User:12AbBa/"Normal" OCFs vs a different version of R function p2

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This is the Mahlo page: OCFs based on a Mahlo cardinal.

Recap
Let's recap the definition of our OCF until now.

The OCF looks like this: \(\psi_\alpha(\beta)\) where \(\alpha\) is a cardinal.

Now, start with {0,1}. We define the nth iteration of the set as follows: For limit n, of course it's just the limit iteration. To define the n+1th iteration, we apply these operations to the nth iteration: +, *, ^, Ω_, I(...) (with multiple arguments or 1 argument), and \(\psi_\alpha(\beta)\). Note that in the case of the \(\psi\), \(\beta\) must be smaller than or equal to n. Now, we define \(\psi_\alpha(\beta)\) as the smallest ordinal that is larger than all members in the set that are smaller than \(\alpha\).

Part VI: Normal Mahlos
To use Mahlo cardinals, we could extend the notation to correspond \(I\) and \(\varphi\). So, we have \(\psi_M\), and change the \(I(\dots)\) back to just \(I_\alpha\). So \(\psi_M(0)=\alpha\to I_\alpha=\psi_{I(1,0)}(0)\). However, we immediately hit a problem: \(\psi_M(1)=\psi_{I(1,0)}(1)\). It is behaving much weaker than expected. So, we invent a new function, \(\chi\) (short for \(\chi_M\)), that only returns 1-inaccessibles. We add this function into our list of operators. \(\chi\) should have two arguments, like this: \(\chi_\alpha(\beta)\). Now \(\alpha\) is a Mahlo.

Now we have to find a point such that \(\chi(\alpha)=\alpha\). This point is not \(\alpha\to\chi(\alpha)=\psi_{I(2,0)}(0)\), but rather \(I(2,0)\) since it is 1-inaccessible and the former is singular. So \(\chi(M)=I(2,0)\). We continue with the \(\chi\) corresponding perfectly with the \(\psi\).

I will be expressing ordinals below \(\psi(M^{M^{\omega^2}})\) with I. The first entry left of the dots is the \(\omega\)th entry.

Now we see that \(\uparrow_{,_{,\uparrow}\uparrow}=M\).

Part VII: Inaccessible Mahlos
Now we have to find a new notation for inaccessible Mahlos. Of course, we have \(M(1,0)\), which is the first inaccessible Mahlo. Note that if so, \(\psi_{M(1,0)}\) and \(\chi_{M(1,0)}\) have bizarre and powerful properties. \(\psi_{M(1,0)}(0)\) is nothing unusual. It is just the first Mahlo fixed point. However, \(\psi_{M(1,0)}(1)\) is much much larger than the second Mahlo fixed point. What is going on?

1. What is \(\psi_{M(1,0)}(1)\), and why is it so large? Recall the definition. We now have \(\chi\) in our list, so \(\psi_{M(1,0)}(1)\) is the next Mahlo fixed point after \(\chi_{M(1,0)}(0)>\psi_{M(1,0)}(0)\).

2. What is \(\chi_{M(1,0)}(0)\)?

Part VIII: \(\alpha\)-Mahlos
Too hard