There are reasonably nice logics

1991 ◽  
Vol 56 (1) ◽  
pp. 300-322 ◽  
Author(s):  
Wilfrid Hodges ◽  
Saharon Shelah
Keyword(s):  
Interpolation Theorem ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Elementary Chain ◽  
Uncountable Cardinal ◽  
Joint Embedding ◽  
Preservation Property ◽  
Strongly Compact Cardinals

A well-known question of Feferman asks whether there is a logic which extends the logic , is ℵ0-compact and satisfies the interpolation theorem. (Cf. Makowsky [M] for background and terminology.)The same question was open when ℵ1 in is replaced by any other uncountable cardinal κ. We shall show that when κ is an uncountable strongly compact cardinal and there is a strongly compact cardinal > κ, then there is such a logic. It is impossible to prove the existence of uncountable strongly compact cardinals in ZFC. However, the logic that we describe has a simple and natural definition, together with several other pleasant properties. For example it satisfies Robinson's lemma, PPP (pair preservation property, viz. the theory of the sum of two models is the sum of their theories), versions of the elementary chain lemma for chains of length < λ, and isomorphism of (suitable) ultralimits.This logic is described in §2 below; we call it 1. It is not a new logic—it was introduced in [Sh, Part II, §3] as an example of a logic which has the amalgamation and joint embedding properties. See the transparent presentation in [M]. But we shall repeat all the definitions. In [HS] we presented a logic with some of the same properties as 1, also based on a strongly compact cardinal λ; but unlike 1, it was not a sublogic of λ,λ.

ON ${\omega _1}$-STRONGLY COMPACT CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 266-278 ◽  
Author(s):  
JOAN BAGARIA ◽  
MENACHEM MAGIDOR
Keyword(s):  
Measurable Cardinal ◽  
Second Order ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Relational Structures ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Strongly Compact Cardinals ◽  
Order Reflection

Abstract An uncountable cardinal κ is called ${\omega _1}$ -strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$ -complete ultrafilter on I. We show that the first ${\omega _1}$ -strongly compact cardinal, ${\kappa _0}$ , cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$ . We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$ . Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$ .

scholarly journals TAMENESS FROM LARGE CARDINAL AXIOMS

2014 ◽  
Vol 79 (4) ◽  
pp. 1092-1119 ◽  
Author(s):  
WILL BONEY
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Strongly Compact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Dual Property ◽  
First Time ◽  
Strongly Compact Cardinals ◽  
Do So

AbstractWe show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS(K) below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, calledtype shortness, and show that it follows similarly from large cardinals.

Arthur W. Apter. On the least strongly compact cardinal. Israel journal of mathematics, vol. 35 (1980), pp. 225–233. - Arthur W. Apter. Measurability and degrees of strong compactness. The journal of symbolic logic, vol. 46 (1981), pp. 249–254. - Arthur W. Apter. A note on strong compactness and supercompactness. Bulletin of the London Mathematical Society, vol. 23 (1991), pp. 113–115. - Arthur W. Apter. On the first n strongly compact cardinals. Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2229–2235. - Arthur W. Apter and Saharon Shelah. On the strong equality between supercompactness and strong compactness.. Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103–128. - Arthur W. Apter and Saharon Shelah. Menas' result is best possible. Ibid., pp. 2007–2034. - Arthur W. Apter. More on the least strongly compact cardinal. Mathematical logic quarterly, vol. 43 (1997), pp. 427–430. - Arthur W. Apter. Laver indestructibility and the class of compact cardinals. The journal of symbolic logic, vol. 63 (1998), pp. 149–157. - Arthur W. Apter. Patterns of compact cardinals. Annals of pure and applied logic, vol. 89 (1997), pp. 101–115. - Arthur W. Apter and Moti Gitik. The least measurable can be strongly compact and indestructible. The journal of symbolic logic, vol. 63 (1998), pp. 1404–1412.

2000 ◽  
Vol 6 (1) ◽  
pp. 86-89
Author(s):  
James W. Cummings
Keyword(s):  
Israel Journal ◽  
American Mathematical Society ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Applied Logic ◽  
Strong Compactness ◽  
Strongly Compact Cardinals

Some results concerning strongly compact cardinals

1985 ◽  
Vol 50 (4) ◽  
pp. 874-880
Author(s):  
Yoshihiro Abe
Keyword(s):  
Order Type ◽  
Supercompact Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Identity Function ◽  
Power Set ◽  
Elementary Embeddings ◽  
Strongly Compact Cardinals

