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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-currypara | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
bj-currypara | ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-animbi 34008 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓))) | |
2 | simpr 488 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
3 | 1, 2 | sylbir 238 | 1 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → 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) |
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