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## Fundamenta Mathematicae

2014 | 226 | 3 | 279-296
Tytuł artykułu

### Easton functions and supercompactness

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Suppose that κ is λ-supercompact witnessed by an elementary embedding j: V → M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) ∀α α < cf(F(α)), and (2) α < β ⇒ F(α) ≤ F(β). We address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while preserving the λ-supercompactness of κ ?
We show that, assuming GCH, if F is any function as above, and in addition for some regular cardinal λ > κ there is an elementary embedding j: V → M with critical point κ such that κ is closed under F, the model M is closed under λ-sequences, H(F(λ)) ⊆ M, and for each regular cardinal γ ≤ λ one has $(|j(F)(γ)| = F(γ))^{V}$, then there is a cardinal-preserving forcing extension in which $2^{δ} = F(δ)$ for every regular cardinal δ and κ remains λ-supercompact. This answers a question of [CM14].
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Tom
Numer
Strony
279-296
Opis fizyczny
Daty
wydano
2014
Twórcy
autor
• Department of Mathematics, and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA 23284, U.S.A.
autor
• Kurt Gödel Research Center, for Mathematical Logic, University of Vienna, Währinger Straße 25, 1090 Wien, Austria
autor
• Kurt Gödel Research Center, for Mathematical Logic, University of Vienna, Währinger Straße 25, 1090 Wien, Austria
• Department of Logic, Charles University, Palachovo nám. 2, 116 38 Praha 1, Czech Republic
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