In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, a tolerant sequence is a sequence
,...,

of formal theories

Formal theory

Formal theory can refer to:* Another name for a theory which is expressed in formal language.* An axiomatic system, something representable by symbols and its operators...

 such that there are consistent extensions
,...,

of these theories with each interpretable

Interpretability

In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.-Informal definition:Assume T and S are formal theories...

 in . Tolerance naturally generalizes from sequences of theories to trees of theories. Weak interpretability

Weak interpretability

Weak interpretability is a special case of the concept of tolerance introduced by Giorgi Japaridze in 1992.Assume T and S are formal theories. Slightly simplified, T is said to be weakly interpretable in S if, and only if, the language of T can be translated into the language of S in such a way...

 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.