|Zero: is a natural number that is used to express that no object meets a certain condition or that a measured value has this quantity with respect to a unit of measure.<_____________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. |
Gottlob Frege on Zero - Dictionary of Arguments
Def Zero/Frege: because nothing falls under the concept "unlike itself", we declare: 0 is the number that is equal to the concept "unlike itself".
>Concept, >Object, >Numbers.
0/1/zero/one/numbers/Frege: 0 is the object that falls under the concept "equal to 0" - thus an object (the zero) falls under the concept. Because an object falls under the concept, the concept is assigned the number 1 (1 object, the zero). On the other hand, 0: no object falls under the concept "equal to 0 but not equal to 0". Hence 0 is the number which corresponds to the concept.
- - -
Subset/element/Frege: a subset must always be distinguished. FregeVsSchröder/FregeVsArea Calculus (German: Gebietekalkül). The zero must not be contained as an element in every class. Otherwise it would depend on the respective manifold. At one point it would be nothing, and then it would be something (e.g. negation of a). - Solution: zero as a subset (empty set).
>Empty set, >Subsets.
Zero/0/empty set/FregeVsSchröder/Frege: the zero must not be contained as an element in another class (>Günter Patzig, introduction to Frege IV), but only subordinate as a class. (+ IV 100/101).
>Element relation._____________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.
Die Grundlagen der Arithmetik Stuttgart 1987
Funktion, Begriff, Bedeutung Göttingen 1994
Logische Untersuchungen Göttingen 1993