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Infinity Axiom: The infinity axiom is an axiom of set theory, which ensures that there are infinite sets. It is formulated in e.g. such a way that a construction rule is specified for the occurrence of elements of a described set. If {x} is the successor of x, the continuation is formed by the union x U {x}. See also set theory, successor, unification, axioms.
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 Infinity Axiom - Dictionary of Arguments

Berka I 122
Definition Number/logical form/extended function calculus/Hilbert: the general number concept can also be formulated logically: if a predicate-predicate φ (F) should be a number, then φ must satisfy the following conditions:
1. For two equal predicates F and G, φ must be true for both or none of them.
2. If two predicates F and G are not equal in number, 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 designate it with Z (φ), then we can say:

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

, >Definitions, >Definability, >Infinity, >Axioms, >Axiom systems, >Predicates, >Properties.

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).
Infinity Axiom/equal numbers/Hilbert: the condition for equal numbers or for the fact that two predicate-predicates φ and ψ define the same number is that, that φ(P) and ψ(P) are true for the same predicates P and false for the same predicates. So that the relationship arises:

(P)(φ(P) ↔ ψ(P))

I 122
Problem: when the object area is finite, all the numbers are made equal which are higher than the number of objects in the individual domain.
>Finiteness/Hilbert, >Finitism, >Finiteness.
For example, if a number is e.g. smaller than 10 to the power of 60 and if we take φ and ψ the predicates which define the numbers 10 to the power of 60+1 and 10 high 60 + 1, then both φ and ψ do not apply to any predicate P.
The relation

(P)(φ(P) ↔ ψ(P))

Is thus satisfied for φ and ψ, that is, φ and ψ would represent the same number.
Solution/Hilbert: infinity axiom: one must presuppose the individual domain as infinite. A logical proof of the existence of an infinite totality is, of course, dispensed with(1).

1. D. Hilbert & W. Ackermann: Grundzüge der theoretischen Logik, Berlin, 6. Aufl. Berlin/Göttingen/Heidelberg 1972, §§ 1, 2.

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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