Tolerant sequence

In mathematical logic, a tolerant sequence is a sequence

:$T\_1$,...,$T\_n$

of formal theories such that there are consistent extensions

:$S\_1$,...,$S\_n$

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 [http://www.csc.villanova.edu/~japaridz/ Japaridze] in 1992, who also proved that, for Peano arithmetic and any stronger theories with effective axiomatizations, tolerance is equivalent to $Pi\_1$-consistency.

See also

*Interpretability
*Cointerpretability
*Interpretability logic

References

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