Description
Axiom of Determinacy (AD): every set of reals is determined (infinite two-player games). AD contradicts AC in full ZFC; AD holds in L(ℝ) under large cardinals. AD implies regularity properties (all sets Lebesgue measurable, etc.). Connections to descriptive set theory and large cardinals.
Source: Wikipedia; Mycielski, Steinhaus; Woodin
Dependency Flowchart
Note: Arrows mean "depends on". Assumes Charts 1–3.
graph TD
ZFC["ZFC\n(Charts 1–3)"]
DefGame["Def: Infinite game\nω^ω, payoff set A"]
DefStrat["Def: Strategy\nσ: ω^ω → ω"]
DefDet["Def: Determined\nA has winning strategy"]
DefAD["Def: AD\nAll A⊆ω^ω determined"]
T1["Thm: AD ⊨ ¬AC\nAD contradicts full AC"]
T2["Thm: AD ⊨ all sets measurable\nRegularity"]
T3["Thm: AD ⊨ perfect set property\nUncountable has perfect subset"]
T4["Thm: Con(ZF+AD) needs large cardinals\nL(ℝ) under Woodin"]
DefLC["Def: Large cardinals\nMeasurable, Woodin"]
T5["Thm: ZF+AD consistent\nif ZFC + large cardinals"]
ZFC --> DefGame
DefGame --> DefStrat
DefStrat --> DefDet
DefDet --> DefAD
DefAD --> T1
DefAD --> T2
DefAD --> T3
ZFC --> DefLC
DefLC --> T4
DefAD --> T4
T4 --> T5
classDef axiom fill:#ff6b6b,color:#fff,stroke:#c0392b
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
class ZFC axiom
class DefGame,DefStrat,DefDet,DefAD,DefLC definition
class T1,T2,T3,T4,T5 theorem
Color Scheme
Red
Foundation (ZFC)
Foundation (ZFC)
Blue
Definitions
Definitions
Teal
Theorems
Theorems
Process Statistics
- Nodes: 12
- Edges: 14
- Axioms: 1
- Definitions: 5
- Theorems: 5