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Supercompactness and failures of GCH

100%

Friedman S.-D., Honzik R.

Fundamenta Mathematicae

2012

tom 219

nr 1

15-36

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.

Easton functions and supercompactness

81%

Cody B., Friedman S.-D., Honzik R.

Fundamenta Mathematicae

2014

tom 226

nr 3

279-296

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].

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Friedman S.-D.

2 Honzik R.

1 Cody B.

1 2014

1 2012

Wyniki wyszukiwania

Supercompactness and failures of GCH

Easton functions and supercompactness