Philosophy Lexicon 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.
 
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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".

F I
G. Frege
Die Grundlagen der Arithmetik Stuttgart 1987

F II
G. Frege
Funktion, Begriff, Bedeutung Göttingen 1994

F IV
G. Frege
Logische Untersuchungen Göttingen 1993


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Ed. Martin Schulz, access date 2017-05-25