Making an OCF is easy.
The idea of pre-inaccessible OCFs (OCFs without inaccessibles) are easy to grasp. Let’s define a function \(\psi(\alpha)\) as the smallest ordinal you can’t name with 0, addition, and \(\psi\) to smaller arguments. The idea is when we run out of names, we just give a name to the next ordinal and keep going.
This post isn’t about the basics of OCFs, but it’s important to see how we get past the limit. We introduce an enormous ordinal \(\Omega_1\), and add that to the set of ordinals we can use to name ordinals. After this, we can add an even larger ordinal \(\Omega_2\) to name enormous ordinals, and keep going.
A sample OCF looks like:
- \(C_\nu(\alpha)\) is the closure of \(\Omega_\nu\) under addition, \(\lambda\beta.\omega^\beta\), and \(\psi\) if the argument is less that \(\alpha\).
- \(\psi_\nu(\alpha)\) is the supremum of \(C_\nu(\alpha)\cap\Omega_{\nu+1}\).
- \(\Omega_0=1\), and otherwise \(\Omega_\nu\) is the least ordinal of cardinality \(\aleph_\nu\).
Adding inaccessibles
The limit of the previous OCF is \(\psi_0(\Omega_{\Omega_\ldots})=\psi_0(\Lambda)\) where \(\Lambda\) is the first omega fixed point. This is a familiar situation…
A good candidate for a huge ordinal to add is the least inaccessible, \(I\). Ideally, we would like \(\psi_I(0)=\Lambda\). It turns out that it is more convenient to have \(\psi_I(1)=\Lambda\), while \(\psi_I(0)=1\).
If we want \(\psi_I(1)=\Lambda\), it is easy to show that we can achieve this by just setting \(\Omega_I=1\) and \(\Omega_{I+1}=I\). (With the old definition \(\Omega_I=I\)). Then \(\psi_I(2)\) is the second fixed point, and so on. Adding \(I\) also allows us to use functions like \(\psi_{I+1}\), \(\psi_{I^I}\) (challenge: figure out how), \(\psi_{\Omega_{I+1}}\), and so on. The limit is \(\psi_0(\Omega_{\Omega_{\ldots_{I+1}}})\), the next omega fixed point after \(I\).
More inaccessibles!
To get past the limit, just add another inaccessible! If \(I_2\) is the second inaccessible, then setting \(\Omega_{I_2}=I\) and \(\Omega_{I_2+1}=I_2\) makes \(\psi_{I_2}(1)\) the first omega fixed point after \(I\). We can then add the function \(\alpha\mapsto I_\alpha\), the enumeration function of the inaccessibles, into the allowed functions. If \(\kappa=I_{\alpha+1}\), define \(\kappa^-=I_\alpha\), and otherwise \(\kappa^-=1\). We define:
- \(\Omega_0=1\).
- If \(\nu\) is inacessible, \(\Omega_\nu=\nu^-\).
- If \(\nu\) is the successor of an inaccessible, \(\Omega_\nu=\nu-1\).
- Otherwise \(\Omega_\nu\) is the least ordinal of cardinality \(\aleph_\nu\).
\(\alpha\)-inaccessibles
A good choice of diagonalizer is \(I(1,0)\), the first 1-inaccessible. A 1-inaccessible is an inaccessible limit of inaccessibles. However, setting \(\Omega_{I(1,0)}=1\) and \(\Omega_{I(1,0)+1}=I(1,0)\) now makes \(\psi_{I(1,0)}(0)\) the fixed point.
By now we should know to add the enumeration function of the 1-inaccessibles. To deal with 1-inaccessibles in subscripts, we just let \(\kappa^-=I(1,\alpha)\) if \(\kappa=I(1,\alpha+1)\), and \(\kappa^-1=1\) for other 1-inaccessibles.
Now to diagonalize over the 1-inaccessibles, we introduce (surprise!) a 2-inaccessible. The process of adding a 2-inaccessible is nearly identical. Add the function \(\alpha\mapsto I(2,\alpha)\), and set \(I(2,\alpha+1)^-=I(2,\alpha)\) and otherwise \(I(2,\alpha)^-=1\).
Adding \(\alpha\)-inaccessibles is essentially identical. We just add the two-argument \(I\) function itself to the allowed functions. Also, \(I(\alpha,\beta+1)^-=I(\alpha,\beta)\) and otherwise \(I(\alpha,\beta)^-=1\).
We could go on and add a third argument to \(I\), but there’s a better way…
Mahlo cardinals
We introduce a Mahlo cardinal, \(M\). We replace the two-argument \(I\) by the 1 argument version. Similarly, if we let \(\Omega_M=1\) and \(\Omega_{M+1}=M\), then our OCF gives \(\psi_M(0)\) as the first 1-inaccessible. \(\psi_M(Mn)\) will be the first \(n+1\)-inaccessible, and then \(\psi_M(M^2)\) diagonalizes over this and is the fixed point of \(\alpha\mapsto I(\alpha,0)\). In fact, \(\psi_M(M\alpha+\beta)\) in our OCF equals \(\chi_{1+\alpha}(\beta)\) in Rathjen’s, for some set of values that I can’t be bothered to calculate. The process of adding the other Mahlos is virtually identical to that of adding the inaccessibles.
Weakly compacts
Roses are red, violets are blue, I’m so bored, and so are you. So let’s just skip ahead. Also, note that the diagonalizer of the weakly compacts, also called \(\Pi^1_1\)-indescribables, is a \(\Pi^1_2\)-indescribable. I trust you to be able to understand the rest.
The final OCF
An inaccessible is a regular limit of regulars. A Mahlo is a cardinal such that the inaccessibles are stationary in itself. Look on Wikipedia for the definition of a \(\Pi^1_n\)-indescribable.
\(\Xi(0,\beta)\) enumerates the inaccessibles. \(\Xi(1,\beta)\) enumerates the Mahlos. \(\Xi(n+1,\beta)\) if \(n>0\) enumerates the \(\Pi^1_n\)-indescribables.
If \(\kappa=\Xi(\alpha,\beta+1)\) for some \(\alpha,\beta\), then \(\kappa^-=\Xi(\alpha,\beta)\). Otherwise, \(\kappa^-=1\).
If \(\nu\) is inaccessible, set \(\Omega_\nu=\nu^-\). If \(\nu=0\), set \(\Omega_\nu=1\). If \(\nu=I+1\) for inaccessible \(I\), set \(\Omega_\nu=I\) Else, set \(\Omega_\nu\) to be the least ordinal of cardinality \(\aleph_\nu\).
\(C_\nu(\alpha)\) is the closure of \(\Omega_\nu\) under addition, \(\lambda\beta.\omega_\beta\), \(\Xi\), and \(\psi\) if the argument is less than \(\alpha\). (I’ve removed \(\lambda\beta.\omega^\beta\) since it doesn’t affect the strength, also added \(\lambda\beta.\omega_\beta\) to get rid of an inconvenient offset.) \(\psi_\nu(\alpha)\) is the supremum of \(C_\nu(\alpha)\cap\Omega_{\nu+1}\).