Tolerant sequence

A sequence $\langle T_{1},\ldots ,T_{n}\rangle$ of formal theories is said to be tolerant iff there are consistent extensions $S_{1}$,\ldots,$S_{n}$ of these theories such that each $T_{i+1}$ is interpretable (see interpretability) in $T_{i}$.

This concept, together with its dual concept of cotolerance, was introduced by Japaridze in 1993, who also proved that, for Peano arithmetic and any stronger theories with effective axiomatizations, tolerance is equivalent to $\Pi _{1}$-consistency.

See also interpretability, interpretability logic.

References

• G.Japaridze, A generalized notion of weak interpretability and the corresponding logic. Annals of Pure and Applied Logic 61 (1993), pp. 113-160.
• G.Japaridze and D. de Jongh, The logic of provability. Handbook of Proof Theory. S.Buss, ed. Elsevier, 1998, pp. 476-546.