Ordinal function: Difference between revisions

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An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s.

Technically speaking and within [[~~ZFC~~]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki.

Technically speaking and within [[ZF]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki.

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	An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s.		An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s.

	Technically speaking and within [[~~ZFC~~]], since [[Burali–Forti paradox\|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki.		Technically speaking and within [[ZF]], since [[Burali–Forti paradox\|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki.

Ordinal function: Difference between revisions

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