In mathematical logic, a tolerant sequence is a sequence

T₁,...,T_n

of formal theories such that there are consistent extensions

S₁,...,S_n

of these theories with each S_{i + 1} interpretable in S_i. Tolerance naturally generalizes from sequences of theories to trees of theories. Weak interpretability can be shown to be a special, binary case of tolerance.

This concept, together with its dual concept of cotolerance, was introduced by Japaridze (http://www.csc.villanova.edu/~japaridz/) in 1992, who also proved that, for Peano arithmetic and any stronger theories with effective axiomatizations, tolerance is equivalent to Π₁-consistency.

See also: interpretability, cointerpretability, interpretability logic.

References

• G.Japaridze (http://www.csc.villanova.edu/~japaridz/), The logic of linear tolerance. Studia Logica 51 (1992), pp. 249-277.
• G.Japaridze (http://www.csc.villanova.edu/~japaridz/), A generalized notion of weak interpretability and the corresponding logic. Annals of Pure and Applied Logic 61 (1993), pp. 113-160.
• G.Japaridze (http://www.csc.villanova.edu/~japaridz/study.html) and D. de Jongh, The logic of provability. Handbook of Proof Theory. S.Buss, ed. Elsevier, 1998, pp. 476-546.