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This page requires knowledge of ordinals up to and including \(\varepsilon_0\) and basic ordinal operators. This page does not present any particularly powerful nonOCF ordinal notations.
OCFs are complicated and you will probably have to reread sections multiple times to get a full grasp of the concepts.
You might be wondering how ordinal collapsing functions work. (If you aren't, then this is not the right page for you.)
You already know how Cantor normal form works (if you don't, then this is also not the right page for you) and that it can represent all ordinals below \(\varepsilon_0\) (if you don't, then you do now) and that ordinals are usually defined as the set of ordinals less than themselves (if this is not how you learned ordinals â€¦
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Note: I'll be using \ for making lambdas since \x.xx is clearer than S(SKK)(SKK). You can represent natural numbers in SKI using Church numerals, but how do you represent ordinals? Using the Church numeral technique doesn't work since w+1 applications is the same as w applications. Instead, I'll be using this representation:
An ordinal is a function that takes a True/False input.
 If the input is True (T), it returns a 3input function.
 If the ordinal is zero, this returns its first input.
 If the ordinal is successor, this returns its second input.
 If the ordinal is limit, this returns its third input.
 If the input is False (F), the behavior depends on the ordinal:
 If the ordinal is zero, the output can be anything.
 If the ordinal is successor, it râ€¦
 If the input is True (T), it returns a 3input function.

Please ignore.
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