User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-35392788-20180813124110

Using Bucholz's psi (or UNOCF's, for that matter), it's very easy to analyse pair sequence system.

(a,b) in PSS means "psi_b at depth a"

Now, the answers I'm going to give you should make a lot of sense :

1. (0,0)(1,1)(2,2)(3,3)

2. (0,0)(1,1)(2,2)(3,3)(4,4)

3. same as 1.

4. same as 2.

5. (0,0)(1,1)(2,2)(3,3)(4,3), corresponds to \(\psi(\psi_1(\psi_2(\psi_3(\psi_2(0)))))\)

6. (0,0)(1,1)(2,2)(3,3)(4,3)(5,2)(6,3), corresponds to \(\psi(\psi_1(\psi_2(\psi_3(\psi_2(\psi_3(\psi_2(0)))))))\)

7. (0,0)(1,1)(2,2)(3,3)(4,3), corresponds to \(\psi(\psi_1(\psi_2(\psi_3(\psi_3(0)))))\)

8. (0,0)(1,1)(2,2)(3,3)(3,2)(4,3), corresponds to \(\psi(\psi_1(\psi_2(\psi_3(0)+\psi_2(\psi_3(0)))))\)

9. (0,0)(1,1)(2,2)(3,3)(3,3), corresponds to \(\psi(\psi_1(\psi_2(\psi_3(0)+\psi_3(0))))\)

By the way, these ordinals are not smaller than the BHO, which is (0,0)(1,1)(2,2). I believe you meant \(\psi(\Omega_\omega)\).