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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
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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 23
ستتعلّم
- 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.
تعريف
- 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.
مبرهنة
- 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.
مفاهيم خاطئة شائعة
- 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.
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اكتشافات اختيارية
- Model theory Inner models such as L, and reflection arguments, are model-theoretic tools used inside set theory.
- ← ZFC: axiomatic set theory
قراءات إضافية
- Thomas Jech, Set Theory (Third Millennium Edition) (2003)
- Kenneth Kunen, Set Theory: An Introduction to Independence Proofs (1980)