Dictionary of Arguments

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Empty set: an empty set is a set without an element. Notation ∅ or {}. There is only one empty set, since without an existing element there is no way to specify a specification of the set. The empty set can be specified as such that each element of the empty set is not identical with itself {x x unequal x}. Since there is no such object, the set must be empty. The empty set is not the number zero, but zero indicates the cardinality of the empty set.

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 Item Summary Meta data
IV 13f
Nothing/FregeVsHeidegger: the nominalization of nothing leads to paradoxes. - E.g. The fact that the empty set is included in every set, also the universal class. - That does not mean that the universal class is identical with the zero class ("the nothing").
IV 98
Subset/Element/Frege: subsets and elements must always be distinguished. FregeVsSchröder/FregeVsRegion Calculus: zero cannot be included as an element in each class, otherwise it would depend on the respective manifold. - Sometimes it would have nothing, sometimes it would be something (E.g. negation of a). - Solution: zero as subset (empty set).
IV 100
Zero/0/Empty Set/FregeVsSchröder/Frege: zero must not be included as an element in another class (> Günter Patzig, Introduction to Frege IV), but only subordinate as a class. (+ IV 100/101). ((s) zero is only included as a subset in any other set, not as an element).
IV 102
Empty Class/Empty Set/Unit Class/Unit Set//FregeVsSchröder: it is not necessary to form a one class - if a is an individual of the manifold, then a is also a class and it is not necessary to admit this class a as a new individual, it is already such. - It is not necessary at all that a class should be given as an individual of a manifold. - It is not about the subter-relation (sic), but about the sub-relation (sic). - ((s) subset, not element.)
IV 108
Zero/Frege/(s): Solution: Zero corresponds to the class of objects that are unequal to themselves. - Then the zero sign has a meaning. - Logical form: "Either there are no self-dissimilar objects or they all coincide with P".

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.
The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

G. Frege
Die Grundlagen der Arithmetik Stuttgart 1987

G. Frege
Funktion, Begriff, Bedeutung Göttingen 1994

G. Frege
Logische Untersuchungen Göttingen 1993

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Ed. Martin Schulz, access date 2019-04-21
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