71 research outputs found
Consecutive singular cardinals and the continuum function
We show that from a supercompact cardinal \kappa, there is a forcing
extension V[G] that has a symmetric inner model N in which ZF + not AC holds,
\kappa\ and \kappa^+ are both singular, and the continuum function at \kappa\
can be precisely controlled, in the sense that the final model contains a
sequence of distinct subsets of \kappa\ 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_\omega\ 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_\omega\ and aleph_{\omega+1} are both singular and the continuum
function at aleph_\omega\ can be precisely controlled. Additionally, we discuss
a result in which we separate the lengths of sequences of distinct subsets of
consecutive singular cardinals \kappa\ and \kappa^+ in a model of ZF. Some open
questions concerning the continuum function in models of ZF with consecutive
singular cardinals are posed.Comment: to appear in the Notre Dame Journal of Formal Logic, issue 54:3, June
201
L-like Combinatorial Principles and Level by Level Equivalence
We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional âL-like â combinatorial principles. In particular, this model satisfies the following properties: 1. âŠÎŽ holds for every successor and Mahlo cardinal ÎŽ. 2. There is a stationary subset S of the least supercompact cardinal Îș0 such that for every ÎŽ â S, €Ύ holds and ÎŽ carries a gap 1 morass. 3. A weak version of €Ύ holds for every infinite cardinal ÎŽ. 4. There is a locally defined well-ordering of the universe W, i.e., for all Îș â„ â”2 a regular cardinal, W H(Îș+) is definable over the structure ăH(Îș+),â ă by a parameter free formula. â2000 Mathematics Subject Classifications: 03E35, 03E55. â Keywords: Supercompact cardinal, strongly compact cardinal, strong cardinal, level by level equivalence between strong compactness and supercompactness, diamond, square, morass, locally defined well-ordering. âĄThe authorâs research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. §The author wishes to thank the referee for helpful comments, suggestions, and corrections which have been incorporated into the current version of the paper. 1 The model constructed amalgamates and synthesizes results due to the author, the author and Cummings, and Aspero Ì and Sy Friedman. It has no restrictions on the structure of its class of supercompact cardinals and may be considered as part of Friedmanâs âouter model programmeâ.
Singular cardinals and strong extenders
We investigate the circumstances under which there exist a singular cardinal
and a short -extender witnessing " is
-strong", such that is singular in \Ult(V, E).Comment: 8 page
A Note on Indestructibility and Strong Compactness
If Îș < λ are such that Îș is both supercompact and indestructible under Îș-directed closed forcing which is also (Îș+,â)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {ÎŽ < Îș | ÎŽ is ÎŽ+ strongly compact yet ÎŽ isnât ÎŽ+ supercompact} must be unbounded in Îș. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal Îș in which no cardinal ÎŽ> Îș is 2ÎŽ = ÎŽ+ supercompact, Îșâs supercom-pactness is indestructible under Îș-directed closed forcing which is also (Îș+,â)-distributive, and for every measurable cardinal ÎŽ, ÎŽ is ÎŽ+ strongly compact iff ÎŽ is ÎŽ+ supercompact.
Inner models with large cardinal features usually obtained by forcing
We construct a variety of inner models exhibiting features usually obtained
by forcing over universes with large cardinals. For example, if there is a
supercompact cardinal, then there is an inner model with a Laver indestructible
supercompact cardinal. If there is a supercompact cardinal, then there is an
inner model with a supercompact cardinal \kappa for which 2^\kappa=\kappa^+,
another for which 2^\kappa=\kappa^++ and another in which the least strongly
compact cardinal is supercompact. If there is a strongly compact cardinal, then
there is an inner model with a strongly compact cardinal, for which the
measurable cardinals are bounded below it and another inner model W with a
strongly compact cardinal \kappa, such that H_{\kappa^+}^V\subseteq HOD^W.
Similar facts hold for supercompact, measurable and strongly Ramsey cardinals.
If a cardinal is supercompact up to a weakly iterable cardinal, then there is
an inner model of the Proper Forcing Axiom and another inner model with a
supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis,
there is an inner model with level by level equivalence between strong
compactness and supercompactness, and indeed, another in which there is level
by level inequivalence between strong compactness and supercompactness. If a
cardinal is strongly compact up to a weakly iterable cardinal, then there is an
inner model in which the least measurable cardinal is strongly compact. If
there is a weakly iterable limit \delta of <\delta-supercompact cardinals, then
there is an inner model with a proper class of Laver-indestructible
supercompact cardinals. We describe three general proof methods, which can be
used to prove many similar results
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