文档介绍：The Consistency of the
GCH with the axioms of
ZF Set Theory
By Jesse Vernon
What is ZF Set Theory?
z Zermelo-Frankel Set Theory
z Axiomatic Formulation of the theory of sets based
on First Order Logic
z Abstracts mathematical notions into the idea of sets
Numbers, Functions, etc… can be represented as sets
mon Foundation theory for describing
mathematics
z Advanced by Cantor in late 1800s
z Introduced concepts of Cardinality and Diagnalization
z Improved by Zermelo, Frankel and Skolem in 1922
The Axioms of ZF
z 1) Axiom of extensionality: Two sets are the same if they have the same
elements.
z 2) Axiom of foundation: Every non-empty set X contains some member y such
that X and y are disjoint sets.
z 3) Axiom schema of separation: If Z is a set and φ is any property relativized
to Z, then there is a subset Y of Z containing those x in Z for which φ(x).
z 4) Axiom of pairing: If X and Y are sets then there exists a set containing both
of them.
z 5) Axiom of union: For any set Z there is a set A containing every set that is a
member of some member of Z.
z 6) Axiom schema of replacement: For every formally defined function f whose
domain is a set there is a set containing the range of f.
z 7) Axiom of infinity: There exists a set X such that the empty set is a member
of X and whenever y is in X, so is S(y).
z 8) Axiom of power set: For any set X there is a set Y that contains every
subset of X.
Axiom of ZFC:
z 9) Axiom of choice: Every set can be well ordered.
What is the GCH?
z The Generalized Continuum Hypothesis
z Motivation: The Cardinality of a Set
z Ordinal Number:
A well ordered set where every element is also a
subset.
z Cardinality of a Set:
(AC) The least ordinal number α such that there is a
bijection between α and the set.
|is defined to be |ω 0א The Cardinal number
+
αאα, namelyאα+1 = the essor ofא
What is the GCH?
z More Motivation:
z Mathematics on a trans-finite