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 the \(\beta\)-set, which is {0,1} if \(\beta=0\) and \(\{k|k<\beta\}\) otherwise. 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 \(\beta\)th iteration of the \(\psi_\alpha(\beta)\)-set that are smaller than \(\alpha\). \begin{eqnarray*} C_0(\alpha,\beta) &=& \{0,1\}\cup\beta \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta,\gamma^\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{I(\gamma_1,\gamma_2\cdots,\gamma_k,\delta)|\gamma_1,\gamma_2\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*}

Part VI: Normal Mahlos
To use Mahlo cardinals, we could extend the notation to correspond \(I\) and \(\varphi\). So, we have \(\psi_M\), change the \(I(\dots)\) back to just \(I_\alpha\), and add \(M_\alpha\). 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 \(\chi_M\), it can have two arguments), that only returns 1-inaccessibles. But that's kind of awkward. So we consider removing the I_ function completely. Now the \(\chi\) function only returns normal inaccessibles, and \(\chi(0)=I\). 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 \(\alpha\to\chi(\alpha)=\psi_{I(1,0)}(0)\), but rather \(I(1,0)\) since it is inaccessible and the former is singular. So \(\chi(M)=I(1,0)\). We continue with the \(\chi\) corresponding with the \(\psi\). Similarly for \(\psi\), \(\chi\) with a non-Mahlo (or things like \(M_\omega\)) as base is contradictory.

\begin{eqnarray*} C_0(\alpha,\beta) &=& \{0\}\cup\beta \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta,\gamma^\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Omega_\gamma,M_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is a Mahlo}\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \chi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is inaccessible}\} \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*}

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, or Mahlos that are limits are Mahlos. Of course, we have \(M(1,0)\), which is the first inaccessible Mahlo. This OCF is quite different from a "normal" OCF, because the "normal" OCF is becoming more and more ill-defined past this point. \begin{eqnarray*} C_0(\alpha,\beta) &=& \{0,1\}\cup\beta \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta,\gamma^\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{M(\gamma_1,\gamma_2\cdots,\gamma_k,\delta)|\gamma_1,\gamma_2\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is a Mahlo}\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \chi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is inaccessible}\} \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*}

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 \(\chi_{M(1,0)}(0)\)? It would be at least \(\psi_{M(1,0)}(0)\), so according to the definition \(\chi_{M(1,0)}(0)\ge I_{\psi_{M(1,0)}(0)+1}\). But now the \(\chi_{M(1,0)}(0)\)-set contains \(\psi_{M(1,0)}(0)+1\), so now we have \(M_{\psi_{M(1,0)}(0)+1}\), \(M_{M_{\psi_{M(1,0)}(0)+1}}\), ... all less than \(\chi_{M(1,0)}(0)\)! Therefore \(\chi_{M(1,0)}(0)\)>the second Mahlo fixed point, and the process starts over again. But it will stop, when we reach a Mahlo fixed point that is also inaccessible, which is equivalent to an inaccessible that is a limit of Mahlos. That is finally \(\chi_{M(1,0)}(0)\).

3. How to represent the second Mahlo fixed point: \(\psi_{\chi_{M(1,0)}(0)}(1)\)

Interesting fact: \(\psi_{M(1,0)}(k)=\psi_{\chi_{M(1,0)}(k)}(1)\), and \(\psi_{\chi_{M(1,0)}(k)}(0)=\psi_{M(1,0)}(0)\).

Oh, and before I begin, I should mention that \(\uparrow_{,_{\uparrow_{,_{,\uparrow},\uparrow}}\uparrow}\uparrow_{,_{,\uparrow},\uparrow}=\psi(\psi_{M(1,0)}(0)2)\ne\uparrow_{,_{,\uparrow}\uparrow_{\uparrow_{,_{,\uparrow},\uparrow}}}\uparrow_{,_{,\uparrow},\uparrow}\) because adding a \(\uparrow_{,_{,\uparrow}\uparrow}\) changes things a lot.

This OCF marks the first time that a well-defined OCF differs from the ill-defined OCF on the cardinal level. Which means, that this OCF and "OCF"s like UNOCF use different cardinal expressions to express the same idea from now on. UNOCF is based on a misunderstanding of large cardinals, which results in a very different strength, even if it had been well-defined.

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