Taranovsky's ordinal notations

Taranovsky's ordinal notations are a collection of ordinal notation systems invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough. They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version.

These were conjectured originally to be very strong, with the main system possibly reaching the full strength of second-order arithmetic and beyond. However, it is believed that, due to missing some bad ordinal structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension.

One of the systems, MP (Main System with Passthrough), is known to be ill-founded.

DoRI

Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal.

It uses a system of degrees so that:

• The term \(C(a,b,c)\) has admissibility degree \(a\).
• Every ordinal has admissibility degree \(0\).
• Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals.
• For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those.
• For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\).

DoR

Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is \(\alpha^{++}\)-stable; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller.

It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI.

Main system

The main system is divided into infinitely many subsystems. The zeroth subsystem has limit \(\varepsilon_0\), the first subsystem has limit BHO, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.