Pierre Matet
We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if $λ\geq 2^κ$, then (a) $I_{κ, λ}$ is not $(λ, 2)$-distributive, and (b) $I_{κ, λ}^+ \rightarrow (I_{κ, λ}^+)^2_ω$ does not hold.
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Suppose that there is a measurable cardinal. If \aleph_ωis a strong limit cardinal, but the power of \aleph_ωis bigger than \aleph_{ω_1}, then there is an inner model with a Woodin cardinal. Modulo the need of the measurable cardinal this answers a question of Gitik and Mitchell.
From large cardinals we show the consistency of normal, fine, $κ$-complete $λ$-dense ideals on $\mathcal{P}_κ(λ)$ for successor $κ$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.
Part A: A revised version of the guide in "Cardinal Arithmetic" ([Sh:g]), with corrections and expanded to include later works. Part B: Corrections to [Sh:g]. Part C: Contains some revised proof and improved theorems. Part D: Contains a list of relevant references.
We present a survey of some results of the pcf-theory and their applications to cardinal arithmetic. We review basics notions (in section 1), briefly look at history in section 2 (and some personal history in section 3). We present main results on pcf in section 5 and describe applications to cardinal arithmetic in section 6. The limitations on independence proofs are discussed in section 7, and in section 8 we discuss the status of two axioms that arise in the new setting. Applications to other areas are found in section 9.
We continue the investigations in the author's book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S_{<= aleph_0}(kappa), subseteq) for kappa real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(lambda, lambda, theta^+,2)=pp(lambda) in more cases even when cf(lambda)= aleph_0 (Section 1), deal with bounds of pp(lambda) for lambda limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6).
This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{<kappa}, pcf-inaccessibility, entangled orders (and narrow Boolean Algebras), measuring prod_{i<kappa}f(i) by a family of ideals and a family of sequences <B_i:i<kappa> with |B_i|<mu_i, and finally we deal with the existence of strongly almost disjoint families.
If cf(kappa) = kappa, kappa^+< cf(lambda) = λ, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing clubs and for each alpha<lambda we have: {C_delta \cap alpha: alpha \in nacc(C_delta)} has cardinality <lambda. Also, we prove that e.g. there is a stationary subset of S_{<aleph_1}(lambda) of cardinality cf(S_{<aleph_1}(lambda),subseteq) Then we prove the existence of nice filters when instead being normal filters on omega_1 they are normal filters with larger domai…
Finite sets, mappings, cardinals, and arithmetic in intuitionistic New Foundations
本文探讨了直觉新基础中有限集、幂集、映射、有限基数及其运算的理论，提出修正后的构造方法，探讨基数运算的可能性及算术溢出问题。
The $κ$-density of a cardinal $μ\geκ$ is the least cardinality of a dense collection of $κ$-subsets of $μ$ and is denoted by $\mathcal D(μ,κ)$. The Singular Density Hypothesis (SDH) for a singular cardinal $μ$ of cofinality $cfμ=κ$ is the equation $\mathcal D(μ,κ)=μ^+$. The Generalized Density Hypothesis (GDH) for $μ$ and $λ$ such that $λ\leμ$ is: $\mathcal D(μ,λ)=μ$ if $cfμ\not=cfλ$ and $\mathcal D(μ,λ)=μ^+$ if $cfμ=cfλ$. Density is shown to satisfy Silver's theorem. The most important case is: Theorem 2.6. If $κ=cfκ<θ=cfμ<μ$ and the set of cardinals $λ<μ$ of cofinality $κ$ that satisfy the \…
In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite set A consider Seq(A), the set of all sequences of A without repetition. We compare |Seq(A)|, the cardinality of this set, to |P(A)|, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF |- for all A: |Seq(A)| not= |P(A)| and we show that this is the best possible result. Furthermore,…
Andrew Wiles' Proof of Fermat's Last Theorem, As Expected, Does Not Require a Large Cardinal Axiom. A Discussion of Colin McLarty's "The Large Structures of Grothendieck Founded on Finite-Order Arithmetic"
本文探讨了 Fermat 大定理的证明是否依赖大基数公理，指出 Colin McLarty 证明了 Grothendieck 的大工具可建立在有限阶算术强度的 ZFC 片段上，从而消除了该证明依赖强于 ZFC 基础的表象。
We look at an old conjecture of A. Tarski on cardinal arithmetic and show that if a counterexample exists, then there exists one of length omega_1 + omega .
