Logical Dreams

  title={Logical Dreams},
  author={Saharon Shelah},
  • S. Shelah
  • Published 26 November 2002
  • Mathematics
We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic. 
Are all natural numbers the same
Recently there have been several attempts to develop forcing axioms analogous to the proper forcing axiom (PFA) for cardinals of the form אn where n > 1. We investigate the difficulties of doing thisExpand
Are all natural numbers the same.
This is a report on state-of-the-art on the question of developing higher analogues of the forcing axiom PFA. Recently there have been several attempts to develop forcing axioms analogous to theExpand
Flow: the Axiom of Choice is independent from the Partition Principle
We introduce a general theory of functions called Flow. We prove ZF, non-well founded ZF and ZFC can be immersed within Flow as a natural consequence from our framework. The existence of stronglyExpand
  • M. Gitik
  • Mathematics, Computer Science
  • The Journal of Symbolic Logic
  • 2019
A new method is presented to blow up the power of a singular in the core model cardinal of uncountable cofinality by making use of overlapping extenders. Expand
Genericity and Arbitrariness
We compare the notions of genericity and arbitrariness on the basis of the realist import of the method of forcing. We argue that Cohen's Theorem, similarly to Cantor's Theorem, can be considered aExpand
A multiverse perspective on the axiom of constructiblity
I shall argue that the commonly held V 6 L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralistExpand
Forcing, Multiverse and Realism
In this article we analyze the method of forcing from a more philosophical perspective. After a brief presentation of this technique we outline some of its philosophical imports in connection withExpand
Mathematical Truth Revisited: Mathematics as a Toolbox
The notion of truth in Mathematics as relative to certain structures is discussed, very much in line with Bernays’s conception of “bezogene Existenz”, and it is argued that even so-called non-standard structures may have their own rationale. Expand
Large Cardinals and Determinacy
The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC.Expand
Foundational implications of the Inner Model Hypothesis
The purpose of this paper is to illustrate the Inner Model Hypothesis, and discuss it with respect to the current debate on the consequences of independence results in set theory. Expand


On nice equivalence relations on λ2
  • S. Shelah
  • Mathematics, Computer Science
  • Arch. Math. Log.
  • 2004
It is investigated whether for E there are many pairwise non equivalent sets for E, which is reasonably definable, and whether any two subsets with symmetric difference of size exactly 1 are not equivalent. Expand
On Logical Sentences in PA
Publisher Summary The chapter discusses the Paris–Harrington partition theorem (PH), which is used to prove the consistency of Peano arithmetic (PA). A finitary π 0 2 combinatorial principleExpand
The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal
The second edition of a well-established monograph on the identification of a canonical model in which the Continuum Hypothesis is false is updated to take into account some of the developments in the decade since the first edition appeared. Expand
Filters, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller Theorem
Two different types of models where, for certain singular cardinals λ of uncountable cofinality, λ → (λ, ω + 1)2, although λ is not a strong limit cardinal are presented. Expand
On power of singular cardinals
  • S. Shelah
  • Mathematics, Computer Science
  • Notre Dame J. Formal Log.
  • 1986
Etude de la puissance des cardinaux singuliers. Etablissement de bornes par des methodes elementaires
The core model
1. Fine Structure 2. Normal Measures 3. Mice 4. The Core Model 5. The Covering Lemma 6. Larger Core Models
Proper and Improper Forcing
This work deals with set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. Consequently, the theory of iteratedExpand
Polish algebras, shy from freedom
Every Polish group is not free whereas some Fσ group may be free. Also, every automorphism group of a structure of cardinality, e.g. ℶω, is not free.
You Can Enter Cantor's Paradise
This paper is based on the talk given by the author after he received the International Bolyai Prize in Mathematics (on November 4, 2000 in Budapest, Hungary).
Accessible Independence Results for Peano Arithmetic
Recently some interesting first-order statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely number-theoretic inExpand