# lo.logic: examples of tests that use induction or recursion in a large recursive ordinal

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?