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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.
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.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980