Editing Tolerant sequence

In [[mathematical logic]], a '''tolerant sequence''' is a sequence 

:<math>T_1</math>,...,<math>T_n</math> 

of [[theory (mathematical logic)|formal theories]] such that there are [[theory (mathematical logic)#Consistency and completeness|consistent]] [[theory (mathematical logic)#Subtheories and extensions|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 [[Giorgi Japaridze| 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]]