In
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, a tolerant sequence is a sequence
:
,...,
of
formal theories such that there are
consistent
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
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
In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nea ...
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
*
Interpretability logic 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, cointerpretabilit ...
References
G. Japaridze ''The logic of linear tolerance''.
Studia Logica
''Studia Logica'' (full name: ''Studia Logica, An International Journal for Symbolic Logic'') is a scientific journal publishing papers employing formal tools from Mathematics and Logic. The scope of papers published in Studia Logica covers all sc ...
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