Philosophy Lexicon of Arguments

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Hilbert, D.
 
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Numbers Berka I 121
Definition Zero/O/Number/Logical Form/Hilbert:

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

"There is no x for which F applies."

Defintion 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"

Definition 2/Two/number/logical form/Hilbert:

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

Definiton 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).
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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.

Brk I
K. Berka/L. Kreiser
Logik Texte Berlin 1983


> Counter arguments against Hilbert
> Counter arguments in relation to Numbers



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Ed. Martin Schulz, access date 2017-03-27