Description
Ordered field: field with total order compatible with + and ·. Completeness: every nonempty bounded above set has supremum. Archimedean property: ℕ unbounded in ℝ.
Dependency Flowchart
Note: Arrows mean "depends on".
graph TD
A1["A1 Field axioms\nR + ·"]
A2["A2 Order axioms\ntotal order, compat"]
D1["Def: Bounded above\n∃M ∀x x≤M"]
D2["Def: Supremum\nleast upper bound"]
D3["Def: Completeness\nsup exists"]
T1["Thm: Archimedean\n∀ε>0 ∃n nε>1"]
T2["Thm: Q dense in R"]
A1 --> D1
A2 --> D1
D1 --> D2
D2 --> D3
D3 --> T1
D3 --> T2
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 A1,A2 axiom
class D1,D2,D3 definition
class T1,T2 theorem
Color Scheme
Red
Axioms
Axioms
Blue
Definitions
Definitions
Teal
Theorems
Theorems
Process Statistics
- Nodes: 7
- Edges: 7
- Axioms: 2
- Definitions: 3
- Theorems: 2