List of ordinals: Difference between revisions
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Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties.
As such, apeirology is linked to:
* [[Set theory]] (which includes study of large cardinals)
* [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α)
* [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α)
* [[Proof theory]] and [[ordinal analysis]] (which assigns recurisve ordinals to theories according to the lengths of the recursive wellorders they can prove well-founded)
* Googology (which translates recursive ordinal notations into systems for constructing large finite numbers).
Below we list some milestone ordinals.
== Countable ordinals ==
In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"-->
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