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In [[mathematical logic]], a '''tolerant sequence''' is a sequence
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:<math>T_1</math>,...,<math>T_n</math>
 
of [[formal theory|formal theories]] such that there are [[consistent extension]]s
 
:<math>S_1</math>,...,<math>S_n</math>
 
of these theories with each <math>S_{i+1}</math> [[interpretability|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 [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>\Pi_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.&nbsp;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.&nbsp;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.&nbsp;476–546.
 
[[Category:Proof theory]]

Latest revision as of 18:05, 23 March 2014

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