Converting TON Expressions to Sequences
In this blog post, I will explain how to convert TON expressions to sequences of numbers, and how the method Hyp cos gave in his blog post relates to this.
- 1 Converting TON expressions to sequences
- 1.1 Step 1
- 1.2 Step 2
- 1.3 Step 3
- 2 Examples
- 3 Code
- 4 Properties of standard TON sequences
- 5 TON fundamental sequence by Hyp cos
In this step, every expression of the form \(C(\alpha_1,C(\alpha_2,...C(\alpha_{n-1},C(\alpha_k,\beta))...))\) for \(k\geq 1\) are converted to \(C(\alpha_k,\alpha_{k-1},...\alpha_2,\alpha_1;\beta)\). \(\alpha_m\) for \(1\leq m\leq k\) and \(\beta\) are \(C\)-expressions.
Note that the order of \(\alpha_k\) are reversed.
In this step, every expressions of the form \(C(\alpha_1,\alpha_2,...,\alpha_k;0)\) are converted to \(D(\alpha_1,\alph…
Converting 2nd System Expressions in Taranovsky's C to Labelled Hydras
Taranovsky's C is a ordinal notation by Dmytro Taranovsky, and even the 2nd system of the notation is believed to be very strong. However, this notation, unlike many other ordinal notations, uses lexicographical ordering and something Taranovsky calls "n-built from below", which makes the analysis extremely difficult. Among the people that understands the notation well, it is mostly accepted that \(C(C(\Omega_22,0),0)= \), but people still do not have an agreed-upon opinion on the behaviour of the notation past \(C(C(\Omega_22,0),0)\).
In this blog post, I will explain the method of converting 2nd system expressions in Taranovsky's C to hydras labelled with 1s and 2s, and how the method relates to the definition of the notation. This method…
I will resume doing googology-related contribution
I had been making googology-related contribution until the beginning of the year, mainly doing Saibian's regiments, but recently I've mostly only been doing things related to trolls, like reverting edits (not necessarily by Edwin Shade sockpuppets) or joining discussions related to Edwin Sade. I was still doing googology, and it's just that I have been doing googology outside this wiki (like discord). Because now the situation related to Edwin Shade are getting more and more serious, I thought I should resume doing googology-related contribution, and I will start by doing Saibian's regiments.
The reason I wrote a blog post announcing this is because I thought this way I will be more serious about doing googology-related contribution because…
Conversation with Edwin Shade
A few minutes ago, I talked with Edwin Shade in the chat of this wiki . I will post the chat log here. (I have Edwin Shade's permission.)
All time shown is UTC. E stands for Edwin Shade and R stands for Rpakr (me).
E (22:22): Hello Rpakr!
E (22:22): How are you doing today?
R (22:23): Fine!
R (22:23): Can I ask you some questions?
E (22:23): Sure.
R (22:24): Firstly, there was the account "AAA121" in googology wiki a few days ago. Is that you?
E (22:24): Yes.
R (22:25): Second question. You vandalized googology wiki in that account. Do you admit this?
R (22:25): The fact that the edits done in that account was vandalism?
E (22:26): Yes.
E (22:26): And you may screenshot this if you want.
R (22:26): Ok.
R (22:26): Third question.
R (22:26): Why did you va…
A Formal Definition of Pair Sequence System using BM1 Rules
- 1 Introduction
- 2 Definitions
- 2.1 Nonnegative integer
- 2.2 Concatenation of two strings
- 2.3 Pair function
- 2.4 Pair
- 2.5 First number
- 2.6 Second number
- 2.7 Adding a number to a pair
- 2.8 Sequence
- 2.9 Expression function
- 2.10 Expression
- 2.11 Sequence function
- 2.12 Bracket function
- 2.13 Length
- 2.14 nth pair
- 2.15 m pairs from nth pair
- 2.16 Adding a number to a sequence
- 2.17 Adding a number to a sequence and repeat
- 3 Definition of BM1 pair sequences
- 3.1 Base case
- 3.2 Successor case
- 3.3 Limit case
- 3.3.1 Bad root
In this blog post, I will define BM1 pair sequences formally without using things like ... which are ambiguous, and defines BM1 number.
In this section I will define some functions, sets, etc. I will be using in the definition. When I use quotation marks, it means the string inside them sh…