In mathematical logic, a tolerant sequence is a sequence
of formal theories such that there are consistent extensions
of these theories with each 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 Japaridze in 1992, who also proved that, for Peano arithmetic and any stronger theories with effective axiomatizations, tolerance is equivalent to -consistency.
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Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella.
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. Assume T and S are formal theories. Slightly simplified, T is said to be interpretable in S if and only if the language of T can be translated into the language of S in such a way that S proves the translation of every theorem of T. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas.