Axiomatic Set Theory — Cantor & Cardinality

graph TD DefEq["Def: Equinumerosity\n|A| = |B| ⟺ ∃ bijection A → B"] DefDen["Def: Countable / denumerable\ninjection into ℕ or |·| ≤ ℵ₀"] C1["Chart 1: Power set 𝒫(S)\n& subset notion"] T1["Thm: ℤ and ℚ countable\nsame cardinality as ℕ"] T2["Thm: ℝ uncountable\n1891 diagonal; cardinality of ℝ exceeds aleph0"] T3["Thm: Cantor’s theorem\npower set strictly larger; no surjection onto 𝒫(S)"] HCH["Hyp: CH\nno cardinal strictly between aleph0 and 2^aleph0"] HGCH["Hyp: GCH\nforall infinite kappa, 2^kappa = kappa+"] C4["Chart 4: CH & GCH\nindependence from ZFC"] DefEq --> DefDen DefEq --> T1 DefDen --> T1 DefEq --> T2 C1 --> T3 T3 --> HCH T2 --> HCH T3 --> HGCH HCH --> C4 HGCH --> C4 classDef definition fill:#b197fc,color:#fff,stroke:#9775fa classDef theorem fill:#51cf66,color:#fff,stroke:#40c057 classDef hypothesis fill:#fd7e14,color:#fff,stroke:#e8590c classDef chartref fill:#868e96,color:#fff,stroke:#495057 class DefEq,DefDen definition class T1,T2,T3 theorem class HCH,HGCH hypothesis class C1,C4 chartref