User blog:Boboris02/Second Order Arithmetic Subsystems in Terms of Taranovsky's C

Yes,I am making a proof for this. It's currently in progress since it's taking much longer than I expected. I have proved that the first system reaches the BHO and now I am writing a section on Degrees of Reflection C,which I will later use for the further proofs. As for now,I made this blog post to give people some basic outline of how TON compares with \(Z_2\) (mainly the subsytems below \(\Pi^{1}_2-\text{CA}_0\)) and other theories to fill the gap in the comparisons. I will provide both an easy to read and understand representation for the ordinals below,and their standard representation. Obviously,when an ordinal uses ordinals equal or bigger than \(\Omega_2\) as constants,the representation will be within the second system,and otherwise,the first system.

We define the proof-theoretic ordinal of a theory \(T\) (denoted with \(|T|_{\text{Con}}\)) as follows:

\(|T|_{\text{Con}} = \text{least ordinal }\alpha\text{ such that } F+EC-TI(\alpha)\vdash \text{Con}(T)\)

Subsystems of SOA and Their PTOs
\(C(3,0) = |\text{ATR}_0*|_{text{Con}}\)

In it's standard representation \(C(3,0)\) is \(C(C(0,C(0,C(0,0))),0)\)

\(C(\omega,0) = |\text{WKL}_0|_{\text{Con}}=|\text{ATR}_0|_{\text{Con}}\)

In it's standard representation \(C(\omega,0)\) is \(C(C(C(0,0),0),0)\)

\(C(\Omega_1,0) = |\text{ACA}_0|_{\text{Con}}\)

\(C(\Omega_1,0)\) is in it's standard representation.

\(C(\Omega^{2}_1,0) = |\text{ATR}_0|_{\text{Con}}\)

In it's standard representation \(C(\Omega^{2}_1,0)\) is \(C(C(C(\Omega_1,\Omega_1),\Omega_1),0)\)

\(C(C(\Omega_2+1,0),0) = |\Pi^{1}_1-\text{CA}_0|_{\text{Con}} = |\Delta^{1}_2-\text{CA}_0|_{text{Con}}\)

In it's standard representation \(C(C(\Omega_2+1,0),0)\) is C(C(C(0,\Omega_2),0),0)

\(C(C(\Omega_2,C(\Omega_2+1,0)),0) = |\Pi^{1}_1-\text{CA}+\text{BI}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2,C(\Omega_2+1,0)),0)\) is \(C(C(\Omega_2,C(C(0,\Omega_2),0)),0)\)

\(C(C(\Omega_2+\varespilon_0,0),0) = |\Delta^{1}_2-\text{CA}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2+\verepsilon_0,0),0)\) is \(C(C(\Omega_2+C(C(\Omega_2,0),0),0),0)\)

\(C(C(\Omega_2 2,0),0) = |\Pi^{1}_1-\text{TR}_0|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2,0),0)\) is C(C(C(\Omega_2,\Omega_2),0),0)

\(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0),0),0) = |\Delta^{1}_2-\text{CA}+\text{BI}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0),0),0)\) is \(C(C(C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),\Omega_2),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^2,0),0),0) = |\Delta^1_2+\text{CA}+\text{BI}+(M)|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0),0),0)\) is \(C(C(C(C(C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),\Omega_2),0),C(\Omega_2,\Omega_2)),0),0)\)

Other Theories
These are theories not interconnected with SOA,but still yeld important results with their consistency strength.

\(C(\Omega_1,0) = |\text{PA}|_{\text{Con}}\)

\(C(\Omega_1,0)\) is in it's standard representation.

\(C(C(\Omega_2,\Omega_1),0) = |\text{KP}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2,\Omega_1),0)\) is \(C(C(\Omega_2,C(\Omega_2,0)),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0),0),0) = |\text{KPi}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0),0),0)\) is \(C(C(C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),\Omega_2),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+1,C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0)),0),0) = |ML_1W|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+1,C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0)),0),0)\) is \(C(C(C(C(C(C(0,\Omega_2),C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),\Omega_2)),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))2,0),0),0) = |\text{KPh}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))2,0),0),0)\) is \(C(C(C(C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),\Omega_2),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^2,0),0),0) = |\text{KPM}|_{\text{Con}}\) (or KPm alternatively)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0),0),0)\) is \(C(C(C(C(C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),\Omega_2),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+1,C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^2,0)),0),0) = |\text{KPM}^+|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+1,C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0)),0)),0),0)\) is \(C(C(C(C(C(C(0,\Omega_2),C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),C(\Omega_2,C(C(\Omega_2),0))),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),\Omega_2)),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^{C(\Omega_2,C(\Omega_2 2,0))},0),0),0) = |\text{KP}+\Pi_3 - \text{Reflection}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^{C(\Omega_2,C(\Omega_2 2,0))},0),0),0)\) is \(C(C(C(C(C(C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),\Omega_2),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^{C(\Omega_2,C(\Omega_2 2,0))^n},0),0),0) = |\text{KP}+\Pi_{n+2}-\text{Reflection}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2 2+C(\Omega_2+C(\Omega_2,C(\Omega_2 2,0))^{C(\Omega_2,C(\Omega_2 2,0))^n},0),0),0)\) is \(C(C(C(C(C(C(C(C(C(\Omega_2,C(C(\Omega_2,\Omega_2),0)),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),C(\Omega_2,C(C(\Omega_2,\Omega_2),0))),\Omega_2),0),C(\Omega_2,\Omega_2)),0),0)\)

\(C(C(\Omega_2,C(\Omega_2 3,0)),0) = |\text{KP}+\Sigma^{1}_1\text{ Monotonic Induction}|_{\text{Con}}\)

In it's standard representation \(C(C(\Omega_2,C(\Omega_2 3,0)),0)\) is \(C(C(\Omega_2,C(C(\Omega_2,C(\Omega_2,\Omega_2)),0)),0)\)

I am not sure about the last one as it was only once stated by Taranovsky and it was about the "Degrees of Reflection" C,which may or may not be as strong as the second system. Plus,I haven't analysed so far and I haven't checked it,so it's just a speculation. (I don't really understand monotonic operators)