Psychology Dictionary of Arguments

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One, number 1: in modern logic it is not possible to introduce the number one directly. It must be introduced indirectly, via existential quantification ("for at least one x ...") and universal quantification ("for all x ..."). In addition, identity is needed. See also definition, identity, logic, elementary logic, number theory, numbers.
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 One (Number 1) - Dictionary of Arguments

Berka I 121
Definition 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) exists 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)]}.

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"(1).
, >Equality, >Identity, >Equations.

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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