Tolerant sequence

In mathematical logic, a tolerant sequence is a sequence

${\displaystyle T_{1}}${\displaystyle T_{1}},...,${\displaystyle T_{n}}${\displaystyle T_{n}}

of formal theories such that there are consistent extensions

${\displaystyle S_{1}}${\displaystyle S_{1}},...,${\displaystyle S_{n}}${\displaystyle S_{n}}

of these theories with each ${\displaystyle S_{i+1}}${\displaystyle S_{i+1}} interpretable in ${\displaystyle S_{i}}${\displaystyle S_{i}}. 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 ${\displaystyle \Pi _{1}}${\displaystyle \Pi _{1}}-consistency.

• Interpretability
• Cointerpretability
• Interpretability logic

• 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.
• G. Japaridze and D. de Jongh, The logic of provability. Handbook of Proof Theory. S. Buss, ed. Elsevier, 1998, pp. 476–546.

• Proof theory