Large Cardinals Research Articles

More on Connector Families

Let Gk,n be the family of all graphs on the same n vertices each having at least k connected components. We are interested in the largest cardinality of a subfamily in which the union of any two of the member graphs has at most k−2 connected components, and determine its exponential asymptotics.

EVERY COUNTABLE MODEL OF SET THEORY EMBEDS INTO ITS OWN CONSTRUCTIBLE UNIVERSE

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations.

Elementary chains and C (n)-cardinals

The C (n)-cardinals were introduced recently by Bagaria and are strong forms of the usual large cardinals. For a wide range of large cardinal notions, Bagaria has shown that the consistency of the corresponding C (n)-versions follows from the existence of rank-into-rank elementary embeddings. In this article, we further study the C (n)-hierarchies of tall, strong, superstrong, supercompact, and extendible cardinals, giving some improved consistency bounds while, at the same time, addressing questions which had been left open. In addition, we consider two cases which were not dealt with by Bagaria; namely, C (n)-Woodin and C (n)-strongly compact cardinals, for which we provide characterizations in terms of their ordinary counterparts. Finally, we give a brief account on the interaction of C (n)-cardinals with the forcing machinery.

Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ⩽κ not only does not collapse κ+ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) [3], Friedman and Halilović (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).

Large cardinals and basic sequences

The purpose of this paper is to present several applications of combinatorial principles, well-known in Set Theory, to the geometry of infinite dimensional Banach spaces, particularly to the existence of certain basic sequences. We mention also some open problems where set-theoretical techniques are relevant.

On Colimits and Elementary Embeddings

AbstractWe give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)-extendible cardinals.

Applications of pcf for mild large cardinals to elementary embeddings

The following pcf results are proved:1. Assume thatκ>ℵ0is a weakly compact cardinal. Letμ>2κbe a singular cardinal of cofinality κ. Then for every regularλ<ppΓ(κ)+(μ)there is an increasing sequence〈λi|i<κ〉of regular cardinals converging to μ such thatλ=tcf(∏i<κλi,<Jκbd).2. Let μ be a strong limit cardinal and θ a cardinal above μ. Suppose that at least one of them has an uncountable cofinality. Then there isσ⁎<μsuch that for everyχ<θthe following holds:θ>sup{suppcfσ⁎-complete(a)|a⊆Reg∩(μ+,χ)and|a|<μ}.As an application we show that:if κ is a measurable cardinal andj:V→Mis the elementary embedding by a κ-complete ultrafilter over κ, then for every τ the following holds:1.ifj(τ)is a cardinal thenj(τ)=τ;2.|j(τ)|=|j(j(τ))|;3.for any κ-complete ultrafilter W on κ, |j(τ)|=|jW(τ)|. The first two items provide affirmative answers to questions from Gitik and Shelah (1993) [2] and the third to a question of D. Fremlin.

Distributive proper forcing axiom and cardinal invariants

In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ? sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.

Aronszajn trees and the successors of a singular cardinal

From large cardinals we obtain the consistency of the existence of a singular cardinal ? of cofinality ? at which the Singular Cardinals Hypothesis fails, there is a bad scale at ? and ? ++ has the tree property. In particular this model has no special ? +-trees.

Subcompact cardinals, squares, and stationary reflection

We generalise Jensen’s result on the incompatibility of subcompactness with □. We show that α +-subcompactness of some cardinal less than or equal to α precludes $${\square _\alpha }$$ , but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling’s hierarchy of weak squares, showing which cases are precluded by α +-subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold.

Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH

Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.

Alleviation Noise Effect Using Flexible Cross Correlation Code in Spectral Amplitude Coding Optical Code Division Multiple Access Systems

A new code for Spectral Amplitude Coding - Optical Code Division Multiple Access (SAC-OCDMA) is proposed and is called Flexible Cross-correlation (FCC) code. The FCC code possesses various advantages; including easy code construction using Tridiagonal matrix compared to other SAC-OCDMA codes such as MDW, MQC and MFH, shorter code length and the important properties of this code is the flexible cross-correlation. Our results reveal that FCC code can effectively alleviated Phase Induced Intensity Noise (PIIN) and allows large cardinality and improve the bit error rate performance.

Criterion for propositional calculi to be finitely generated

The paper is concerned with propositional calculi having arbitrary modus inference operations that are analogous to the modus ponens operation. For these calculi the question on the existence and cardinality of sets of generators is examined. A criterion for propositional calculi with arbitrary modus inference operations to be finitely generated is put forward. For some calculi, the existence of a finite complete system of tautologies is shown to imply the existence of a basis of arbitrarily large finite cardinality. We show the existence of propositional calculi with countable basis, without any basis, and calculi having a complete subsystem without any basis.

