User blog comment:Bubby3/Guesses about the strength of BMS and the Catching Hierarchy./@comment-32697988-20180419222547/@comment-27513631-20180423084107

A potential example of a way to make much stronger functions in inconvenient but simpler ways would be collapsing terms of the type [Math Processing Error], whose existence is incompatible with LEM.

A type is the version of 'set' that type theory has. Type theories aren't based on FOL, but have their own logic incorporated into them, and this logic is usually intuitionistic. As such, the law of excluded middle usually fails, which can actually be a good thing. For example, we no longer necessarily have \(|A|<|A\to B|\) for arbitrary types \(A\), \(B\) with an injection \(2\to B\).

For instance, standard \(ML_{<\omega}W\) has arbitrary function sets and is weaker than KPM, whereas if we added LEM, the theory would be equiconsistent with ZFC+"there are at least n inaccessible cardinals (schematically in n)".

Now, we can define a type \(Tree(A)\) in this theory, such that there are two ways to construct elements of \(Tree(A)\): a \(leaf \in Tree(A)\) and a \(A\)-indexed branch \(branch \in (A \to Tree(A)) \to Tree(A)\). Note that in ZFC, we could take a quotient of \(Tree(A)\) to obtain \(A^+\).

Now, because we don't have LEM, it's possible to make a minimal type \(BigOrd\) which is isomorphic to \(Tree^2(BigOrd)\). This would be an example of a non-strictly-positive inductive type and have a relatively complex definition, but it's certainly doable.

\(BigOrd\) is useful as we cannot define it in classical logic. Moreover, we can build up an initial segment of the universe (corresponding, heuristically, to inaccessibles, Mahlos, Greatly Mahlos, which can be reached strictly from below) which \(BigOrd\) must be above, as it should be able to encode these.

That said, I haven't actually worked with a definition of this type, but the potential for an \(\alpha\) which behaves like \(\alpha^{++}\) is hopefully clear.