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is a…
In this paper we investigate some properties of first order theories which prevent them from having universal models under certain cardinal arithmetic assumptions. Our results give a new syntactical condition, oak property, which is a sufficient condition for a theory not to have universal models in cardinality lambda when certain cardinal arithmetic assumptions implying the failure of GCH (and close to the failure of SCH) hold.
We prove that for regular $λ$ above a strong limit singular $μ$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $λ$ and $μ$ there are coboundedly many regular $κ<μ$ such that $\clubsuit^-(S^λ_κ)$ holds whenever $λ=λ^{<κ}$.
Let $Z_2$, $Z_3$, and $Z_4$ denote $2^{\rm nd}$, $3^{\rm rd}$, and $4^{\rm th}$ order arithmetic, respectively. We let Harrington's Principle, {\sf HP}, denote the statement that there is a real $x$ such that every $x$--admissible ordinal is a cardinal in $L$. The known proofs of Harrington's theorem "$Det(Σ_1^1)$ implies $0^{\sharp}$ exists" are done in two steps: first show that $Det(Σ_1^1)$ implies {\sf HP}, and then show that {\sf HP} implies $0^{\sharp}$ exists. The first step is provable in $Z_2$. In this paper we show that $Z_2 \, + \, {\sf HP}$ is equiconsistent with ${\sf ZFC}$ and th…
We give an application of our extender based Radin forcing to cardinal arithmetic. Using a preparation forcing and interleaving of Cohen and Levy forcings in the normal Radin sequence we get a model with a power function having a fixed finite gap on all cardinals.
We investigate pseudopowers of singular cardinals, and show that deduce some consequences for cardinal arithmetic. For example, we show that in {\sf ZFC} that $$\cov(μ,μ,θ,σ)=\cov(μ,μ,(\cfμ)^+,σ)+\cov(μ,μ,θ,σ^+)$$ whenever $\aleph_1\leqσ=\cf(σ)<\cf(μ)<θ<μ$, and use recent work of Gitik to show that both summands in the equation are required.
Force a set model of $Z_3$ + Harrington's Principle
本文在存在remarkable基数且其上有弱不可达基数的假设下，通过无需重塑的集合力迫方法，构造了满足三阶算术（$Z_3$）加Harrington原理（HP）的集合模型。
Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<ł<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $ł$ and its existence is not a result of …
We show that some cardinal arithmetic configurations related to the negation of the Shelah Weak Hypothesis and natural from the forcing point of view are impossible.
Under some cardinal arithmetic assumptions, we prove that every stationary subset of lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond- colouring restrictions. We then deal with semi- saturated (normal) filters.
Covering matrices were introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of his work and in subsequent work with Sharon, he isolated two reflection principles, $\mathrm{CP}$ and $\mathrm{S}$, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of $\mathrm{CP}$ and $\mathrm{S}$ and variations on Jensen's square principle. We prove that, for a regular cardinal $λ> ω_1$, assuming large cardinals, $\square(λ, 2)$ is consistent with $\mathrm{C…
We prove that assuming suitable cardinal arithmetic, if B is a Boolean algebra every homomorphic image of which is isomorphic to a factor, then B has locally small density. We also prove that for an (infinite) Boolean algebra B, the number of subalgebras is not smaller than the number of endomorphisms, and other related inequalities. Lastly we deal with the obtainment of the supremum of the cardinalities of sets of pairwise incomparable elements of a Boolean algebra.
A club guessing toolbox I
本文综述文献中各种 club-guessing 形式，并系统研究 club-guessing 序列的改进方式，特别是猜测频率的校准，为不熟悉该领域的读者提供独立说明章节。
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