Axiomatic Set Theory (ZFC)

Zermelo–Fraenkel set theory with the Axiom of Choice. Dependency graphs showing axioms, definitions, and key theorems. Charts 1–3: ZFC foundations. Chart 4: Continuum Hypothesis & independence (Gödel, Cohen). Chart 5: Forcing (posets, generic filters, M[G]). Chart 6: Axiom of Determinacy (AD) and large cardinals. Chart 7: Cantor & cardinality (countability, diagonal argument, Cantor’s theorem, bridge to CH/GCH).

Cantor & cardinality (historical thread). See Chart 7 for the dedicated flowchart. Cantor showed that the integers ℤ and the rationals ℚ are countable (same cardinality as ℕ), while ℝ is uncountable (Cantor’s diagonal argument, 1891; earlier non-denumerability of ℝ in 1874). Cantor’s theorem states |𝒫(S)| > |S| for every set S—in particular 2^ℵ₀ > ℵ₀. These results motivate the Continuum Hypothesis (CH: there is no cardinal strictly between ℵ₀ and |ℝ| = 2^ℵ₀) and the Generalized Continuum Hypothesis (GCH: for every infinite cardinal κ, 2^κ = κ⁺). CH and GCH are independent of ZFC (Gödel’s constructible universe L; Cohen’s forcing).