Large Cardinals Research Articles

Eastonʼs theorem and large cardinals from the optimal hypothesis

The equiconsistency of a measurable cardinal with Mitchell order o(κ)=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell (1984) [13] and M. Gitik (1989) [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see Gitik, 1993 [8]).In Friedman and Honzik (2008) [5], we formulated and proved Eastonʼs (1970) theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik (we used the assumption that the relevant target model contains H(μ), for a suitable μ, instead of the cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik (2008) [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals.

TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ+). Then with further forcing we show that it is consistent that GCH fails at ℵω, ℵω strong limit, while there is a lightface definable wellorder of H(ℵω+1) (“definable failure” of the singular cardinal hypothesis at ℵω). The large cardinal hypothesis used is the existence of a κ++-strong cardinal, where κ is κ++-strong if there is an embedding j: V → M with critical point κ such that H(κ++) ⊆ M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ++-strong cardinal is used throughout.

Violating the singular cardinals hypothesis without large cardinals

We extend a transitive model V of ZFC+GCH cardinal preservingly to a model N of ZF + “GCH holds below ℵ ω ” + “there is a surjection from the power set of ℵ ω onto λ”, where λ is an arbitrarily high fixed cardinal in V. The construction can be described as follows: add ℵ n +1 many Cohen subsets of ℵ n+1 for every n < ω, and adjoin λ many subsets of ℵ ω which are unions of ω-sequences of those Cohen subsets; then let N be a choiceless submodel generated by equivalence classes of the λ subsets of ℵ ω modulo an appropriate equivalence relation.

A local relaxation method for the cardinality constrained portfolio optimization problem

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a number of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints present in practice. The proposed local relaxation algorithm explores the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using a suitable heuristic method such as the centroids of initial clusters or be warm started based on the last output. Results, using a basket of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm can lead to significant performance gain over popular branch-and-cut methods. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency.

External automorphisms of ultraproducts of finite models

Let $${\fancyscript{L}}$$ be a finite first-order language and $${\langle{\fancyscript{M}_n} \,|\, {n < \omega}\rangle}$$ be a sequence of finite $${\fancyscript{L}}$$ -models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ? 2 is non-empty, then there is a non-principal ultrafilter $${\fancyscript{U}}$$ over ? such that the corresponding ultraproduct $${\prod_\fancyscript{U}\fancyscript{M}_n}$$ has an automorphism that is not induced by an element of $${\prod_{n<\omega}{\rm Aut}(\fancyscript{M}_n)}$$ .

現代集合論における巨大基数(あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

現代集合論における巨大基数(<特集>あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

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 κ for which 2κ = κ +, another for which 2κ = κ ++ 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 κ, such that \({H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}\). 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 δ of <δ-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.

Coincidences and Estimation of Entropies of Random Variables with Large Cardinalities

We perform an asymptotic analysis of the NSB estimator of entropy of a discrete random variable. The analysis illuminates the dependence of the estimates on the number of coincidences in the sample and shows that the estimator has a well defined limit for a large cardinality of the studied variable. This allows estimation of entropy with no a priori assumptions about the cardinality. Software implementation of the algorithm is available.

Codes for MIMO transmission based on orthogonal sequences

We consider MIMO communication systems with Rayleigh fading. We propose a new coded modulation based on orthogonal sequences and state a new decodability condition. We introduce concepts and constructions of permutation free (PF) and permutation and repetition free (PRF) codes. We also propose a construction of PRF codes with sign manipulation, whose code rate can exceed 1. For better analysis and construction of these codes we introduce a one-to-one mapping that transforms signal matrices to vectors over a finite field. We propose construction algorithms for PF and PRF codes. We build PF and PRF codes with large cardinality, which in several case achieve the maximum cardinality. Simulation of the constructed codes and estimation of their performance was done in Simulink environment. Results show high error-correcting capability, which often reaches that of STBC codes with full transmit diversity.

An Empirical Bayesian Approach for Identifying Differential Coexpression in High‐Throughput Experiments

A common goal of microarray and related high-throughput genomic experiments is to identify genes that vary across biological condition. Most often this is accomplished by identifying genes with changes in mean expression level, so called differentially expressed (DE) genes, and a number of effective methods for identifying DE genes have been developed. Although useful, these approaches do not accommodate other types of differential regulation. An important example concerns differential coexpression (DC). Investigations of this class of genes are hampered by the large cardinality of the space to be interrogated as well as by influential outliers. As a result, existing DC approaches are often underpowered, exceedingly prone to false discoveries, and/or computationally intractable for even a moderately large number of pairs. To address this, an empirical Bayesian approach for identifying DC gene pairs is developed. The approach provides a false discovery rate controlled list of significant DC gene pairs without sacrificing power. It is applicable within a single study as well as across multiple studies. Computations are greatly facilitated by a modification to the expectation-maximization algorithm and a procedural heuristic. Simulations suggest that the proposed approach outperforms existing methods in far less computational time; and case study results suggest that the approach will likely prove to be a useful complement to current DE methods in high-throughput genomic studies.

