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## Fundamenta Mathematicae

2012 | 219 | 1 | 15-36
Tytuł artykułu

### Supercompactness and failures of GCH

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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.
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Rocznik
Tom
Numer
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15-36
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wydano
2012
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autor
• Kurt Gödel Research Center, for Mathematical Logic, Währinger Strasse 25, 1090 Wien, Austria
autor
• Department of Logic, Charles University, Celetná 20, Praha 1, 116 42, Czech Republic
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