Shelah's pcf Theory and Its Applications

2002 ◽  
Vol 8 (2) ◽  
pp. 307
Author(s):  
Menachem Kojman ◽  
Maxim R. Burke ◽  
Menachem Magidor
Keyword(s):  
Pcf Theory

scholarly journals Shelah's pcf theory and its applications

1990 ◽  
Vol 50 (3) ◽  
pp. 207-254 ◽  
Author(s):  
Maxim R. Burke ◽  
Menachem Magidor
Keyword(s):  
Pcf Theory

scholarly journals A ZFC Dowker space in $\aleph _{\omega +1}$: An application of pcf theory to topology

1998 ◽  
Vol 126 (8) ◽  
pp. 2459-2465 ◽  
Author(s):  
Menachem Kojman ◽  
Saharon Shelah
Keyword(s):  
Pcf Theory ◽  
Dowker Space

Applications of PCF theory

2000 ◽  
Vol 65 (4) ◽  
pp. 1624-1674 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Independent Sets ◽  
Boolean Algebras ◽  
Pcf Theory ◽  
Cardinal Invariants

AbstractWe deal with several pcf problems: we characterize another version of exponentiation: maximal number of k-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further TD{f). The sections can be read independently, although there are some minor dependencies.

Singular Cardinals and the PCF Theory

1995 ◽  
Vol 1 (4) ◽  
pp. 408-424 ◽  
Author(s):  
Thomas Jech
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Natural Generalization ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Quarter Century ◽  
Pcf Theory ◽  
Cardinal Numbers ◽  
Singular Cardinals ◽  
The Rich

§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 (“pcf” stands for “possible cofinalities”). 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(κ), 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.

SQUARES, SCALES AND STATIONARY REFLECTION

2001 ◽  
Vol 01 (01) ◽  
pp. 35-98 ◽  
Author(s):  
JAMES CUMMINGS ◽  
MATTHEW FOREMAN ◽  
MENACHEM MAGIDOR
Keyword(s):  
Continuum Hypothesis ◽  
Pcf Theory ◽  
Consistency Results ◽  
Singular Cardinals ◽  
Combinatorial Principles ◽  
Stationary Set ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic ◽  
The Continuum

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

Applications of pcf-Theory

1999 ◽  
pp. 249-268
Author(s):  
M. Holz ◽  
K. Steffens ◽  
E. Weitz
Keyword(s):  
Pcf Theory
Sign in / Sign up

Export Citation Format

Share Document