16 lines
396 B
Plaintext
16 lines
396 B
Plaintext
|
Applications of compactness theorem:
|
||
|
|
2-colorability of graphs
|
||
|
|
|
||
|
|
Goal: Given a graph G=(V,E) it is 2-colorable iff every finite subset of G is 2-colorable.
|
||
|
|
|
||
|
|
Proof outline:
|
||
|
|
Given a graph G=(V,E) construct a set \Sigma of wffs such that
|
||
|
|
G is 2-colorable
|
||
|
|
iff (step 1)
|
||
|
|
\Sigma is satisfiable.
|
||
|
|
iff
|
||
|
|
(by CT) \Sigma is finitely satisfiable
|
||
|
|
iff (step 2)
|
||
|
|
Each finite subset of G is 2-colorable.
|
||
|
|
|