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For all the people who want to find some result of PTOs regarding the subject.
I will add references and sources to proofs or papers that mention it as I discover them.
ProofTheoretic Ordinal (in 'standard' notations) ProofTheoretic Ordinal (in TON) Arithmetical Theories Set Theories Reference(s) Notes
\(\omega\) \(C(1,0)\)
\(\text{KP}^0,Q\)
\(\omega^3\) \(C(3,0)\) \(\text{RCA}^{*}_0\)
[1]
\(\omega^\omega\) \(C(\omega,0)\) \(\text{RCA}_0,\text{WKL}_0\)
[1 ]
\(\varepsilon_0\) \(C(\Omega,0)\) \(\text{PA},\text{ACA}_0,\text{RCA},\text{WKL},\Delta^1_1\text{CA}_0\) \(\text{KP}\backslash\text{Infinity}\) [1]
\(\varepsilon_\omega\) \(C(\Omega+1,0)\) \(\text{ACA}_0+\) "for all \(n\), there exists an \(n\)th Turing jump"
\(\varepsilon_{\varepsilon_0}â€¦
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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 otherwiâ€¦
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In this blog I will try to explain and familiarize people to Taranovsky's notation as well as make bounds for ordinals describable within his n=1 and n=2 systems. Note that I have some problems with understanding this notation fully myself,so if anyone reading this believes to understand it better than me,that please be sure to correct me for any mistake I make.
The actual definition of the notation is quite complicated,in my opinion. So I will try to break it down and simplify it.
Let's denote a binary relation of "\(\alpha\) is \(n\)built from below by \(\beta\)" and a unary relation of "standard form" to ordinals.
\(\alpha\) is 0built from below by \(\beta\) if \(\alpha
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Part 1  http://googology.wikia.com/wiki/User_blog:Boboris02/Measuring_the_strength_of_ABHAN_part_1
Part 2  http://googology.wikia.com/wiki/User_blog:Boboris02/Measuring_the_strength_of_ABHAN_part_2
(NOTE: This section is mostly wrong! Expect a fixed version soon.)
\(\{0,1\{1\bullet_1 1\}1\}\) \(\psi(\psi_I(I))\)
\(\{0,1\{1\bullet_1 1\}2\}\) \(\psi(\psi_I(I+\psi_I(I)))\)
\(\{0,1\{1\bullet_1 1\}3\}\) \(\psi(\psi_I(I+\psi_I(I+\psi_I(I))))\)
\(\{0,1\{1\bullet_1 1\}0,1\}\) \(\psi(\psi_I(I2))\)
\(\{0,1\{1\bullet_1 1\}0,2\}\) \(\psi(\psi_I(I2+\psi_I(I2)))\)
\(\{0,1\{1\bullet_1 1\}0,0,1\}\) \(\psi(\psi_I(I3))\)
\(\{0,1\{1\bullet_1 1\}0\{1\}1\}\) \(\psi(\psi_I(I\omega))\)
\(\{0,1\{1\bullet_1 1\}0\{0\backslash 1\}1\}\) \(\psi(\psi_I(I\Omega))\)
\(\{0,1\{1\bâ€¦
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Considering I've spent the past few days measuring the strength of a notation that was first defined on a website,that no longer exists,I've decided to redefine it to avoid confusion.
For this definition I am going to use these symbols a lot:
A  An array. Can have one or any finite amount of entries.
\(\#\)  A segment of an array. It can also be nothing,for example b(a,b,#,c) = b(a,b,c).
a,b,c,d,.....x,y,z  Arbitrary integers.
/A/  The array "A" is solved "Normally",aka the same way it's solved in b(a,b,A).
ABHAN (Another Boris' Hyper Array Notation) is expressed in the form "b(A)",where "A" is an array.
Arrays and array seperators have ranks.
\(A_1 < A_2 \iff b(a,b,A_1) < b(a,b,A_2) \forall a,b\)
Seperator ranks are measured in R().
If \(A_1\) â€¦
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