $2^{λ}$ (i.e. we do not force the failure of GCH at λ as a consequence of having $2^{κ}$ large enough). The direct manipulation of $2^{λ}$, where λ 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 λ-supercompactness.

Kategorie tematyczne

• 03E55: Large cardinals
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2012
• Tom: 219
• Numer: 1
• Strony: 15-36

Daty

wydano

2012

Twórcy

autor

Sy-David Friedman

• Kurt Gödel Research Center, for Mathematical Logic, Währinger Strasse 25, 1090 Wien, Austria

autor

Radek Honzik

• Department of Logic, Charles University, Celetná 20, Praha 1, 116 42, Czech Republic

Identyfikatory

DOI

10.4064/fm219-1-2