Forcing axioms and the continuum hypothesis

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π2-sentences over the structure (H(ω2), ∈, NSω1), in the sense that its (H(ω2), ∈, NSω1) satisfies every Π2-sentence σ for which (H(ω2), ∈, NSω1) ⊨ σ can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π2-sentences over the structure (H(ω2), ∈, ω1) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $. In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.

Degree-Doubling Graph Families

Let ${\cal G}$ be a family of $n$-vertex graphs of uniform degree 2 with the property that the union of any two member graphs has maximum degree 4. We determine the leading term in the asymptotics of the largest cardinality of such a family. Several analogous problems are discussed.

Solving Capacitated Dominating Set by using covering by subsets and maximum matching

The Capacitated Dominating Set problem is the problem of finding a dominating set of minimum cardinality where each vertex has been assigned a bound on the number of vertices it has capacity to dominate. Cygan et al. showed in 2009 that this problem can be solved in O(n3mnn/3) or in O∗(1.89n) time using maximum matching algorithm. An alternative way to solve this problem is to use dynamic programming over subsets. By exploiting structural properties of instances that cannot be solved fast by the maximum matching approach, and “hiding” additional cost related to considering subsets of large cardinality in the dynamic programming, an improved algorithm is obtained. We show that the Capacitated Dominating Set problem can be solved in O∗(1.8463n) time.

A Metric Between Probability Distributions on Finite Sets of Different Cardinalities and Applications to Order Reduction

In this paper we define a metric distance between probability distributions on two distinct finite sets of possibly different cardinalities. The metric is defined in terms of a joint distribution on the product of the two sets, which has the two given distributions as its marginals, and has minimum entropy. Computing the metric exactly turns out to be NP-hard. Therefore an efficient greedy algorithm is presented for finding an upper bound on the distance. We then study the problem of optimal order reduction in the metric defined here. It is shown that every optimal reduced-order approximation must be an aggregation of the original distribution, and that optimal reduced order approximation is equivalent to finding an aggregation with maximum entropy. This problem also turns out to be NP-hard, so again a greedy algorithm is constructed for finding a suboptimal reduced order approximation. Taken together, all the results presented here permit the approximation of an independent and identically distributed (i.i.d.) process over a set of large cardinality by another i.i.d. process over a set of smaller cardinality. In future work, attempts will be made to extend this work to Markov processes over finite sets.

Partial near supercompactness

A cardinal κ is nearly θ-supercompact if for every A⊆θ, there exists a transitive M⊨ZFC− closed under <κ sequences with A,κ,θ∈M, a transitive N, and an elementary embedding j:M→N with critical point κ such that j(κ)>θ and j″θ∈N.22Here, we use ZFC− to mean the theory of ZFC without the Powerset axiom, but where ZFC is understood to be axiomatized with Collection instead of Replacement. This concept strictly refines the θ-supercompactness hierarchy as every θ-supercompact cardinal is nearly θ-supercompact, and every nearly 2θ<κ-supercompact cardinal κ is θ-supercompact. Moreover, if κ is a θ-supercompact cardinal for some θ such that θ<κ=θ, we can move to a forcing extension preserving all cardinals below θ++ where κ remains θ-supercompact but is not nearly θ+-supercompact. We will also show that if κ is nearly θ-supercompact for some θ⩾2κ such that θ<θ=θ, then there exists a forcing extension preserving all cardinals at or above κ where κ is nearly θ-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result. A forcing poset is <κ-directed closed if it is γ-directed closed for all γ<κ in the sense of Jech (2003) [13, Def. 21.6]. We will prove that if κ is nearly θ-supercompact for some θ⩾κ such that θ<θ=θ, then there is a forcing extension where its near θ-supercompactness is preserved and indestructible by any further <κ-directed closed θ-c.c. forcing of size at most θ. Finally, these cardinals have high consistency strength. Specifically, we will show that if κ is nearly θ-supercompact for some θ⩾κ+ for which θ<θ=θ, then AD holds in L(R). In particular, if κ is nearly κ+-supercompact and 2κ=κ+, then AD holds in L(R).

Polish Set Theorists and Large Cardinals

Polish Set Theorists and Large Cardinals

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

AbstractLetRbe a commutative ring with identity, and letMbe a unitary module overR. We callMH-smaller (HS for short) if and only ifMis infinite and |M/N| &lt; |M| for every nonzero submoduleNofM. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: supposeMis faithful overR,Ris a domain (we will show that we can restrict to this case without loss of generality), andKis the quotient field ofR. IfMis HS overR, thenRis HS as a module over itself,R⊆M⊆K, and there exists a generating setSforMoverRwith |S| &lt; |R|. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.