Ordinal notation system: Difference between revisions
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An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. |
An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences of other such objects, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. |
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Notable ordinal notation systems include: |
Notable ordinal notation systems include: |
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- [[Y sequence]] (as long as it is well-ordered) |
- [[Y sequence]] (as long as it is well-ordered) |
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As can be seen in this list, proposed ordinal notation systems need to be [[Proving well-orderedness | proven]] well-ordered in order to be considered ordinal notation systems with certainty, and there are notable cases where this is not proven yet. |
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Revision as of 19:01, 11 July 2023
An ordinal notation system (also called an ordinal notation informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences of other such objects, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems.
Notable ordinal notation systems include:
- Ordinal notation systems associated to ordinal collapsing functions
- Taranovsky's ordinal notations (the ones that are well-ordered)
- Y sequence (as long as it is well-ordered)
As can be seen in this list, proposed ordinal notation systems need to be proven well-ordered in order to be considered ordinal notation systems with certainty, and there are notable cases where this is not proven yet.