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P(x) = "x is Provable"

axiom 1 : P(x)→x "Statement x can be proven true."

1. (x∧¬x) consider a contradiction

2. x simplification(1)

3. ¬x simplification (1)

4. x∨∀sP(s) addition (2)

5. ∀sP(s) disjunctive syllogism (3,4)

6. (x∧¬x)→∀sP(s) conditional proof (1,5) "Anything is provable if it follows from a contradiction."

7. P(x) universal instantiation (5)

8. P(¬x) universal instantiation (5)

9. x axiom 1 (7)

10. ¬x axiom 1 (8)

11. (x∧¬x) conjunctional introduction (9,10)

12. ∀sP(s)→(x∧¬x) conditional proof (5,11)

"If any statement is a provable, then one can prove a contradiction."

13. ∀sP(s)↔(x∧¬x) biconditional introduction (6,12)

14. ∃s¬P(s)↔(¬x∨x) contraposition (13)

•13. "(All statements are provable) is false."

•14. "(There exists a statement that is unprovable) is true."

Does this also imply "There exists no unprovable statement that is false."?

axiom 1 : P(x)→x "Statement x can be proven true."

1. (x∧¬x) consider a contradiction

2. x simplification(1)

3. ¬x simplification (1)

4. x∨∀sP(s) addition (2)

5. ∀sP(s) disjunctive syllogism (3,4)

6. (x∧¬x)→∀sP(s) conditional proof (1,5) "Anything is provable if it follows from a contradiction."

7. P(x) universal instantiation (5)

8. P(¬x) universal instantiation (5)

9. x axiom 1 (7)

10. ¬x axiom 1 (8)

11. (x∧¬x) conjunctional introduction (9,10)

12. ∀sP(s)→(x∧¬x) conditional proof (5,11)

"If any statement is a provable, then one can prove a contradiction."

13. ∀sP(s)↔(x∧¬x) biconditional introduction (6,12)

14. ∃s¬P(s)↔(¬x∨x) contraposition (13)

•13. "(All statements are provable) is false."

•14. "(There exists a statement that is unprovable) is true."

Does this also imply "There exists no unprovable statement that is false."?

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