Axiomatic set theory: hierarchy, cardinal arithmetic and L

Builds on

Required prerequisites, all the way down: 22

You will learn to

• Define the cumulative hierarchy V_α by transfinite recursion and show, using foundation, that every set lies in some V_α.
• Compute with cofinality and prove König's theorem that cf(2^κ) > κ.
• Outline Gödel's proof that if ZF is consistent then so is ZFC + GCH, using the constructible universe L.

Definition

• V₀ = ∅, V_{α+1} = P(V_α), and V_λ = ⋃_{α<λ} V_α for limit ordinals λ.
• The cofinality cf(κ) is the least order type of a cofinal subset of κ; κ is regular when cf(κ) = κ.
• L is the union of levels L_α, where L_{α+1} consists of the subsets of L_α definable over L_α with parameters.
• The generalised continuum hypothesis (GCH) states that 2^κ = κ⁺ for every infinite cardinal κ.

Theorem

• König: cf(2^κ) > κ for every infinite cardinal κ; in particular 2^ℵ₀ ≠ ℵ_ω.
• Gödel: L satisfies ZFC + GCH, so if ZF is consistent then so is ZFC + GCH.
• Reflection: for each finite set of formulas, ZF proves that some V_α reflects them.

Common misconceptions

• Thinking L is the universe of all sets; V = L is an extra axiom, consistent with ZFC but not provable from it (if ZFC is consistent).
• Confusing a relative consistency proof with an absolute consistency proof.

Leads to

Optional discoveries

• Model theory Inner models such as L, and reflection arguments, are model-theoretic tools used inside set theory.
• ← ZFC: axiomatic set theory

Further reading

• Thomas Jech, Set Theory (Third Millennium Edition) (2003)
• Kenneth Kunen, Set Theory: An Introduction to Independence Proofs (1980)