On the expressive power of naive set theory based on substructural logics

Cantor's naive set theory is characterized by the unrestricted comprehension principle, saying that for every formula A(x), there exists a set {x|A(x)} such that A(t) ↔ t ∈ {x|A(x)} for any term t. The theory is intuitive, elegant, powerful, but unfortunately inconsistent (as witnessed by Russell's paradox). While it is common to somehow restrict the comprehension scheme (as in ZFC), there is another way to avoid inconsistency. It is observed by Grishin [Gri81] that the existence of Russell's formula does not lead to inconsistency when the underlying logic lacks the contraction inference rule. Since then, naive set theory has been investigated in the framework of contraction-free logics such as BCK [OK85]. By BCK set theory, we mean quantificational BCK logic enriched with the unrestricted comprehension scheme. It enjoys cut-elimination, hence it is provably consistent. The equality relation is given by (t = u) ≡ ∀x.(t ∈ x − • u ∈ x). Then we can define in BCK set theory basic concepts such as the empty set, singletons, pairs and ordered pairs in the standard way. Numerals are also definable, by letting 0 ≡ ∅ ≡ {x|x = x} and S(t) ≡ ≡∅, t. The most fundamental property is the fixpoint theorem: for every formula A(x, y) there exists a term f such that x ∈ f • − • A(x, f) is provable in BCK set theory. Based on the fixpoint theorem, we can show that every recursively enumerable predicate is weakly numeralwise representable (a result essentially proved by Shirahata [Shi99]). Thus BCK set theory is undecidable. The last result shows that BCK set theory is descriptively very rich. On the other hand, it is computationally (and proof-theoretically) too weak, as cut-elimination can be done in quadratic steps in the absense of contraction. In some sense, BCK set theory may be compared with Robinson's system Q in arithmetic; both are numeralwise expressive enough, but yet to be extended to gain suitable computational power. In [Gir98], light linear logic (LLL) is introduced as a subsystem of linear logic and it is proved that its proofs precisely correspond to the polynomial time functions under the Curry-Howard correspondence. Being a proper subsystem of linear logic, LLL only admits a substantially restricted form of contraction (in other words, the modality governing structural inferences is properly weaker than the S4-modality of linear logic). Hence there is a possibility to adapt Grishin's idea and to …