In
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, a tolerant sequence is a sequence
:
,...,
of theory (mathematical logic), formal theories such that there are theory (mathematical logic)#Consistency and completeness, consistent theory (mathematical logic)#Subtheories and extensions, extensions
:
,...,
of these theories with each
interpretability, interpretable in
. 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
-consistency.
See also
*Interpretability
*Cointerpretability
*Interpretability logic
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.
and D. de Jongh, ''The logic of provability''. Handbook of Proof Theory. S. Buss, ed. Elsevier, 1998, pp. 476–546.
Proof theory