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Zermelo fraenkel set Theory - Printable Version

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Zermelo fraenkel set Theory - elim - 03-14-2017 02:05 PM

0. (i) There exists at least one thing. (ii) Every thing is a set.

1. Extensionality.$\;\;\forall a\forall b(\forall x(x\in a\iff x\in b)\implies a = b)$

2. Separation or Aussonderung.$\underset{\,}{.}$
$\quad\;\forall a\exists b\forall x(x\in b\iff x\in a\vee E(x))$

$\quad\;\varnothing =_{\text{def}}\{x\in a\mid x\ne x\}$ is thus a set, called empty set.

$\quad\; a\cap b =_{\text{def}} \{x\in a\mid x\in b\} = \{x\mid x\in a\wedge x\in b\}$ is a set.
$\qquad\quad$(called the intersection of $a\,$and$\,b$)

3. Pairing.$\;\forall a\forall b\exists c\forall x(x\in c\Leftrightarrow x=a\vee x=b)$

4. Sumset.$\;\forall a\exists b\forall x(x\in b\iff \exists y(x\in y \wedge y\in a)$
$\qquad\;$That is:$\;\;\{x\mid \exists y(x\in y\wedge y\in a)\}$ is a set.

Definition$\quad\displaystyle{\bigcup a = \bigcup_{y\in a}y =_{\text{def}} \{x\mid \exists y\in a(x\in y)\}}$
$\qquad\; a\cup b =_{\text{def}}\bigcup \{a,b\}$ is called the union of $a,\,b.$
$\qquad\;\{a_0,\ldots,a_n\} =_{\text{def}}\{a_0\}\cup\cdots\cup\{a_n\}$

5. Powerset.$\;\;\forall a\exists b\forall x(x\in b\iff x\subset a)$
$\qquad\;$where $x\subset a\equiv_{\text{def}}\forall y(y\in x\implies y\in a)$
$\qquad\;$that is:$\;\;\mathcal{P}(a)=_{\text{def}}\{x\mid x\subset a\}$ is a set.

6. Substitution or Replacement.
$\qquad\;\exists b\forall x(x\in b\iff \exists x\in a(y = F(x)))$
$\qquad\;F(a)=_{\text{def}}\{F(x)\mid x\in a\} = \{y\mid \exists x\in a(y=F(x))\}$

7. Infinity. $\exists a\,((\varnothing\in a)\wedge (x\in a\implies x\cup\{x\}\in a))$
$\qquad\;$or without (4.):
$\quad\;\small\exists a\,((\varnothing\in a)\wedge (x\in a\Rightarrow \exists y\in a\forall z(z\in y\iff z\in x\vee z = x)))$

8. Regularity or Foundation.
$\qquad\;\forall a(a\ne\varnothing\implies \exists x\in a(x\cap a = \varnothing))$