This paper consists of two parts. In §1 we mention the first strongly compact cardinal. Magidor proved in [6] that it can be the first measurable and it can be also the first supercompact. In [2], Apter proved that Con(ZFC + there is a supercompact limit of supercompact cardinals) implies Con(ZFC + the first strongly compact cardinal κ is ϕ(κ)-supercompact + no α < κ is ϕ(α)-supercompact) for a formula ϕ which satisfies certain conditions.We shall get almost the same conclusion as Apter's theorem assuming only one supercompact cardinal. Our notion of forcing is the same as in [2] and a trick makes it possible.In §2 we study a kind of fine ultrafilter on Pκλ investigated by Menas in [7], where κ is a measurable limit of strongly compact cardinals. He showed that such an ultrafilter is not normal in some case (Theorems 2.21 and 2.22 in [7]). We shall show that it is not normal in any case (even if κ is supercompact). We also prove that it is weakly normal in some case.We work in ZFC and much of our notation is standard. But we mention the following: the letters α,β,γ… denote ordinals, whereas κ,λ,μ,… are reserved for cardinals. R(α) is the collection of sets rank <α. φM denotes the realization of a formula φ to a class M. Except when it is necessary, we drop “M”. For example, M ⊩ “κ is φ(κ)-supercompact” means “κ is φM(κ)-supercompact in M”. If x is a set, |x| is its cardinality, Px is its power set, and . If also x ⊆ OR, denotes its order type in the natural ordering. The identity function with the domain appropriate to the context is denoted by id. For the notation concerning ultrapowers and elementary embeddings, see [11]. When we talk about forcing, “⊩” will mean “weakly forces” and “p < q” means “p is stronger than q”.

Ideals and combinatorial principles

1997 ◽  
Vol 62 (1) ◽  
pp. 117-122 ◽  
Author(s):  
Douglas Burke ◽  
Yo Matsubara
Keyword(s):  
Regular Cardinal ◽  
Stationary Subset ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Uncountable Cardinal ◽  
Combinatorial Principles ◽  
Stationary Subsets

It is well known that if σ is a strongly compact cardinal and λ a regular cardinal ≥ σ, then for every stationary subset X of {α < λ: cof (α) = ω} there is some β < λ such that X ⋂ β is stationary in β. In fact the existence of a uniform, countably complete ultrafilter over λ is sufficient to prove the same conclusion about stationary subsets of {α < λ: cof (α) = ω}. See [13] or [10]. By analyzing the proof of this theorem as presented in [10], we realized the same conclusion will follow from the existence of a certain ideal, not necessarily prime, on . Throughout we will assume that σ is a regular uncountable cardinal and use the word “ideal” to mean fine ideal.

scholarly journals Exactly controlling the non-supercompact strongly compact cardinals

2003 ◽  
Vol 68 (2) ◽  
pp. 669-688 ◽  
Author(s):  
Arthur W. Apter ◽  
Joel David Hamkins
Keyword(s):  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Measurable Cardinals ◽  
Strong Cardinals ◽  
Strongly Compact Cardinals

AbstractWe 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 Theorems 1 and 2 of [5], due to the first author.

Laver indestructibility and the class of compact cardinals

1998 ◽  
Vol 63 (1) ◽  
pp. 149-157 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Limit Point ◽  
Supercompact Cardinal ◽  
Generic Extension ◽  
Ground Model ◽  
Strongly Compact Cardinal ◽  
Joint Work ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
The Universe ◽  
Strongly Compact Cardinals

AbstractUsing an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every κ ∈ K is a supercompact cardinal indestructible under ∈-directed closed forcing, and every κ a measurable limit point of K is a strongly compact cardinal indestructible under κ-directed closed forcing not changing ℘(κ). We then derive as a corollary a model for the existence of a strongly compact cardinal κ which is not κ+ supercompact but which is indestructible under κ-directed closed forcing not changing ℘(κ) and remains non-κ+ supercompact after such a forcing has been done.

On measurable limits of compact cardinals

1999 ◽  
Vol 64 (4) ◽  
pp. 1675-1688
Author(s):  
Arthur W. Apter
Keyword(s):  
Supercompact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Strongly Compact Cardinals

AbstractWe extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and also a limit of strongly compact cardinals having no non-trivial degree of supercompactness. We also show that it is consistent for the least supercompact cardinal κ to be a limit of strongly compact cardinals and be so that every measurable limit of compact cardinals below κ has a non-trivial degree of supercompactness. In this model, the only compact cardinals below κ with a non-trivial degree of supercompactness are the measurable limits of compact cardinals.

scholarly journals WOODIN FOR STRONG COMPACTNESS CARDINALS

2019 ◽  
Vol 84 (1) ◽  
pp. 301-319
Author(s):  
STAMATIS DIMOPOULOS
Keyword(s):  
Identity Crisis ◽  
Large Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Strong Compactness ◽  
Definition Of ◽  
Vopěnka Cardinals

AbstractWoodin and Vopěnka cardinals are established notions in the large cardinal hierarchy and it is known that Vopěnka cardinals are the Woodin analogue for supercompactness. Here we give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor’s identity crisis theorem for the first strongly compact cardinal.

Sign in / Sign up

Export Citation Format

Share Document