Zermelo–Fraenkel Set Theory with the Axiom of Choice

Tags: axiom of choice, axiom of infinity, axiom of completeness, set, ZF

set
set theory

Fundamental Notions

• Equality predicate: $=$ or ${=}_{\text{Set}}$
• Membership relation: $\in$ — binary predicate expressing set membership.

pure set

• Membership predicate: $\in$
• Equality: $=$ — as defined for sets.

Axioms of ZFC

ZFC consists of the following axioms:

1. extensionality axiom
2. foundation axiom
3. empty axiom (sometimes considered redundant)
4. pairing axiom
5. union axiom
6. separation axiom
7. replacement axiom
8. power set axiom
9. infinity set
10. axiom of choice

Axioms 1–9 form ZF, while the addition of 10 makes it ZFC.

Note: Axioms [4–9] can be viewed as constrained forms of the disproven comprehension axiom.

Interpretations of ZFC

• Platonic picture
• von Neumann universe
• Multiuniverse view
• formalism interpretation
• pragmatism interpretation
• finitism interpretation

ZFC uses:

• Logical quantifiers and connectives
• Variables that represent sets or statements
• Terms: may be variables, the empty set, or function outputs — always representing sets in ZFC

Related Axioms

• determinacy axiom
• constructibility axiom
• large cardinal axioms — strengthen the foundational framework

Applications

• Infinite combinatorics
• large cardinals

Advanced Concepts

• simple type theory
• first-order theory
• material set theory
• A rigorous reformulation of naïve set theory
• Axiomatization of set theory as the foundation of mathematics
• Variants:
  • Constructive ZFC
  • Class ZFC
• Cumulative hierarchy: universe built via ordinal-valued rank function
• Concepts like:
  • The empty set
  • Power set and union operations
  • Iterative conception of sets
  • algebraic set theory

Related Theories

• ZFA – an early variant allowing urelements
• NBG – conservative, finitely axiomatizable extension of ZFC
• MK – stronger, not finitely axiomatizable unless restricted; allows meta-classes

Alternative Foundations

• Predicative mathematics
• Constructive mathematics
• ETCS — structural set theory
• NFU — impredicative material set theory

Historical Figures

1. Ernst Zermelo
2. Abraham Fraenkel
3. Thoralf Skolem
4. John von Neumann

References

6. The ZFC axioms define a widely accepted foundation for mathematics. These include:

• axiom of extensionality
• axiom of infinity
• axiom of choice

[[Brown2008_PhilosophyMathematicsContemporaryIntroductionWorldProofsPictures]]  
  - CHAPTER 11: How to Refute the Continuum Hypothesis  
[[Harris.etal2008_CombinatoricsGraphTheory]]  
  - Section 3.3