User blog comment:Ubersketch/Lexicographic ordinal notation up to the Bachmann-Howard ordinal/@comment-30869823-20190913225449/@comment-35392788-20190917181715

I mean, it is true in buchholz's OCF, in that \( \psi_\nu(\alpha+1)\) is the smallest additively principal ordinal about \( \psi_\nu(\alpha)\), which is probably a fancier way of saying \( \omega^{\psi_\nu(\alpha)+1} \). However that works with the restriction that \( \alpha \in C_\nu(\alpha) \), which is not true for \( \alpha = \varepsilon_0 \). This works with \( \alpha = \Omega \) though, and we get the same ordinal because of Buchholz's OCF being constant on intervals like this one.

              = min { γ  ∈ P  :  ψ   ν      <span class="mo" id="MathJax-Span-341" style="font-family: MathJax_Main;">( <span class="mi" id="MathJax-Span-342" style="font-family: MathJax_Math; font-style: italic;">α <span class="mo" id="MathJax-Span-343" style="font-family: MathJax_Main;">) <span class="mo" id="MathJax-Span-344" style="font-family: MathJax_Main; padding-left: 0.278em;">< <span class="mi" id="MathJax-Span-345" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">γ  <span class="mo" id="MathJax-Span-346" style="font-family: MathJax_Main;">} <span class="mtext" id="MathJax-Span-347" style="font-family: MathJax_Main;"> if  <span class="mi" id="MathJax-Span-348" style="font-family: MathJax_Math; font-style: italic;">α <span class="mo" id="MathJax-Span-349" style="font-family: MathJax_Main; padding-left: 0.278em;">∈  <span style="position: absolute; clip: rect(3.149em, 1000.76em, 4.189em, -1000em); top: -4.01em; left: 0em;"><span class="mi" id="MathJax-Span-351" style="font-family: MathJax_Math; font-style: italic;">C    <span class="mi" id="MathJax-Span-352" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">ν      <span class="mo" id="MathJax-Span-353" style="font-family: MathJax_Main;">( <span class="mi" id="MathJax-Span-354" style="font-family: MathJax_Math; font-style: italic;">α <span class="mo" id="MathJax-Span-355" style="font-family: MathJax_Main;">)      <span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" style="position: relative;" data-mathml="" role="presentation">  <span style="position: absolute; clip: rect(1.333em, 1021.42em, 2.646em, -1000em); top: -2.24em; left: 0em;">   <span style="position: absolute; clip: rect(3.16em, 1000.63em, 4.372em, -1000em); top: -4.01em; left: 0em;"><span class="mi" id="MathJax-Span-324" style="font-family: MathJax_Math; font-style: italic;">ψ   <span class="mi" id="MathJax-Span-325" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">ν      <span class="mo" id="MathJax-Span-326" style="font-family: MathJax_Main;">( <span class="mi" id="MathJax-Span-327" style="font-family: MathJax_Math; font-style: italic;">α <span class="mo" id="MathJax-Span-328" style="font-family: MathJax_Main; padding-left: 0.222em;">+ <span class="mn" id="MathJax-Span-329" style="font-family: MathJax_Main; padding-left: 0.222em;">1 <span class="mo" id="MathJax-Span-330" style="font-family: MathJax_Main;">) <span class="mo" id="MathJax-Span-331" style="font-family: MathJax_Main; padding-left: 0.278em;">= <span class="mtext" id="MathJax-Span-332" style="font-family: MathJax_Main; padding-left: 0.278em;">min <span class="mo" id="MathJax-Span-333" style="font-family: MathJax_Main;">{ <span class="mi" id="MathJax-Span-334" style="font-family: MathJax_Math; font-style: italic;">γ  <span class="mo" id="MathJax-Span-335" style="font-family: MathJax_Main; padding-left: 0.278em;">∈ <span class="mi" id="MathJax-Span-336" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">P  <span class="mo" id="MathJax-Span-337" style="font-family: MathJax_Main; padding-left: 0.278em;">:  <span style="position: absolute; clip: rect(3.16em, 1000.63em, 4.372em, -1000em); top: -4.01em; left: 0em;"><span class="mi" id="MathJax-Span-339" style="font-family: MathJax_Math; font-style: italic;">ψ   <span class="mi" id="MathJax-Span-340" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">ν      <span class="mo" id="MathJax-Span-341" style="font-family: MathJax_Main;">( <span class="mi" id="MathJax-Span-342" style="font-family: MathJax_Math; font-style: italic;">α <span class="mo" id="MathJax-Span-343" style="font-family: MathJax_Main;">) <span class="mo" id="MathJax-Span-344" style="font-family: MathJax_Main; padding-left: 0.278em;">< <span class="mi" id="MathJax-Span-345" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;">γ  <span class="mo" id="MathJax-Span-346" style="font-family: MathJax_Main;">} <span class="mtext" id="MathJax-Span-347" style="font-family: MathJax_Main;"> if  <span class="mi" id="MathJax-Span-348" style="font-family: MathJax_Math; font-style: italic;">α <span class="mo" id="MathJax-Span-349" style="font-family: MathJax_Main; padding-left: 0.278em;">∈  <span style="position: absolute; clip: rect(3.149em, 1000.76em, 4.189em, -1000em); top: -4.01em; left: 0em;"><span class="mi" id="MathJax-Span-351" style="font-family: MathJax_Math; font-style: italic;">C    <span class="mi" id="MathJax-Span-352" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">ν      <span class="mo" id="MathJax-Span-353" style="font-family: MathJax_Main;">( <span class="mi" id="MathJax-Span-354" style="font-family: MathJax_Math; font-style: italic;">α <span class="mo" id="MathJax-Span-355" style="font-family: MathJax_Main;">)