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: a non-trivial theory may prove 0 = 1 and not prove everything — though Meyer's finite model shows R# proves neither.

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# (1976) is the programme's strongest result: Gödel's second theorem blocks a finitary consistency proof for PA but does not block a finitary non-triviality proof for its relevant counterpart, because non-triviality is strictly weaker than consistency — a theory can prove 0 = 1 without proving everything. Friedman and Meyer, "Whither relevant arithmetic?" (1992), give the strongest result against it: R# does not contain PA, so the escape from Gödel comes from not proving classical arithmetic. Priest's collapsing lemma is the cleanest technical device: for a classical model M and a congruence on it, the quotient is an LP-model of every sentence true in M, which yields finite inconsistent models satisfying all theorems of PA while identifying some n with n+1. Those models are finite, hence the theories they determine are decidable and complete, giving up consistency and gaining exactly the properties Gödel showed cannot accompany it. Brady's non-triviality proofs (1971, 1989) do the same work for naive set theory over a relevant base, where Russell's set exists and its contradictory status does not propagate.

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 arithmetic, and the ω-rule fails in R#. The paraconsistent bases lose inference rules rather than axioms — LP invalidates disjunctive syllogism and modus ponens for the material conditional, both used constantly in ordinary proof, so classical arguments must be rewritten rather than reinterpreted. Priest's finite inconsistent models are quotients of classical ones and add no independent mathematics: the collapsing lemma establishes that the inconsistent models exist because the consistent ones do. Naive set theory over a contraction-free base incurs the costs recorded under noncontractive logic — unrestricted induction is unavailable and the known consistent systems are proof-theoretically very weak — so the difficulty is relocated rather than removed. And the programme has produced no theorem that classical mathematics wanted and could not obtain.

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