arXiv subjects

Brent Cody

Publications and source records attributed to Brent Cody.

26 records · Page 2Linked to original sources

arXiv2013

Easton functions and supercompactness

Suppose $κ$ is $λ$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ 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) $\forallα$ $α<\textrm{cf}(F(α))$ and (2) $α<β$ $\Longrightarrow$ $F(α)\leq F(β)$. In this article 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 $j:V\rightarrow M$ with critical point $κ$ such that $κ$ is closed under $F$, the model $M$ is closed under $λ$-sequences, $H(F(λ))\subseteq M$, and for each regular cardinal $γ\leq λ$ 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 B. Cody, M. Magidor, On supercompactness and the continuum function, Ann. Pure Appl. Logic, (2013).

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arXiv2013

On supercompactness and the continuum function

Given a cardinal $κ$ that is $λ$-supercompact for some regular cardinal $λ\geqκ$ and assuming $\GCH$, we show that one can force the continuum function to agree with any function $F:[κ,λ]\cap\REG\to\CARD$ satisfying $\forallα,β\in\dom(F)$ $α<\cf(F(α))$ and $α<β$ $\implies$ $F(α)\leq F(β)$, while preserving the $λ$-supercompactness of $κ$ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding $j:V\to M$ with critical point $κ$ such that $M^λ\subseteq M$ and $j(κ)>F(λ)$. Our argument extends Woodin's technique of surgically modifying a generic filter to a new case: Woodin's key lemma applies when modifications are done on the range of $j$, whereas our argument uses a new key lemma to handle modifications done off of the range of $j$ on the ghost coordinates. This work answers a question of Friedman and Honzik [FH2012]. We also discuss several related open questions.

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arXiv2013

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $θ$-supercompact

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $θ$-supercompact, for any desired $θ$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $θ_κ$-supercompact cardinals $κ$, for nearly any desired function $κ\mapstoθ_κ$. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.

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arXiv2012

Consecutive singular cardinals and the continuum function

We show that from a supercompact cardinal κ, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, κ and κ^+ are both singular, and the continuum function at κ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of κ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of ZF + not AC_ω in which either (1) aleph_1 and aleph_2 are both singular and the continuum function at aleph_1 can be precisely controlled, or (2) aleph_ω and aleph_{ω+1} are both singular and the continuum function at aleph_ω can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals κ and κ^+ in a model of ZF. Some open questions concerning the continuum function in models of ZF with consecutive singular cardinals are posed.

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arXiv2012

Easton's Theorem for Ramsey and Strongly Ramsey cardinals

We show that, assuming GCH, if $κ$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $κ$ and obeying the constraints of Easton's theorem, namely, $F(α)\leq F(β)$ for $α\leqβ$ and $α<\cf(F(α))$, then there is a cofinality preserving forcing extension in which $κ$ remains Ramsey or strongly Ramsey respectively and $2^δ=F(δ)$ for every regular cardinal $δ$.

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arXiv2012

Some results on large cardinals and the continuum function

Given a Woodin cardinal $δ$, I show that if $F$ is any Easton function with $F"δ\subseteqδ$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $λ$-supercompact cardinal $κ$ such that $\GCH$ fails at $λ$ is equiconsistent with the existence of a cardinal $κ$ that is $λ$-supercompact and $λ^{++}$-tall. I generalize a theorem on measurable cardinals due to Levinski, which says that given a measurable cardinal, there is a forcing extension preserving the measurability of $κ$ in which $κ$ is the least regular cardinal at which $\GCH$ holds. Indeed, I show that Levinski's result can be extended to many other large cardinal contexts. This work paves the way for many additional results, analogous to the results stated above for Woodin cardinals and partially supercompact cardinals.

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arXiv2012

Easton's Theorem in the presence of Woodin cardinals

Under the assumption that $δ$ is a Woodin cardinal and $\GCH$ holds, I show that if $F$ is any class function from the regular cardinals to the cardinals such that (1) $κ<\cf(F(κ))$, (2) $κ<λ$ implies $F(κ)\leq F(λ)$, and (3) $δ$ is closed under $F$, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. Unlike the analogous results for supercompact cardinals [Men76] and strong cardinals [FH08], there is no requirement that the function $F$ be locally definable.

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arXiv2012

The failure of GCH at a degree of supercompactness

We determine the large cardinal consistency strength of the existence of a $λ$-supercompact cardinal $κ$ such that GCH fails at $λ$. Indeed, we show that the existence of a $λ$-supercompact cardinal $κ$ such that $2^λ\geq θ$ is equiconsistent with the existence of a $λ$-supercompact cardinal that is also $θ$-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.

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