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. |
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. |
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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, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]]<ref>D. Probst, |
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, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]]<ref>D. Probst, [https://boris.unibe.ch/108693/1/pro17.pdf#page=153 A modular ordinal analysis of metapredicative subsystems of second-order arithmetic] (2017), p.153</ref> 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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Revision as of 21:51, 29 August 2023
An ordinal function refers to a function from ordinals to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include continuous functions and normal functions.
Technically speaking and within ZF, since ordinals don't form a set, one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an inaccessible ordinal or even an uncountable[1] 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.
- ↑ D. Probst, A modular ordinal analysis of metapredicative subsystems of second-order arithmetic (2017), p.153