Axiom/Field: a required law can easily be proven by adding it as an axiom - Vs: but then you need for each pair of distinct predicates an axiom that says that the first one and the second does not, e.g. "The distance between x and y is r times that between z and w". - Everything that substantivalism or heavy-duty Platonism may introduce as derived theorems, Relationism must introduce as axioms ("no empty space"). - That leads to no correct theory. - Problem of quantities. - The axioms used would precisely be connectable if also non-moderate characterizations are possible. - The modal circumstances are adequate precisely then when they are not needed.
Axiom/Mathematics/Necessity/Field: axioms are not logically necessary, otherwise we would only need logic and no mathematics.
Axioms/Field: we then only accept those that have disquotationally true modal translations. - (Because of conservativism). Conservatism: is a holistic property, not property of the individual axioms. - Acceptability: of the axioms: depends on the context. - Another theory (with the same Axiom) might not be conservative. - Disquotational truth: can be better explained for individual axioms, though.
E.g. Set theory plus continuum hypothesis and set theory without continuum hypothesis can each be true for their representatives. - They can attribute different truth conditions. - This is only non-objective for Platonism. - The two representatives can reinterpret the opposing view, so that it follows from their own view. (>Gödel: relative consistency).
Axiom/(s): not part of the object language. - Scheme formula: can be part of the object language. - Field: The scheme formulacaptures the notion of truth better.
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
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
Theories of Truth, Paul Horwich Aldershot 1994
|Continuum Hypothesis||Hilbert||Berka I 295
Definition continuum hypothesis/Cantor/Berka: (Cantor, 1884): if an infinite set of real numbers is not countable, then it is equal to the set of real numbers R itself. The term "continuum hypothesis" emerged later.
Gödel: (1938) Gödel proved the relative consistency in the continuity hypothesis.
Independence/Cohen: (1963, 64): Cohen proved that the negation of continuum hypothesis is also consistent with the axioms of set theory, that is, he proved the independence of the continuum hypothesis from the set theory.(1)
1. D. Hilbert, „Mathematische Probleme“ in: Ders. Gesammelte Abhandlungen (1935) Bd. III S. 290-329 (gekürzter Nachdruck v. S 299-301)
Logik Texte Berlin 1983