외국사이트에 올린 글 그대로 올려봄

Math destruction 



Proof of the relationship between contradiction and falsity 

Contradiction<->(p and not p)<->False 

Contradiction<->False 

The contrapositive is 

No contradiction<->True 

------------------------------------------------------------- 

Proof by contradiction 

1. (A is false->Contradiction)->(A has proof) 

The contrapositive of 1 is 2. 

2. (A has no proof)->(A is false and no contradiction) 

No contradiction<->True 

3. (A has no proof)->(A is false) 

The contrapositive of 3 is 4 

4. (A is true)->(A has proof) 

------------------------------------------------------------------------------- 

5. (A is an axiom)->(A has no proof) 

Connecting 3 and 5 is 6 

6. (A is an axiom)->(A is false) 

The contrapositive of 6 is 7 

7. (A is true)->(A is not an axiom) 

-------------------------------------------------------------------------------- 

8. (A is an axiom)->(A is true) 

Connecting 4 and 8 is 9. 

9. (A is an axiom)->(A has proof) 

Contrapositive of 9 is 10 

10. (A has no proof)->(A is not an axiom) 

-------------------------------------------------------------------------------- 

5 and 9 are contradictory.  

6 and 8 are contradictory. 

explosion. 

----------------------------------------------------------------------------- 

9 is false 

8 is also false 

Because Connecting 4 and 8 is 9 

4 must be true if "Proof by contradiction" is true 

Therefore, the negation of 9 and 8 is true 

The negation of 9 is 11 

11. A is an axiom and A has no proof 

The negation of 8 is 12 

12. A is an axiom and A is false 

13 is true 

13. (A is an axiom) and (A has no proof) and (A is false) 

6 and 7 is also true

공리는 거짓이라는 증명임