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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.

Author Concept Summary/Quotes Sources

David Hilbert on Numbers - Dictionary of Arguments

Berka I 121
Def Zero/O/Number/Logical Form/Hilbert:

0(F) : ~(Ex)F(x)

"There is no x for which F applies."

Def 1/one/number/logical form/Hilbert:

1(F) : (Ex)[F(x) & (y)(F(y) > ≡ (x,y)].

Hilbert: "There is an x for which F(x) exists, and every y for which F(y) consists is identical with this x."

Def 2/two/number/logical form/Hilbert:

2(F) :(Ex)(Ey) {~≡(x,y) & F(x) & F(y) & (z)[F(z) > ≡ (x,z) v ≡ (y,z)]}.

I 122
"There are two different x and y to which F applies, and every z for which F(z) exists is identical with x or y".

Definition number equality/logical form/Hilbert: equal numbers of two predicates F and G can be regarded as an individual predicate-predicate Glz (F, G). It means nothing else than that the objects to which F and the objects to which G apply are reversibly relatable to each other. Therefore the logical form can be represented as follows:

(ER){(x)[F(x) > (Ey) (R(x,y) & G(y))] & (y)[G(y) >
> (Ex) (R(x,y) & F(x)] & (x)(y)(z) [(Rx,y) & R(x,z) >
> = (y,z) & (R(x,z) & R(y,z) > = (x,y)]}.

Def number/logical form/extendend function calculus/Hilbert: also the general number term can be formulated logically: If a predicate-predicate φ (F) should represent a number, then φ must satisfy the following conditions:
1. For two equal predicates F and G, φ must be true for both or neither.
2. If two predicates F and G are not equal in numbers, φ can only be true for one of the two predicates F and G.
Logical form:

(F)(G){(φ(F) & φ(G) > Glz (F,G) & [φ(F) & Glz (F,G) > φ(G)]}.

The entire expression represents a property of φ. If we denote this number with Z (φ), then we can say:

A number is a predicate-predicate φ that has the property Z (φ).

Problem/infinity axiom/Hilbert: a problem occurs when we ask for the conditions under which two predicate-predicates φ and ψ define the same number with the properties Z (φ) and Z (ψ)(1).

1. D. Hilbert & W. Ackermann: Grundzüge der Theoretischen Logik, Berlin, 6. Aufl. Berlin/Göttingen/Heidelberg 1972, §§ 1, 2.
>Infinity axiom
Berka I 295
Real numbers/Hilbert: the epitome of real numbers is not the totality of all possible decimal-fraction developments, nor is it the totality of all possible laws according to which the elements of a fundamental series can proceed, but a system of things whose mutually interrelated relations are defined by the Axioms, and for which all and only the facts are true, which can be inferred from the axioms by a finite number of logical inferences.
Existence/real numbers/Hilbert: the concept of the continuum, or the concept of the system of all functions, exists in the same sense as the system of the whole rational numbers, or even the higher Cantor numerical classes and magnitudes(1).
>Real numbers, >Existence/Hilbert.

1. D. Hilbert: Mathematische Probleme, in: Ders. Gesammelte Abhandlungen, 1935, Bd. III, pp. 290-329 (gekürzter Nachdruck v. pp. 299-301).

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. Translations: Dictionary of Arguments
The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

Berka I
Karel Berka
Lothar Kreiser
Logik Texte Berlin 1983

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