It is consistent that, for every n ≥ 2, every stationary subset of ω n consisting of ordinals of cofinality ω k, where k = 0 or k ≤ n - 3, reflects fully in the set of ordinals of cofinality ω n - 1. We also show that this result is best possible.
§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals (...) problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory. The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4.In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.§2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P, the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ. (shrink)
This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure.Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results (...) provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article.§1. Complete Boolean algebras and weak distributivity. ABoolean algebrais a setBwith Boolean operationsa˅b,a˄b and −a, partial orderinga≤bdefined bya˄b=aand the smallest and greatest element,0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S−a, ordered by inclusion, with0= ∅ and1=S.Complete Boolean algebras and weak distributivity.A Boolean algebrais a setBwith Boolean operationsa˅b,a˄b and -a, partial orderinga≤bdefined bya˄b=aand the smallest and greatest element.0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S-a, ordered by inclusion, with0= ϕ and1=S. (shrink)
A stationary subset S of a regular uncountable cardinal κ reflects fully at regular cardinals if for every stationary set $T \subseteq \kappa$ of higher order consisting of regular cardinals there exists an α ∈ T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection (...) holds at every λ ≤ κ and κ is measurable. (shrink)
There exists a family $\{B_\alpha\}_{\alpha of sets of countable ordinals such that: (1) max B α = α, (2) if α ∈ B β then $B_\alpha \subseteq B_\beta$ , (3) if λ ≤ α and λ is a limit ordinal then B α ∩ λ is not in the ideal generated by the $B_\beta, \beta , and by the bounded subsets of λ, (4) there is a partition {A n } ∞ n = 0 of ω 1 such that for (...) every α and every n, B α ∩ A n is finite. (shrink)
It is shown (in ZF) that every hereditarily countable set has rank less than ω 2 , and that if ℵ 1 is singular then there are hereditarily countable sets of all ranks less than ω 2.
It is consistent that, for every $n \geq 2$, every stationary subset of $\omega_n$ consisting of ordinals of cofinality $\omega_k$, where $k = 0$ or $k \leq n - 3$, reflects fully in the set of ordinals of cofinality $\omega_{n - 1}$. We also show that this result is best possible.
We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.
Reviewed Works:John R. Steel, A. S. Kechris, D. A. Martin, Y. N. Moschovakis, Scales on $\Sigma^1_1$ Sets.Yiannis N. Moschovakis, Scales on Coinductive Sets.Donald A. Martin, John R. Steel, The Extent of Scales in $L$.John R. Steel, Scales in $L$.
If F is a normal filter on a regular uncountable cardinal κ, let |f| be the F-norm of an ordinal function f. We introduce the class of positive ordinal operations and prove that if F is a positive operation then |F(f)| ≥ F(|f|). For each $\eta let f η be the canonical ηth function. We show that if F is a Σ operation then F(f η ) = f F(η) . As an application we show that if κ is greatly (...) Mahlo then there are normal filters on κ of order greater than κ +. (shrink)
We investigate the system TRC of untyped illative combinatory logic that is equiconsistent with New Foundations. We prove that various unstratified combinators do not exist in TRC.
