10,264 Pages

# Boboris02

## aka Boris Dimitrov

My favorite wikis
• I live in Veliko Tarnovo,Bulgaria
• I was born on August 15
• I am male
• ## Algorithm for Generating LUCOs From TON Expressions + Intuitive Analysis

May 6, 2018 by Boboris02

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 $a=C(\Omega_22,0)$ $\pi_+=C(\Omega_2,\pi)$ \[\kappa=\text{some ordinal }

• ## Breakthrough! Traditional OCF definition for TON

April 14, 2018 by Boboris02

While I was on holidays I had a lot of time to think about other ways of defining Taranovsky's notation. The problem is that TON is that it's not defined the same way other ordinal collapsing functions are, but rather you are given a lexicographic of strings and ways to compare them. Then you are given requirements for the strings to be valid. Finally, an ordinal function is introduced and is defined from those strings where the lexicographic ordering on a code of strings becomes a normal increasing order on ordinals. The main problem here is that, because the function is defined from lexicographic strings, we cannot be certain that it's well founded - aka that every valid string corresponds to a real ordinal below the limit of TON and vis…

• ## 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