On the independence property of first order theories and indiscernible sequences

We refute the strong version of Shelah’s conjecture about models of large cardinalities, the independence property, and indiscernible sequences. We find necessary and sufficient conditions for a theory to lack the independence property and present applications of these conditions.

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω 2 . Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.

Ramsey-like cardinals

AbstractOne of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V = L.

Ramsey-like cardinals II

AbstractThis paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

Midpoint sets contained in the unit sphere of a normed space

The midpoint set M(S) of a set S of points is the set of all midpoints of pairs of points in S. We study the largest cardinality of a midpoint set M(S) in a finite-dimensional normed space, such that M(S) is contained in the unit sphere, and S is outside the closed unit ball. We show in three dimensions that this maximum (if it exists) is determined by the facial structure of the unit ball. In higher dimensions no such relationship exists. We also determine the maximum for euclidean and sup norm spaces.

Development of a new code family based on SAC-OCDMA system with large cardinality for OCDMA network

We have proposed a new Multi-Diagonal (MD) code for Spectral Amplitude – Coding Optical Code Division Multiple Access (SAC-OCDMA). Although this new MD code has many properties, one of the important properties of this code is that the cross correlation is always zero. Simplicity in code construction and flexibility in cross correlation control has made this code a compelling candidate for future OCDMA applications. The Multiple access interference (MAI) effects have been successfully and completely eliminated. Based on the theoretical analysis MD code is shown here to provide a much better performance compared to Modified Quadratic Congruence (MQC) code and Random Diagonal (RD) code. Proof-of-principle simulations of encoding with 5 and 10 users with 622 Mb/s data transmission at a BER of 10 −12 have been successfully demonstrated together with the DIRECT detection scheme.

Coding into HOD via normal measures with some applications

We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Roads to infinity: the mathematics of truth and proof

The Diagonal Argument Counting and Countability Does One Infinite Size Fit All? Cantor's Diagonal Argument Transcendental Numbers Other Uncountability Proofs Rates of Growth The Cardinality of the Continuum Historical Background Ordinals Counting Past Infinity The Countable Ordinals The Axiom of Choice The Continuum Hypothesis Induction Cantor Normal Form Goodstein's Theorem Hercules and the Hydra Historical Background Computability and Proof Formal Systems Post's Approach to Incompleteness Godel's First Incompleteness Theorem Godel's Second Incompleteness Theorem Formalization of Computability The Halting Problem The Entscheidungs problem Historical Background Logic Propositional Logic A Classical System A Cut-Free System for Propositional Logic Happy Endings Predicate Logic Completeness, Consistency, Happy Endings Historical Background Arithmetic How Might We Prove Consistency? Formal Arithmetic The Systems PA and PAomega Embedding PA in PAomega Cut Elimination in PAomega The Height of This Great Argument Roads to Infinity Historical Background Natural Unprovable Sentences A Generalized Goodstein Theorem Countable Ordinals via Natural Numbers From Generalized Goodstein to Well-Ordering Generalized and Ordinary Goodstein Provably Computable Functions Complete Disorder Is Impossible The Hardest Theorem in Graph Theory Historical Background Axioms of Infinity Set Theory without Infinity Inaccessible Cardinals The Axiom of Determinacy Largeness Axioms for Arithmetic Large Cardinals and Finite Mathematics Historical Background

Condensation and large cardinals

Wir definieren lokale Clubmengenkondensation (Local Club Condensation), ein Prinzip, welches Eigenschaften von Godels Kondensationsprinzip isoliert und verallgemeinert. Wir zeigen, dass wir uber einem beliebigen Modell der Mengenlehre durch die Erzwingungsmethode zu einem Modell der Mengenlehre gelangen konnen, welches lokale Clubmengenkondensation erfullt und zugleich verschiedene grosse Kardinalzahlen erhalten werden konnen; insbesondere zeigen wir, dass lokale Clubmengenkondensation mit der Existenz einer omega-superstarken Kardinalzahl konsistent ist. Wir gehen ahnlich fur Acceptability vor, ein weiteres Prinzip welches Aspekte von Godels Kondensationsprinzip isoliert und verallgemeinert. Dies setzt das Outer Model Program (zu deutsch Programm der ausseren Modelle) von Sy Friedman fort. Wir fuhren auch eine mogliche Zukunftsanwendung in Bezug auf das Proper Forcing Axiom PFA oben beschriebener Ergebnisse am Ende der Arbeit auf.