Tolerant sequence
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In mathematical logic, a tolerant sequence is a sequence
- <math>T_1<math>,...,<math>T_n<math>
of formal theories such that there are consistent extensions
- <math>S_1<math>,...,<math>S_n<math>
of these theories with each <math>S_{i+1}<math> interpretable in <math>S_i<math>. 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 <math>\Pi_1<math>-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.