Taranovsky's ordinal notations

Taranovsky's \(C\) is a series of recursively ordered sets descrived by Dmytro Taranovsky in a self-published web page the following general frame work to construct a binary function \(C\) from a well-ordered set \((S,<)\) equipped with an additional structure \(D \subset S \times S\) satisfying a suitable condition explained later. Whether the recursively ordered sets form ordinal notations or not, i.e. the well-foundedness of them, is an unsolved problem in mathematics.

Explanation
Let \(0_S\) denote the least element of \((S,<)\). For an \(a \in S\), we put \(C_a := \{c \in S : (c,a) \in D\}\), \(a+1 := \min \{c \in S : a < c\}\) (assume the existence), and \((a) := \{c \in S : c < a\}\). For an \(S' \subset S\), we denote by \(\lim(S') \subset S\) the subset of limits of elements of \(S'\) in \((S,<)\). We say that \(D\) is a degree for \((S,<)\) if the following hold:


 * \(C_{0_S} = S\)
 * \(\forall a \in S: a \neq 0_S \Rightarrow 0_S \notin C_a\)
 * \(\forall a \in \lim(S): C_a = \bigcup_{b < a} C_b\).
 * \(\forall a \in S: C_{a+1} = \lim(C_a) \lor \exists d \in \lim(S) \cap (a+1) \land C_{a+1} = \lim(C_a) \cup (d+1)\)

If \(D\) is a degree for \((S,<)\), then set \(C(a,b) := \min\{c \in C_a : b < c\}\). In particular, this construction works for a limit ordinal \(\eta\) equipped with a degree \(D \subset \eta \times \eta\) for \((\eta,\in)\). Since \(D\) is not unique, the resulting function \(C\) heavily depends on the choice of \(D\). Taranovsky introduced several explicit examples of degrees.

Define a partial ordinal notation system \(O\) as a partial mapping from ordinals below \(\eta\) to finite strings comprised of symbols and ordinals such that \(O(a)\) is undefined if \(a\) occurs in a string in the range of \(O\). Given a partial ordinal notation system \(O\) and a degree \(D\) for \((\eta,\in)\), we define Taranovsky's notation for \(a\):


 * If \(O(a)\) is defined, then we simply use the string \(O(a)\) to notate \(a\).
 * Otherwise, let \(b,c\) be ordinals so that \(a = C(b, c) \wedge b'>b \Rightarrow C(b', c) > C(b, c) \wedge c' < c \Rightarrow C(b, c') < C(b, c)\). We then use the string "\(C(b,c)\)."

Second-order arithmetic
One implementation Taranovsky conjectures to reach further than the proof-theoretic ordinal of second-order arithmetic, which if true would make it an exceptionally strong system.

For \(k\in\mathbb{N}\), define the binary relation "\(a\) is \(k\)-built from below by \(b\)" over ordinals as follows:


 * \(a\) is \(0\)-built from below by \(b\) iff \(a < b\).
 * \(a\) is \(k+1\)-built from below by \(b\) iff the standard representation of \(a\) does not use ordinals above \(a\) except as a subterm of an ordinal that is \(k\)-built from below by \(b\).

Taranovsky's notation, then, is a countably infinite family of notations indexed by positive integer \(n\) defined individually as follows:


 * The language consists of constants "\(0\)" and "\(\Omega_n\)" and a binary function "\(C\)" written in reverse Polish notation.
 * Ordering is lexicographic with "\(C\)" < "\(0\)" < "\(\Omega_n\)".
 * The strings "\(0\)" and "\(\Omega_n\)" are in standard form.
 * The string "\(abC\)" is in standard form iff all the following are true:
 * "\(a\)" and "\(b\)" are in standard form.
 * If "\(a\)" is of the form "\(cdC\)", \(b \leq d\) according to the aforementioned lexicographic ordering.
 * "\(b\)" is \(n\)-built from below by "\(abC\)".

For \(n = 1\), Taranovsky showed that the system reaches the Bachmann-Howard ordinal.