User:P進大好きbot/List of Articles

I list my contributions in googology.

= Large Numbers =

I list my largest numbers defined under specific regulations.

Computable
I list three regulations:
 * 1) One-Ruled Style
 * 2) Elementary Style
 * 3) Free Style

One-Ruled Style
Kyodaisuutan System

The regulation only allows a computable function defined by a single rule containing no function other than \({+},{-},{\times},{/},\textrm{^},!\).

I actually used \({+},{\times},{/},\textrm{^}\).

It is a \(3\)-ary function, and the growth rate of its \(1\)-ary diagonalisation is precisely \(f_{\omega}\) in FGH.

Elementary Style
Elementary Large Number

The regulation only allows a computable function defined by \({+},{-},{\times},{/}\) and other functions whose definitions are explicitly written by the creator only using those accepted functions.

I actually used \({+},{-},{\times},{/}\) and four other functions \(\textrm{^}\) (usual power), composition, \(\downarrow(x)\), and \(\uparrow(x)\) by writing explicit definitions only using those accepted functions.

It is a \(1\)-ary function, and its growth rate is greater than \(f_{\varepsilon_0}\) in FGH.

Free Style
Ordinal Notation with the PTO of ZFC

The regulation allows any computable function.

I diagonalised provably well-founded recursive relations with provable comparison among them.

It is a \(1\)-ary function, and its growth rate is greater than \(f_{\alpha}\) for any recursive ordinal \(\alpha\) equipped with a computable system of fundamental sequences such that whose specific properties (e.g. well-foundedness) are provable under restrictions of the length of formal proofs in FGH. Through a rough estimation the length of the formal proof of the well-foundedness of the standard ordinal notation system with weakly Mahlo cardinal under \(\textrm{ZFC}\), I guess that it is greater than \(f_{\psi_{\chi_0(0)}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))}\) in FGH. I expect that it is also greater than well-defined functions corresponding to other known OCF-based recursive ordinals in FGH. Also, it is conjecturally bounded by \(T(2 \uparrow^8 10^{100}(n+1))\), where \(T\) is the Transcendental Integer System.

Uncomputable
I list three regulations:
 * 1) ZFC Style
 * 2) Free Style

ZFC Style
最小の証明を書けなくても戦え数

The regulation allows any natural number defined in \(\textrm{ZFC}\) set theory.

For example, Rayo's number, which is originally defined in an unspecified second order logic, is not allowed in this regulation, because the truth predicate for coded \(\textrm{ZFC}\) set thoery is not formalised in \(\textrm{ZFC}\) set theory even if we use Platonist universe. For more detail, see this.

I diagonalised effectively axiomised consistent first order formal theories including Peano arithmetic.

It is a \(1\)-ary function, and its growth rate is uncomparably much greater than Busy Beaver function. It should be compared to another greater large function, but I do not know an appropriate one because there are few uncomputable large functions well-defined in \(\textrm{ZFC}\)-set theory,

Free Style
New Large Number beyond MK set theory

The regulation allows any natural number defined in a formal theory whose meta-theoretic consistency is strongly believed in mathematics.

I introduced a conservative extension of an \(\textrm{MK}\) set theory, and formalised the truth predicate for the coded counterpart.

I guess that it is the greatest large number in the world, i.e. it is the greatest among all known well-defined large numbers.

= Survey Articles =

I list survey articles explaning several topics in googology.

List of Mistakes
List of common mistakes on formal logic appearing in googology

I listed common mistakes on formal logic.

Bashicu Matrix System
I list explanations on BMS.

History
Summary on historical background of BMS

I explained historical background of BMS.

Proof
ペア数列の停止性の証明

I verified the termination of a specific version of PSS.

Large Cardinal
I list explanations on googological topics with large cardinals.

Introduction
Guideline on How to Use Large Cardinals for Ordinal Notations

I introduced a way to use large cardinals in googology in order to resolve the common confusion of them and placeholders.

OCF
Relation between an OCF and an Ordinal Notation

I explained relation between an OCF and an ordinal notation system in order to resolve the common confusion of them.