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Axiomatic set theory: hierarchy, cardinal arithmetic and L

Graduate set theory: the cumulative hierarchy, transfinite recursion, cofinality and cardinal arithmetic, reflection, and Gödel's constructible universe L, which shows that AC and GCH are consistent relative to ZF.

math.found.axiomatic-set-theory

Onyesha kila kinachohitajika kufikia mada hii

Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.

Inajengwa juu ya

The axiom of choice and its equivalents Completeness, compactness and Löwenheim–Skolem

Masharti ya awali ya lazima 22 (hadi chini kabisa)

Utajifunza

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.

Fasili

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.

Nadharia

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.

Mawazo potofu ya kawaida

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.

Inaelekea kwa

Forcing and independence

Ugunduzi wa hiari

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

Usomaji zaidi

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