Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-currypara Structured version   Visualization version   GIF version

Theorem bj-currypara 34009
 Description: Curry's paradox. Note that the proof is intuitionistic (use ax-3 8 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where 𝜑 is the self-referential sentence "If this sentence is true, then 𝜓", so that one can prove everything. Therefore, a consistent system cannot allow the formation of such self-referential sentences. This has lead to the study of logics rejecting contraction pm2.43 56, such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023.)
Assertion
Ref Expression
bj-currypara ((𝜑 ↔ (𝜑𝜓)) → 𝜓)

Proof of Theorem bj-currypara
StepHypRef Expression
1 bj-animbi 34008 . 2 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
2 simpr 488 . 2 ((𝜑𝜓) → 𝜓)
31, 2sylbir 238 1 ((𝜑 ↔ (𝜑𝜓)) → 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator