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Theorem bj-babygodel 31567
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 31562 . 2 Prv ¬ Prv ⊥
3 bj-babygodel.s . . . . . . . . . 10 (𝜑 ↔ ¬ Prv 𝜑)
43biimpi 204 . . . . . . . . 9 (𝜑 → ¬ Prv 𝜑)
54prvlem1 31565 . . . . . . . 8 (Prv 𝜑 → Prv ¬ Prv 𝜑)
6 ax-prv3 31564 . . . . . . . 8 (Prv 𝜑 → Prv Prv 𝜑)
7 pm2.21 118 . . . . . . . . 9 (¬ Prv 𝜑 → (Prv 𝜑 → ⊥))
87prvlem2 31566 . . . . . . . 8 (Prv ¬ Prv 𝜑 → (Prv Prv 𝜑 → Prv ⊥))
95, 6, 8sylc 62 . . . . . . 7 (Prv 𝜑 → Prv ⊥)
109con3i 148 . . . . . 6 (¬ Prv ⊥ → ¬ Prv 𝜑)
1110, 3sylibr 222 . . . . 5 (¬ Prv ⊥ → 𝜑)
1211prvlem1 31565 . . . 4 (Prv ¬ Prv ⊥ → Prv 𝜑)
1312, 9syl 17 . . 3 (Prv ¬ Prv ⊥ → Prv ⊥)
141, 13mto 186 . 2 ¬ Prv ¬ Prv ⊥
152, 14pm2.24ii 115 1
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wfal 1479  Prv cprvb 31561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-prv1 31562  ax-prv2 31563  ax-prv3 31564
This theorem depends on definitions:  df-bi 195
This theorem is referenced by: (None)
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