|Description Levels: Levels result from dividing a domain into sub-domains, for which different rules for making statements are valid. Thus, e.g. other statements are made about sets than about their elements. See also metalanguage, object language, theories, metatheory, metalogic, metasemantics, meta-ethics, meta-level, paradoxes, order, 2nd order logic, higher order logic, HOL, completeness._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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Indefiniteness 2nd order: it is unclear whether an undecidable sentence has a particular truth value.
Logic/second order logic/Field/(s): excludes non-standard models better than theory 1st order. - 2nd order has no impredicative comprehension scheme.
Theory of the 1st order/Field: E.g. the theory of the space-time points - (s) E.g. theory which only uses functions but does not quantify via them.
Theory 2nd order/Field: E.g. theory of real numbers, because it quantifies via functions. - Quantities of higher order: are used for the definition of continuity and differentiability.
Theory of 1st order/2nd order/Hilbert/Field. Variables 1st order: via points, lines, surfaces. - 2nd order: Quantities of ... - Solution/Field: quantification 2nd order in Hilbert's geometry as quantification via regions. - only axiom 2nd order: Dedekind's continuity axiom.
III 95 f
Logic 2nd order/Field: E.g. Quantors like "there are only finitely many". - ((s) quantified via quantities). - Also not: E.g. "There are less Fs than Gs". - ((s) Fs and Gs only definable as sets or properties.)
Extension of the logic: preserves us from a huge range of additionally assumed entities. - E.g., what obeys the theory of gravity - QuineVs: rather accept abstract entities than to expand the logic. (Quine in this case pro Platonism).
Platonism 1st order/Field: accepts abstract entities, but no logic 2nd order. - Problem: but it needs this (because of the power quantifiers)._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
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