In
mathematical logic
Mathematical logic is the study of 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 formal ...
, a tolerant sequence is a sequence
:
,...,
of
formal theories
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to:
Dress code and events
* Formal wear, attire for formal events
* Semi-formal atti ...
such that there are
consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is 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.
See also
*
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. Slightly simplified, '' ...
*
Cointerpretability In mathematical logic, cointerpretability is a binary relation on formal theories: a formal theory ''T'' is cointerpretable in another such theory ''S'', when the language of ''S'' can be translated into the language of ''T'' in such a way that ''S ...
*
Interpretability logic
References
G. Japaridze ''The logic of linear tolerance''.
Studia Logica
''Studia Logica'' (full name: Studia Logica, An International Journal for Symbolic Logic), is a scienific journal publishing papers employing formal tools from Mathematics and Logic. The scope of papers published in Studia Logica covers all scient ...
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