User blog:Boboris02/Algorithm for Generating LUCOs From TON Expressions + Intuitive Analysis

Preface
LUCO = Large Unrecursive Countable Ordinal

Typically appears when trying to find models \(L_\alpha\) of theories. Become increasingly important for stronger theories. In this blog post I will be using various LUCO notions, such as \(\Pi_n\)-reflection and stability, so some background understanding will be required to extract the essence of my calculations.

For this blog post I will use constants \[a=C(\Omega_22,0)Boboris02 (talk)\pi_+=C(\Omega_2,\pi)Boboris02 (talk)\kappa=\text{some ordinal }<a\]

Introduction
I get a lot of doubts from people when I try to analyse Taranovsky's C for unorthodoxically large ordinals when trying to derive some sort of conclusion or intuition for the analysis of strong TON expressions. Here, I will give an explicid algorithm for getting LUCOs from TON expressions. Once you reach the least ordinal \(\alpha\) such that \(L_\alpha\models T\) (or alternatively \(L_\alpha\cap\mathcal{P}(\omega)\) for arithmetical theories), then the proof-theoretic ordinal of T should be roughly a collapse of that ordinal. This method however, is:

(1) not yet proven to work;

(2) not working for some theories (although those are specific exeptions) such as theories that assume certain recursive well-orders to exist or pairs of theories with equal PTOs and different minimal model height and vise versa; (3) very informal;

I should have mentioned right from the start that this blog post won't contain any actual proofs, so you well-defindness-proof-maniacs can leave if you wish.

Degrees of Reflection
http://web.mit.edu/dmytro/www/other/OrdinalNotation.htm#A3

In this section of his paper, Taranovsky describes a notation, similar to the main one but with some differences.

I won't be getting into the details, but it is essentially the second system, but without the 1-shiftedness property, but with other ways of collecting strength by iterating ordinals built from below. It has certian differences with the n=2 system of the main notation, such as having a different way of checking whether ordinals in it are standard and therefore some ordinals for the n=2 system are standard in C but not in the "Degrees of Reflection" C and vise versa.

For this notation we use \(\Omega\) as a diagonalizer.

The reader is advised to read the section of the paper before trying to figure out how to work with the algorithm suggested below.

The Algorithm
From here, take \(C(\alpha,\beta)\) to be a "Degrees of Reflection" C expression.

The as an approach at constructing LUCOs from it we define:

\[C(\alpha,\beta)=\begin{cases}\beta+\omega^\alpha & \alpha<\Omega\lor\beta\geq\Omega\\ \text{next admissible }>\beta & \alpha=\Omega\land\beta<\Omega \\ \beta+\Omega^\gamma & \Omega<\alpha=\Omega+\gamma<\Omega2\end{cases}\]

When \(\alpha\geq\Omega2\) the expression should be collapsed like it normally would be in the "Degrees of Reflection" C.

Once you have the LUCO assigned to the expression, to convert the DoR C expression into one of the n=2 main system you operate the following steps:
 * 1) Take \(C(\alpha,\beta)\)
 * 2) If \(\Omega\) is nowhere to be found in the standard representation of the expresssion, then leave it as it is.
 * 3) If \(\alpha\) or \(\beta\) are atom expressions, all you need to change is \(\Omega\) into \(\Omega_2\) and leave the zeros as they are.
 * 4) If \(\Omega<\alpha<\Omega2\) use the sub-algorithm provided above.
 * 5) If \(\alpha=C(\Omega(1+\gamma+1),\delta_1)^{\delta_2}\delta_3+\delta_4\) for some \(\gamma\) then change it into \(\alpha' = C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))\gamma+C(\Omega_22,0)^{\delta_2}\delta_3+\delta_4,\delta_1)\). Once you have \(\alpha'\), look at the representation of \(\delta_{1,2,3,4}\), respectively, and convert them as well. Denote the version of \(\alpha'\) with all four ordinals converted as \(\alpha"\). Finally, take \(\alpha'"=C(\Omega_22+\alpha",0)\) as the last modification of \(\alpha\), if \(\alpha>0\) and \(\alpha=C(\Omega_22,0)\) otherwise. As the last part, take \(C(\alpha,\beta)\) and look at the standard representation of \(\beta\). Repeat this part on \(\beta\) if needed.
 * 6)  If \(\alpha=C(\Omega\gamma,\delta_1)^{\delta_2}\delta_3+\delta_4\) for some limit ordinal \(\gamma\), then define \(\alpha'=C(\Omega_2+C(\Omega_2,C(\Omega_22,0))\gamma,\delta_1)^{\delta_2}\delta_3+\delta_4\). Look at the standard representation of \(\delta_{1,2,3,4}\) and convert all four. Once they are converted, call this verision of \(\alpha'\) \(\alpha"\). Finally, take \(\alpha"'=C(\Omega_22+\alpha",0)\) and \(C(\alpha"',\beta)\). If necessary, repeat this step on \(\beta\).
 * 7) If all steps are repeated a finite number of times untill you've converted the DoR C expression into a n=2 system one, then you are done!

