10,004 Pages

Boboris02

aka Boris Dimitrov

My favorite wikis
• I live in Veliko Tarnovo,Bulgaria
• I was born on August 15
• I am male
• Ordinal Analysis of Theories

February 12, 2018 by Boboris02

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.

Proof-Theoretic Ordinal (in 'standard' notations) Proof-Theoretic 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"

• Analysis of Taranovsky's Ordinal Notation with "standard OCFs."

January 12, 2018 by Boboris02

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 0-built from below by $$\beta$$ if $$\alpha Read more > • Measuring the strength of ABHAN part 3 December 13, 2017 by Boboris02 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\b… Read more > • Redefining ABHAN December 11, 2017 by Boboris02 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 $$A_1$$ …