Philosophy Lexicon of Arguments

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.
Author Item Excerpt Meta data

Books on Amazon
I 96
Def Strong Consistency/Strong consistent/Field: a mathematical theory M is st. c. , if it causes that the conjunction with a consistent non-mathematical theory T is still consistent. - T + M = consistent - Punch line: although st.c. does not follow from truth, it follows from necessary truth. - st.c. is however weaker than necessary truth because st.c. theories need not be true. - Purely mathematical theories (without math. entities): for them consistency involves strong consistency. - Non-pure: E.g. set theory with basic elements. - Urelement: Element of the lowest level, e.g. real numbers.
I 240
Consistency/Consistent/Mathematics/FieldVs: is untenable as a condition for the quality of mathematics: a consistent mathematical theory can be largely inadequate - Consistent (without contradiction) here means semantically consistent, i.e. satisfiable.

Fie I
H. Field
Realism, Mathematics and Modality Oxford New York 1989

Fie II
H. Field
Truth and the Absence of Fact Oxford New York 2001

H. Field
Science without numbers Princeton New Jersey 1980

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Ed. Martin Schulz, access date 2017-05-29