Description
Compactness: T has model iff every finite subset has model. Downward Löwenheim-Skolem: countable theory has countable model. Upward L-S: infinite model has arbitrarily large elementary extension.
Dependency Flowchart
Note: Arrows mean "depends on". Assumes Charts 1–2.
graph TD
D1["Def: Consistent\nT has model"]
T1["Thm: Compactness\nT model ⟺ fin sat"]
T2["Thm: Downward L-S\ncountable submodel"]
T3["Thm: Upward L-S\nlarge elementary ext"]
T4["Thm: Completeness\nprovable ⟺ valid"]
D1 --> T1
T1 --> T2
T1 --> T3
T1 --> T4
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
class D1 definition
class T1,T2,T3,T4 theorem
Color Scheme
Blue
Definitions
Definitions
Teal
Theorems
Theorems
Process Statistics
- Nodes: 5
- Edges: 4
- Definitions: 1
- Theorems: 4