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: ,

, ,   « ». 1897 . , .

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. :

  • x , x x: \mathrm{Trans}(x)\Leftrightarrow\forall t(t\in x\to t\subseteq x).
  • , , : \mathrm{Ord}(x)\Leftrightarrow\mathrm{Trans}(x)\wedge\forall t(t\in x\to\mathrm{Trans}(t)).

, , ,   .

\alpha, \beta, \dots. .

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  • \alpha  , \alpha  .
  • \alpha, \beta : \alpha \in \beta, \alpha = \beta, \beta \in \alpha.
  • \in ( , , , \in), \bigcap x  x, \bigcup x  , x. \alpha < \beta \alpha \in \beta . , \in.
  • x , x ( , , ).
  • \alpha , , \alpha.
  • .
  • \varnothing  ( , ).
  • \alpha  (: ), \varnothing, \beta; , \beta < \alpha, \beta < \gamma < \alpha. , \alpha , \beta, : \alpha = \beta \dot+ 1 ( \alpha = \beta + 1, ).
  • , , ( \varnothing ).
  • \alpha \dot+ 1 = \alpha\cup\{\alpha\}.
  • , , . , . :
\begin{align}
&0=\varnothing;\\
&1=\{0\}=0\cup\{0\}=\{\varnothing\};\\
&2=\{0,1\}=1\cup\{1\}=\{\varnothing,\{\varnothing\}\};\\
&3=\{0,1,2\}=2\cup\{2\}=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\};\\
&\dots
\end{align}
  • \omega.  ( ) . \omega \dot+ 1 = \omega\cup\{\omega\}.
  • \alpha \alpha < \omega , , \alpha \in \omega.
  • , . , .
  • A , \sup A. A \subseteq \sup A.
  • \alpha \varnothing, \sup \alpha = \alpha, \sup \alpha < \alpha.
  • .
  •  (.).

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[]

  • :
\begin{align}
&\alpha + 0 = \alpha\\
&\alpha + (\beta \dot+ 1) = (\alpha + \beta) \dot+ 1\\
&\alpha + \gamma = \sup \{ \alpha + \beta | \beta < \gamma \},
\end{align}
, \gamma .
  • , :
\begin{align}
&\alpha \cdot 0 = 0\\
&\alpha \cdot (\beta \dot+ 1) = \alpha \cdot \beta + \alpha\\
&\alpha \cdot \gamma = \sup \{ \alpha \cdot \beta \mid \beta < \gamma \}.
\end{align}
  • , :
\begin{align}
&\alpha^0 = 1\\
&\alpha^{\beta \dot+ 1} = \alpha^\beta \cdot \alpha\\
&\alpha^\gamma = \sup \{ \alpha^\beta \mid \beta < \gamma \}.
\end{align}

[]

  • ; , 1 + \omega = \omega \ne \omega + 1.
  • : \alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma,\, .
  • : \beta_1 > \beta_2\, \alpha + \beta_1 > \alpha + \beta_2\, \beta_1 + \alpha \geqslant \beta_2 + \alpha.
  • \alpha \geqslant \beta, \,\gamma, \beta + \gamma = \alpha.\,
  • ; , 2 \cdot \omega = \omega \ne \omega \cdot 2.
  • : \alpha \cdot (\beta \cdot \gamma)=(\alpha \cdot \beta) \cdot \gamma, .
  • : \alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma.
  • \alpha + 0 = 0 + \alpha = \alpha.\,
  • \alpha + 1 = \alpha \dot+ 1.
  • \alpha \in \omega \leftrightarrow \alpha + \omega = \omega.
  • \alpha \cdot 0 = 0 \cdot \alpha = 0.
  • \alpha \cdot 1 = 1 \cdot \alpha = \alpha.
  • \alpha \in \omega \land \alpha \ne 0 \leftrightarrow \alpha \cdot \omega = \omega.
  • \alpha + \beta = 0 \leftrightarrow \alpha = 0 \land \beta = 0.
  • \alpha \cdot \beta = 0 \leftrightarrow \alpha = 0 \lor \beta = 0.
  • \alpha^0 = 1.\,
  • \alpha^1 = \alpha.\,
  • \alpha \ne 0 \leftrightarrow 0^\alpha = 0.
  • 1^\alpha = 1.\,
  • \alpha \in \omega \land \alpha > 1 \leftrightarrow \alpha^\omega = \omega.
  • \alpha^\beta \cdot \alpha^\gamma = \alpha^{\beta + \gamma}.
  • (\alpha^\beta)^\gamma = \alpha^{\beta \cdot \gamma}.
  • \alpha > 1 \land \beta > \gamma \leftrightarrow \alpha^\beta > \alpha^\gamma.
  • \beta \in \omega \to \alpha + \beta = \alpha \underbrace{\dot+ 1 \dot+ 1 \dot+ \dots \dot+ 1}_\beta.
  • \beta \in \omega \to \alpha \cdot \beta = 0 \underbrace{+\alpha+\alpha+\dots+\alpha}_\beta.
  • \beta \in \omega \to \alpha^\beta = 1 \underbrace{\cdot \alpha \cdot \alpha \cdot \dots \cdot \alpha}_\beta.
  • , ( ).
  • , .

[] .

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