Analysis
Okay, done with the boring stuff. Now it's time for the analysis! Up until Mahlo-fixed-opoints we work in this notation:

\[D_0(\alpha,\beta) = \{0,\Omega_\alpha\}\] \[B_0(\alpha,\beta)=\{0,M_\alpha\}\]

\[B_{n+1}(\alpha,\beta)=\{\gamma+\delta,\omega^\gamma,\omega^{CK}_\gamma,I_\gamma,M_\epsilon,\chi_\epsilon(\gamma),\chi_\alpha(\eta)|\gamma,\delta,\epsilon,\eta\in B_n(\alpha,\beta)\land\epsilon\in\alpha\land\eta\in\beta\}\] \[D_{n+1}(\alpha,\beta)=\{\gamma+\delta,\omega^\gamma,\omega^{CK}_\epsilon,\psi_\epsilon(\gamma),\psi_\alpha(\eta)|\gamma,\delta,\epsilon,\eta\in B_n(\alpha,\beta)\land\epsilon\in\alpha\land\eta\in\beta\}\]

\[B(\alpha,\beta)=\bigcup_{n\in\omega}B_n(\alpha,\beta)\] \[D(\alpha,\beta)=\bigcup_{n\in\omega}D_n(\alpha,\beta)\]

\[\chi_\alpha(\beta)=\begin{cases}\min\{\pi|B(\alpha,\beta)\cap M_\alpha\subseteq\pi\land\pi\text{ is admissible}\}&\alpha\notin\text{Lim}\\ \min\{\pi|B(\alpha,\beta)\cap M_{\alpha+1}\subseteq\pi\land\pi\text{ is admissible}\}&\alpha\in\text{Lim}\]

\[\psi^{CK}_\alpha(\beta)=\min\{\pi|D(\alpha,\beta)\cap"\alpha\text{th admissible}"\subseteq\pi\}\]

Here, we use \(I_\alpha\) and \(M_\alpha\) to denote the \(\alpha\)th recursively inaccessible and recursively Mahlo ordinals, respectively. We the label \(\chi_1(\alpha)\) as \(\chi(\alpha)\) and, if \(\alpha<M\) we can label it as \(I(1,\alpha)\) or the \(1+\alpha\)th recursively hyperinaccessible ordinal.

Recal that we use \[a=C(\Omega_22,0)Boboris02 (talk)\pi_+=C(\Omega_2,\pi)Boboris02 (talk)0<\kappa<a\] as shortcuts. I use \(M(....,\alpha,\beta,\gamma)\) and \(\Xi(....,\alpha,\beta,\gamma)\) as the "recursive" analogs of what Deedlit uses them for in this blog post.

The following are true when the ordinals are collapsed in \(C(\Omega_22+\text{__},0)\)

Up to Fully Stable Ordinals
Note: this analysis is in progress. More parts will be added and more elements will be added in the tables. If it becomes too much for the page to render I will split this into a series.