Axiom of the empty set

The axiom (くうしゅうごうのこうり British: axiom of empty set) of the empty set is one of the axioms of ZF set theory and KP set theory and insists on "the meeting that any meeting does not include existing". But the formulation of the ZF axiom system not to adopt this axiom exists [¹].

Table of contents

Definition

"A certain meeting x exists, and, for any y, y is not an element of x.";, in other words,

${\displaystyle \exists x\,\forall y\,\lnot (y\in x)}$ 

Property

By the axiom of the extensionality, it is revealed that the meeting insisted on exists uniquely by an axiom. I call the meeting "an empty set" and usually express it in a sign of {} and ∅. I prepare for a fixed number sign to express an empty set beforehand and may describe ZF. In that case, ∅ emerging in an infinite axiom is merely a sign to express some kind of meetings, and it is guaranteed that it is empty only after it depends on the axiom of the empty set.

It is thought that claim in itself of this axiom is clear, but may not add it to an axiom because I can derive it from the first-order predicate logic and a substituted axiom [²].

Footnote, source

[Help]

1. For ^ Kenneth キューネン "guidance Hiroshi Fujita reason to set theory independency proof", Japanese criticism company, 2,008 years, it is ISBN 978-4-535-78382-9
2. ^ Metamath Proof Explorer, Theorem axnul

References

• Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN 3-540-44085-2 

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