aka Boris Dimitrov

  • I live in Veliko Tarnovo,Bulgaria
  • I was born on August 15
  • My occupation is Cookies
  • I am male
  • 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 }

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  • 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…

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  • 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
    \(\omega\) \(C(1,0)\)

    \(\omega^3\) \(C(3,0)\) \(\text{RCA}^{*}_0\)

    \(\omega^\omega\) \(C(\omega,0)\) \(\text{RCA}_0,\text{WKL}_0\)
    [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"


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  • Boboris02

    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 otherwi…

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  • 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

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