Theorem bj-babygodel 31761

Description: See the section header comments for the context.

The first hypothesis reads "𝜑 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 𝜑 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 𝜑, 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 ⊥ 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.)

Hypotheses

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

Assertion

Ref	Expression
bj-babygodel	⊢ ⊥

Proof of Theorem bj-babygodel

Step	Hyp	Ref	Expression
1		bj-babygodel.1	. . 3 ⊢ ¬ Prv ⊥
2	1	ax-prv1 31756	. 2 ⊢ Prv ¬ Prv ⊥
3		bj-babygodel.s	. . . . . . . . . 10 ⊢ (𝜑 ↔ ¬ Prv 𝜑)
4	3	biimpi 205	. . . . . . . . 9 ⊢ (𝜑 → ¬ Prv 𝜑)
5	4	prvlem1 31759	. . . . . . . 8 ⊢ (Prv 𝜑 → Prv ¬ Prv 𝜑)
6		ax-prv3 31758	. . . . . . . 8 ⊢ (Prv 𝜑 → Prv Prv 𝜑)
7		pm2.21 119	. . . . . . . . 9 ⊢ (¬ Prv 𝜑 → (Prv 𝜑 → ⊥))
8	7	prvlem2 31760	. . . . . . . 8 ⊢ (Prv ¬ Prv 𝜑 → (Prv Prv 𝜑 → Prv ⊥))
9	5, 6, 8	sylc 63	. . . . . . 7 ⊢ (Prv 𝜑 → Prv ⊥)
10	9	con3i 149	. . . . . 6 ⊢ (¬ Prv ⊥ → ¬ Prv 𝜑)
11	10, 3	sylibr 223	. . . . 5 ⊢ (¬ Prv ⊥ → 𝜑)
12	11	prvlem1 31759	. . . 4 ⊢ (Prv ¬ Prv ⊥ → Prv 𝜑)
13	12, 9	syl 17	. . 3 ⊢ (Prv ¬ Prv ⊥ → Prv ⊥)
14	1, 13	mto 187	. 2 ⊢ ¬ Prv ¬ Prv ⊥
15	2, 14	pm2.24ii 116	1 ⊢ ⊥

Syntax hints:  ¬ wn 3   ↔ wb 195  ⊥wfal 1480  Prv cprvb 31755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-prv1 31756  ax-prv2 31757  ax-prv3 31758

This theorem depends on definitions:  df-bi 196

This theorem is referenced by: (None)