Theorem bj-babygodel

Description: See the section header comments for the context.

The first hypothesis reads " ph is true if and only if it is not provable in T" (and having this first hypothesis means that we can prove this fact in T). The wff ph is a formal version of the sentence "This sentence is not provable". The hard part of the proof of Gödel's theorem is to construct such a ph , called a "Gödel–Rosser sentence", for a first-order theory T which is effectively axiomatizable and contains Robinson arithmetic, through Gödel diagonalization (this can be done in primitive recursive arithmetic). The second hypothesis means that F. is not provable in T, that is, that the theory T is consistent (and having this second hypothesis means that we can prove in T that the theory T is consistent). The conclusion is the falsity, so having the conclusion means that T can prove the falsity, that is, T is inconsistent.

Therefore, taking the contrapositive, this theorem expresses that if a first-order theory is consistent (and one can prove in it that some formula is true if and only if it is not provable in it), then this theory does not prove its own consistency.

This proof is due to George Boolos,Gödel's Second Incompleteness Theorem Explained in Words of One Syllable, Mind, New Series, Vol. 103, No. 409 (January 1994), pp. 1--3.

(Contributed by BJ, 3-Apr-2019)

	Ref	Expression
Hypotheses	bj-babygodel.s	⊢ ( 𝜑 ↔ ¬ Prv 𝜑 )
	bj-babygodel.1	⊢ ¬ Prv ⊥
Assertion	bj-babygodel	⊢ ⊥

Proof

Step	Hyp	Ref	Expression
1		bj-babygodel.s	⊢ ( 𝜑 ↔ ¬ Prv 𝜑 )
2		bj-babygodel.1	⊢ ¬ Prv ⊥
3	2	ax-prv1	⊢ Prv ¬ Prv ⊥
4	1	biimpi	⊢ ( 𝜑 → ¬ Prv 𝜑 )
5	4	prvlem1	⊢ ( Prv 𝜑 → Prv ¬ Prv 𝜑 )
6		ax-prv3	⊢ ( Prv 𝜑 → Prv Prv 𝜑 )
7		pm2.21	⊢ ( ¬ Prv 𝜑 → ( Prv 𝜑 → ⊥ ) )
8	7	prvlem2	⊢ ( Prv ¬ Prv 𝜑 → ( Prv Prv 𝜑 → Prv ⊥ ) )
9	5 6 8	sylc	⊢ ( Prv 𝜑 → Prv ⊥ )
10	9	con3i	⊢ ( ¬ Prv ⊥ → ¬ Prv 𝜑 )
11	10 1	sylibr	⊢ ( ¬ Prv ⊥ → 𝜑 )
12	11	prvlem1	⊢ ( Prv ¬ Prv ⊥ → Prv 𝜑 )
13	3 12 9	mp2b	⊢ Prv ⊥
14	13 2	pm2.24ii	⊢ ⊥