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In [[mathematical logic]], a '''cotolerant sequence''' is a sequence | |||
:<math>T_1, \ldots, T_n</math> | |||
of [[formal theory|formal theories]] such that there are [[consistent extension]]s <math>S_1, \ldots, S_n</math> of these theories with each <math>S_{i+1}</math> is [[cointerpretability|cointerpretable]] in <math>S_i</math>. Cotolerance naturally generalizes from sequences of theories to trees of theories. | |||
This concept, together with its dual concept of [[tolerance (in logic)|tolerance]], 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 <math>\Sigma_1</math>-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/study.html 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. | |||
[[Category:Logic]] | |||
{{logic-stub}} | |||
Revision as of 16:26, 25 December 2013
In mathematical logic, a cotolerant sequence is a sequence
of formal theories such that there are consistent extensions of these theories with each is cointerpretable in . Cotolerance naturally generalizes from sequences of theories to trees of theories.
This concept, together with its dual concept of tolerance, was introduced by Japaridze in 1992, who also proved that, for Peano arithmetic and any stronger theories with effective axiomatizations, tolerance is equivalent to -consistency.
See also
References
- G.Japaridze, The logic of linear tolerance. Studia Logica 51 (1992), pp. 249–277.
- 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.