There are many tests of induction or recursion use in $ omega $, or in an arbitrary ordinal (may be uncountable). Are there any good examples of tests that use a large but computable ordinal?
The original proof of Ramsey's theorem and Hales-Jewett's theorem uses induction in $ omega ^ 2 $, but the use is not essential, because Erdos and Shelah have given better limits when using induction only in $ omega $. Plus $ omega ^ 2 $ It should not be considered large.
A typical use of large ordinal induction is to demonstrate the consistency of axiom systems, for example, using $ varepsilon_0 $-induction to test the consistency of PA. This is a kind of examples.
The existence of the Goodstein function uses induction in $ varepsilon_0 $, and I think it's just a direct explanation of how recursion works in ordinal.
Are there more examples?