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Theorem bj-babygodel 32572
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
StepHypRef Expression
1 bj-babygodel.1 . . 3 ¬ Prv ⊥
21ax-prv1 32567 . 2 Prv ¬ Prv ⊥
3 bj-babygodel.s . . . . . . . . . 10 (𝜑 ↔ ¬ Prv 𝜑)
43biimpi 206 . . . . . . . . 9 (𝜑 → ¬ Prv 𝜑)
54prvlem1 32570 . . . . . . . 8 (Prv 𝜑 → Prv ¬ Prv 𝜑)
6 ax-prv3 32569 . . . . . . . 8 (Prv 𝜑 → Prv Prv 𝜑)
7 pm2.21 120 . . . . . . . . 9 (¬ Prv 𝜑 → (Prv 𝜑 → ⊥))
87prvlem2 32571 . . . . . . . 8 (Prv ¬ Prv 𝜑 → (Prv Prv 𝜑 → Prv ⊥))
95, 6, 8sylc 65 . . . . . . 7 (Prv 𝜑 → Prv ⊥)
109con3i 150 . . . . . 6 (¬ Prv ⊥ → ¬ Prv 𝜑)
1110, 3sylibr 224 . . . . 5 (¬ Prv ⊥ → 𝜑)
1211prvlem1 32570 . . . 4 (Prv ¬ Prv ⊥ → Prv 𝜑)
1312, 9syl 17 . . 3 (Prv ¬ Prv ⊥ → Prv ⊥)
141, 13mto 188 . 2 ¬ Prv ¬ Prv ⊥
152, 14pm2.24ii 117 1
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wfal 1487  Prv cprvb 32566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-prv1 32567  ax-prv2 32568  ax-prv3 32569
This theorem depends on definitions:  df-bi 197
This theorem is referenced by: (None)
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