User:Hyp cos/Taranovsky's various ordinal notations

"Taranovsky's ordinal notation" (TON for short) usually refers to an ordinal notation defined at Taranovsky's self-published web page, but there are many systems. They share part of definitions and properties, but the rest of definitions differ. Some of them are very strong, beyond knowledge of many googologists, thus it is difficult to compare with other notations. Here I will show their definitions and the differences between, and also comparisons between the systems.

Syntax
The syntax of most systems are constructed using two constants and one function. The constants include 0, and one of Ω or Ωn (where n is a fixed natural number within the system). The function, usually denoted by C, is binary.
 * 1) \(0\) and \(\Omega\) (or \(\Omega_n\)) are terms
 * 2) If \(a\) and \(b\) are terms, then \(C(a,b)\) is a term

Comparison
Terms can be compared and connected with ">", "<" or "=".

Firstly, write terms in postfix form, i.e. delete all the "(", ")" and "," and then reverse the string. Secondly, compare postfix forms in lexicographical order, where "C" < "0" < "Ω" (or Ωn being largest single letter).

Standard terms
In a system, a part of the terms are standard. To check "built-from-below condition", we need to go through the syntax tree of \(a\). In the definitions (see below),
 * Constants are standard.
 * \(C(a,b)\) is standard if all the following 3 are true.
 * Both \(a\) and \(b\) are standard
 * If \(b=C(c,d)\), then \(a\le c\)
 * This condition differs between systems, usually called the "built-from-below condition"
 * Quantifiers are not over terms, but over them with the position inside \(a\), so identical terms at different positions are treated differently
 * \(x\sqsubseteq y\) means x is a subterm of y (also \(y\sqsupseteq x\)). Using position index (e.g. the three 0's in C(C(C(C(Ω,0),C(Ω,Ω)),0),0) are at position (1,1,1,2), (1,2) and (2)), the position index of y is an initial substring of the position index of x
 * \(x\sqsubset y\) means x is a proper subterm of y (also \(y\sqsupset x\)). That is \(x\sqsubseteq y\land x\neq y\)

Ordinals
One standard term means one ordinal, and different standard terms mean different ordinals. The ordering of ordinals is defined to be exactly the ordering of standard terms.

So the least standard term, 0, corresponds to the least ordinal, 0. The standard term \(a\) larger than \(b_1,\ b_2,\cdots\) corresponds to an ordinal larger than what \(b_1,\ b_2,\cdots\) correspond to.

The definitions do not ensure well-foundedness, so it require proofs. Currently, only one system is fully proved to be well-founded.

Shared Properties
The third can help convertion between standard terms and Cantor Normal Form (CNF). With Ω being a "large" ordinal such that \(\omega^\Omega=\Omega\), convertion between standard terms and base-Ω CNF can also be done.
 * \(C(a,b)>b\)
 * \(C(a,b)\) is monotonic in both \(a\) and \(b\), and continuous in \(a\)
 * \(C(a,b)=b+\omega^a\) iff \(C(a,b)\ge a\)

Particular Definitions
The built-from-below condition is where the systems differ. It is combined from 3 concepts: the built-from-below style, the passthrough, and reflection configuration. Built-from-below style is introduced first, then the passthrough, and the reflection configuration is the latest. So three concepts make 3 generations of TON.

Built-from-below style
There are 3 built-from-below styles: "Degrees of Reflection", "Main", and "Iteration" (latter is stronger).

Degrees of Reflection
The "Degrees of Reflection" built-from-below style treats terms \(<\Omega\) and \(\ge\Omega\) differently, with only one layer of built-from-below (see "Main" built-from-below style for comparison).

System Degrees of Reflection is the one with "Degrees of Reflection" built-from-below style, without other two concepts.

To check C(a,b) standard, its built-from-below condition is \(a\prec C(a,b)\), defined as \[\forall x\forall y\sqsubset x(x<y<\Omega\rightarrow\exists z\sqsupseteq y(z<\Omega\land(z\sqsupset x\lor z<C(a,b))))\] (note again that quantifiers range over terms with their positions in the syntax tree of \(a\))

Main
The "Main" built-from-below style treats all terms uniformly, with fixed n layers of built-from-below, where the n comes from the system.

System Main Ordinal Notation System is the one with "Main" built-from-below style, without other two concepts. When used, we need to specify a natural number, n, as the subscript in Ωn.

To check C(a,b) standard, its built-from-below condition is \(a\prec_nC(a,b)\), defined as
 * \(a\prec_0b\leftrightarrow a< b\)
 * \(a\prec_{m+1}b\leftrightarrow\forall x(x>a\rightarrow\exists y\sqsupseteq x(y\prec_mb))\) (quantifiers range over terms with their positions in the syntax tree of \(a\))

This one is the simplest one among all the systems. In plain text, it is:
 * a is 0-built from below from b if a<b
 * a is (m+1)-built from below from b if a does not use terms above a, except as a subterm of an ordinal m-built from below from b

Iteration
The "Iteration" built-from-below style treats terms \(<\Omega\) and \(\ge\Omega\) differently, with n layers of built-from-below, where the n comes from the term itself.

System Iteration of n-built from below (variation without 4b) is the one with "Iteration" built-from-below style, without other two concepts.

To check C(a,b) standard, firstly we need to find all the a' (syntactic widest \(<\Omega\) child, \(\forall x\sqsupset a'(x\ge\Omega)\)) of a, then derive n for every a' as follows: And the built-from-below condition is that for all a', syntactic widest \(<\Omega\) child of a, a' is n(a')-built from below from C(a,b), i.e. \(a'\prec_{n(a')}C(a,b)\), defined as
 * 1) Write a in prefix form (term without "(", ")" and ",")
 * 2) Change a' into Ω
 * 3) Delete everything to the left of a', then add correct amount of "C" to the left to form a term
 * 4) Replace C(c,C(d,e)) with C(c,e) where c>d until there is no such subterm.
 * 5) Let alim be the resulting term after step 1 ~ 4
 * 6) Let a be the longest common ending substring of original a and alim (Note: a is standard if a is standard, but alim may not)
 * 7) Let n(a') be the largest natural number such that the prefix form of a'' can be written as \(CC\cdots C\underbrace{C0C0\cdots C0}_{n\quad C0's}X\) (where X is a string)
 * \(a\prec_0b\leftrightarrow a< b\)
 * \(a\prec_{m+1}b\leftrightarrow\forall x(x>a\rightarrow\exists y\sqsupseteq x(y\prec_mb))\) (quantifiers range over terms with their positions in the syntax tree of \(a\))

Passthrough
Passthrough is the second concept among the systems. There are 3 kind of passthroughs: no passthrough, a -passthrough and a''-passthrough (latter is stronger).