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Arthur W. Apter

Publications and source records attributed to Arthur W. Apter.

10 recordsLinked to original sources
arXiv2026

The number of measures on very large measurable cardinals

We study the possible number of normal measures on a measurable cardinal in settings where inner model techniques are unavailable. Instead, we exploit consequences of the Ultrapower Axiom to obtain our theorems. We show that the classical Kimchi-Magidor result -that the first $n$ measurable cardinals can be strongly compact- can be combined with an arbitrary prescribed pattern for the number of normal measures they carry. We also prove that the first measurable cardinal above a supercompact cardinal can carry any given number of normal measures; the same conclusion is established for the first measurable limit of supercompact cardinals. As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda. Our analysis circumvents both the reliance of Friedman--Magidor on core model methods and the limitations of the Prikry-type forcing iterations of Gitik--Kaplan.

math.LO↗
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.

math.LO↗
arXiv2012

Singular cardinals and strong extenders

We investigate the circumstances under which there exist a singular cardinal $μ$ and a short $(κ, μ)$-extender $E$ witnessing "$κ$ is $μ$-strong", such that $μ$ is singular in $\Ult(V, E)$.

math.LO↗
arXiv2006

Large cardinals with few measures

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa+ many normal measures on the least measurable cardinal kappa. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of lambda strong compactness or lambda supercompactness measures on P_kappa(lambda) can be exactly lambda+, if lambda>kappa is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

math.LO↗
arXiv2003

Exactly controlling the non-supercompact strongly compact cardinals

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.

math.LO↗
arXiv2001

Indestructibility and the level-by-level agreement between strong compactness and supercompactness

Can a supercompact cardinal kappa be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above kappa, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.

math.LO↗
arXiv1999

Patterns of Compact Cardinals

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model of ``ZFC + Omega is the least inaccessible limit of measurable limits of supercompact cardinals + f :Omega-->2 is a function'', then there is a partial ordering P in V so that for Vbar = V^P, Vbar_Omega models ``ZFC + There is a proper class of compact cardinals + If f(alpha) = 0, then the alpha-th compact cardinal isn't supercompact + If f(alpha) = 1, then the alpha-th compact cardinal is supercompact''. We then prove a generalized version of this theorem assuming kappa is a supercompact limit of supercompact cardinals and f:kappa-->2 is a function, and we derive as corollaries of the generalized version of the theorem the consistency of the least measurable limit of supercompact cardinals being the same as the least measurable limit of non-supercompact strongly compact cardinals and the consistency of the least supercompact cardinal being a limit of strongly compact cardinals.

math.LO↗
arXiv1998

Universal Indestructibility

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.

math.LO↗