## Psychology Dictionary of ArgumentsHome | |||

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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 |
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Alfred Tarski on Numbers - Dictionary of Arguments Berka I 539 Ordinals/ordinal numbers/Tarski: we need them to classify characters of infinite order. - It is a generalization of the concept of natural numbers. - These are the smallest ordinal number. Transfinite ordinal numbers: since there are figures for every infinite sequence of ordinal numbers that are larger than any member of the sequence, there are particular numbers that are greater than all natural numbers. In every non-empty class of ordinal numbers there is a smallest number. Omega/ω: in particular there is the smallest transfinite ordinal number ω (Omega). The next largest is ω + 1 and so on. - The language of the general set theory is of order ω. ^{(1)}>Numbers, >Set theory. 1. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935 _____________ 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. |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |