|Numbers: whether numbers are objects or concepts, has been controversial in the philosophical discussion for millennia. The most widely accepted definition today is given by G. Frege (G. Frege, Grundlagen der Arithmetik 1987, p. 79ff). Frege-inspired notions represent numbers as classes of classes, or as second-level terms, or as that with one measure the size of sets. Up until today, there is an ambiguity between concept and object in the discussion of numbers. See also counting, sets, measurements, mathematics, abstract objects, mathematical entities, theoretical entities, number, platonism._____________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. |
Number/statement/property/Mates: every statement in which a number occurs, represents a property of the corresponding number Denkform - e.g. no person under 18 receives permission- property of the number 18, that it is a number k with the property, that no person obtains a permit under k years - therefore complete induction can be applied to such statements.
Numbers/Frege/Mates: Definition cardinal number: of a set a: set of all sets which are numerically equivalent to a - Definition one/1: the set of all sets a, satisfying the condition (x) (Ey) (y e a <> y = x). - Definition Two/2: set of all sets which satisfy the condition (x)(y)(x not equal y u (z)(z e a <> (z = x v z = y))). ((s) Excluding right?) -
Definition sum: the sum p + q of two integers p and q is the set of all sets g which satisfying the condition (a)(Eb)(a e p u b e q u a U b = g u a D b = L) - Definition set of all positive numbers: the average ((s) Common) of all sets a which satisfying the condition
1 e a u (n)(n e a > n + 1 e a). ((S) successor) - thus Frege shows that arithmetic can be lead back entirely on logic, and thus that it is part of the logic._____________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.
Elementare Logik Göttingen 1969
Skeptical Essays Chicago 1981