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 my best large numbers in three regulations:

One-Ruled Style
Article: 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
Article: Elementary Large Number beyond ψ_0(Ω_ω)

The regulation only allows a computable function defined by \({+},{-},{\times},{/},\textrm{^}\) and other \(1\)-ary functions whose definitions are explicitly written by the creator only using those accepted functions.

I actually used \({+},{-},{\times},{/},\textrm{^}\), and for other \(1\)-ary functions \(v(x)\), \(\triangleleft(x)\),\(\downarrow(x)\), and \(\Delta(x)\) by writing explicit definitions only using those accepted functions.

It is a \(1\)-ary function, and its growth rate is greater than \(f_{\psi_0(\Omega_{\omega})}\) in FGH, where \(\psi\) denotes Buchholz's OCF.

Free Style
Article: 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, where \(\psi\) denotes Rathjen's standard OCF with weakly Mahlo.

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 my best large numbers in two regulations:

ZFC Style
Article: 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
Article: New Large Number beyond MK set theory

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

I introduced a conservative extension of an \(\textrm{MK}\) set theory, and formalised "a system of truth predicates" 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 Common Mistakes
Article: 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
Article: Summary on historical background of BMS

I explained historical background of BMS.

Proof
Article: ペア数列の停止性の証明

I verified the termination of a specific version of PSS.

Interpretation
Article: BMOCF

I constructed a notation system "BMOCF" of predicate logic form which directly interprets the expansion rule of BM2.3 and hence is essentially equivalent to BM2.3. In particular, the termination of BMOCF (which is still open) implies the termination of BM2.3.

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

Introduction
Article: 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.

Analysis
I list explanations on googological topics related to analysis.

OCF
Article: 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.

Evaluation of the Reprodibility
Article: Evaluation of Analysis

I explained how to write an evaluative analysis which is reproducible in order to explain why almost all of analyses of BMS in this community do no make sense at all.

Axiomatic Estimation
Article: 解析チートシート

I gave a systematic approach to verify a sufficiently effective upperbound for large numbers defined in several specific ways.