Gödel’s Landmark Paper

The Consistency of the Axiom of Choice and the Generalized Continuum-Hypothesis with the Axioms of Set Theory

Background

  • Set Theory and ZFC: The Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC) form the standard framework for modern set theory.
    • The Axiom of Choice (AC) states that for any collection of non-empty sets, there exists a choice function that selects one element from each set.
    • The Generalized Continuum Hypothesis (GCH) postulates that for any infinite cardinal κ, 2^κ = κ+ (the next cardinal).
  • Hilbert’s Program: David Hilbert asked about the consistency of mathematics, including AC and GCH within ZFC.
  • Motivation for Gödel’s Work: To show that AC and GCH are consistent with ZFC, assuming ZFC itself is consistent.

Gödel’s Results

  • Inner Models and Constructibility: Gödel introduced the constructible universe L, a class of sets built in a step-by-step, definable manner.
  • Axiom of Choice (AC): Gödel showed that AC holds within L.
  • Generalized Continuum Hypothesis (GCH): Gödel proved that 2^κ = κ+ for all infinite cardinals κ in L.
  • Consistency Proof: Gödel concluded that if ZFC is consistent, then ZFC + AC + GCH is also consistent.

Methodology

  • The Constructible Universe: Gödel defined L as a hierarchy indexed by ordinals:
    • L0 contains all hereditarily finite sets.
    • Lα+1 includes subsets of Lα that are definable from parameters.
    • Lλ for limit ordinals λ is the union of all Lα for α < λ.
  • Relative Consistency: Gödel showed that if ZFC is consistent, then so is ZFC + AC + GCH.

Impact and Subsequent Developments

  • Completeness vs. Independence: Paul Cohen (1963) showed that AC and GCH are independent of ZFC, meaning they can neither be proved nor disproved from ZFC.
  • Foundation of Modern Set Theory: Gödel’s methods laid the groundwork for later developments in set theory.
  • Philosophical Implications: Gödel’s work highlighted the limitations of formal systems and the existence of different “universes” of set theory.

Conclusion

Gödel’s 1940 paper demonstrated the relative consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the axioms of set theory. This foundational work had a profound impact on set theory and mathematical logic, influencing both its philosophy and technical practice.