normal (ordinal) function

Definition.

A function $F\\colon\\operatorname{\\mathbf{On}}\\to\\operatorname{\\mathbf{On}}$ is continuous if and only if for each $u\\subset\\operatorname{\\mathbf{On}}$ such that $u\eq\\emptyset$ it holds that $F(\\sup(u))=\\sup\\{F(\\alpha)|\\alpha\\in u\\}$ .

Definition.

A function $F\\colon\\operatorname{\\mathbf{On}}\\to\\operatorname{\\mathbf{On}}$ is order preserving if and only if for each $\\alpha,\\beta\\in\\operatorname{\\mathbf{On}}$ such that $\\alpha<\\beta$ it follows that $F(\\alpha)<F(\\beta)$ .

Definition.

A function $F\\colon\\operatorname{\\mathbf{On}}\\to\\operatorname{\\mathbf{On}}$ is a normal function if and only if $F$ is continuous and order preserving.

单词	NormalordinalFunction
释义	normal (ordinal) function Definition. A function $F\\colon\\operatorname{\\mathbf{On}}\\to\\operatorname{\\mathbf{On}}$ is continuous if and only if for each $u\\subset\\operatorname{\\mathbf{On}}$ such that $u\eq\\emptyset$ it holds that $F(\\sup(u))=\\sup\\{F(\\alpha)\|\\alpha\\in u\\}$ . Definition. A function $F\\colon\\operatorname{\\mathbf{On}}\\to\\operatorname{\\mathbf{On}}$ is order preserving if and only if for each $\\alpha,\\beta\\in\\operatorname{\\mathbf{On}}$ such that $\\alpha<\\beta$ it follows that $F(\\alpha)<F(\\beta)$ . Definition. A function $F\\colon\\operatorname{\\mathbf{On}}\\to\\operatorname{\\mathbf{On}}$ is a normal function if and only if $F$ is continuous and order preserving.
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