|Consistency, philosophy, logic: within a system, consistency may be demonstrated, but not beyond the boundaries of this system, since the use of the symbols and the set of possible objects are only defined for this system.|
Within mathematics, and only there applies that the mathematical objects, which are mentioned in consistent formulas, exist (Hilbert, Über das Unendliche, 1926). See also falsification, verification, existence, well-formed.
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Consistency/Mates: can be doubly checked: a) semantically: by specifying an interpretation in which all axioms are true - b) syntactically: by showing without referring to an interpretation that there is no statement j such that both j and ~j can be derived from the axioms
Elementare Logik Göttingen 1969