Inconsistent Mathematics

Origin. Robert K. Meyer's relevant arithmetic R# (1976) and his conjecture that it proves its own non-triviality; Ross Brady's non-triviality proof for naive set theory (1971, 1989); Chris Mortensen's Inconsistent Mathematics (1995), which named the programme; Graham Priest, "Inconsistent models of arithmetic" (1997); Zach Weber's Paradoxes and Inconsistent Mathematics (2021) for the current state.

Models. Do the mathematics, keep the naive axioms, and change the logic underneath so the contradictions do not spread. Naive comprehension is consistent — non-trivial, rather — over a suitable paraconsistent base; arithmetic over relevant logic has models with a top element where everything is both true and false. The programme's claim is that the classical restrictions (the cumulative hierarchy, the ban on unrestricted comprehension) were responses to a logic problem and can be lifted once the logic is fixed.

Formalism.

Relevant arithmetic R#: Peano's axioms over the relevant logic R rather than classical logic. Meyer (1976): R# is non-trivial, and the proof is finitary — so Hilbert's programme succeeds for R#, where it fails for PA by Gödel II. The catch is what "non-trivial" means: R# may prove 0 = 1 and not prove everything.

Priest's collapsing lemma: Take a classical model M of arithmetic and a congruence ~ on it. The quotient M/~ is an LP-model of every sentence true in M. Finite inconsistent models of arithmetic exist: collapse ℕ above some n. Every classical theorem survives; the model is finite and inconsistent.

Inconsistent models of arithmetic: For each n, a model of size n satisfying all of PA's theorems, in which n = n+1. The arithmetic is complete, decidable, and inconsistent — trading consistency for the properties Gödel showed cannot coexist with it.

Naive set theory (Brady, Weber): Full comprehension: ∃A ∀x (x ∈ A ↔ φ(x)), no restriction. Over a relevant or paraconsistent base with contraction absent, non-trivial (Brady 1989). Russell's set exists and is both a member and not a member of itself; nothing follows. Weber develops cardinal arithmetic and a version of the ordinals inside it.

The Friedman–Meyer result (1992): R# does not contain PA. Some classical arithmetic theorems are underivable in relevant arithmetic — the ω-rule fails, and Meyer's programme does not deliver classical number theory. The strongest negative result the field has against itself.

Symbols.

Metatheory. Meyer's finitary non-triviality proof for R# is the programme's strongest card: Gödel's second theorem blocks a finitary consistency proof for PA and does not block a finitary non-triviality proof for its relevant counterpart, because non-triviality is weaker than consistency. Friedman and Meyer's 1992 result is the strongest card against: R# does not contain PA, so the escape from Gödel is bought by not doing classical arithmetic. Priest's collapsing lemma is the cleanest technical contribution and cuts both ways — it manufactures inconsistent models cheaply, which shows they exist and also that they are quotients of classical ones rather than independent objects.

Applies to. Paraconsistent foundations. Naive set theory and naive truth, where the same move works. The limits of Gödel's theorems, which the programme probes by changing the logic rather than the theory. Dialetheism, for which this is the constructive half — the part that builds rather than diagnoses.

Limitations. The Friedman–Meyer result is the standing objection: an arithmetic that does not prove the theorems of arithmetic has not replaced it. Priest's finite inconsistent models are quotients of classical models, so they add no independent mathematics — the collapsing lemma shows the inconsistent models exist because the consistent ones do. Naive set theory over a contraction-free base pays the price recorded under Noncontractive Logic: without contraction, ordinary reasoning that reuses a hypothesis needs repair, and the repairs are where the difficulty was relocated rather than removed. And the programme has produced no theorem that classical mathematics wanted and could not get.

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