# 空集公理

## 正式表述

${\displaystyle (N)}$  — ${\displaystyle (\exists A)(\forall x)[\neg (x\in A)]}$

${\displaystyle \vdash (\exists !x)[\,(\forall y)(y\not \in x)\,]}$

${\displaystyle (\forall y)(y\not \in x)}$
${\displaystyle (\forall y)(y\not \in t)}$

${\displaystyle \neg (y\in x)}$
${\displaystyle \neg (y\in t)}$

${\displaystyle \vdash (y\not \in x)\Rightarrow [\,(y\in x)\Rightarrow (y\in t)\,]}$
${\displaystyle \vdash (y\not \in t)\Rightarrow [\,(y\in t)\Rightarrow (y\in x)\,]}$

${\displaystyle (y\in x)\Rightarrow (y\in t)}$
${\displaystyle (y\in t)\Rightarrow (y\in x)}$

${\displaystyle (y\in x)\Leftrightarrow (y\in t)}$

${\displaystyle (\forall y)(y\in x)\Leftrightarrow (y\in t)}$

${\displaystyle (\forall y)(y\not \in x),\,(\forall y)(y\not \in t)\vdash x=t}$

${\displaystyle \vdash (\forall x)(\forall t){\big \{}[(\forall y)(y\not \in x)\wedge (\forall y)(y\not \in t)]\Rightarrow (x=t)\}}$

${\displaystyle (N^{\prime })}$  — ${\displaystyle (\forall y)(y\not \in \varnothing )}$

## 引用

• Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.