User blog comment:Bubby3/Guesses about the strength of BMS and the Catching Hierarchy./@comment-32697988-20180419222547/@comment-27513631-20180422214846

Can I point out a mistake about stable ordinals, please?

Stable ordinals use a notion of \(\prec_{\Sigma_1}\), which roughly says that for sets \(A\), \(B\), if \(A\prec_{\Sigma_1}B\) then all of the 'there exists' statements about stuff in \(A\) are true in \(A\) iff they are true in \(B\).

Now, a stable ordinal \(\alpha\) is an ordinal such that \(L_\alpha\prec_{\Sigma_1}L_{\omega_1}\). However, these stable ordinals are much too strong for current ordinal collapsing functions directly. In particular, if \(L\) 'gets bigger', then the stable ordinals do too.

The usual concept is a \(\beta\)-stable ordinal, which says that \(L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}\), and is much weaker. In particular, a nonprojectible ordinal \(\kappa\) is a limit of \(\kappa\)-stable ordinals, and corresponds to \(\Pi_1^2-\text{CA}\), and the least nonprojectible is much smaller than the smallest stable ordinal. The first stable ordinal you mentioned is actually the first 1-stable ordinal.

Also, I think you might be overpraising the complexity of ordinal collapsing functions for specific large ordinals. They're tremendously complex, but much of their complexity (beyond \(\Pi_3\) reflection) comes from being easier to analyse proof-theoretically, which is of course not related to being easier to understand by most people.

I believe (but may be very wrong) that there are simpler, much stronger ordinal collapsing functions that collapse ordinals in less convenient ways. Of course, actually showing that this is the case is tantamount to embedding one into the other, and if that were easy they'd be simple in the first place.

A potential example of a way to make much stronger functions in inconvenient but simpler ways would be collapsing terms of the type \(\mu(\lambda A.\mu \lambda B.1+A\to B)^2\), whose existence is incompatible with LEM.