User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-27513631-20180129224338

Interesting. I'm looking at http://web.mit.edu/dmytro/www/other/OrdinalNotation.htm#A8, by the way, which is apparently a fragment (I misread earlier, thinking it was an extension, but still dispute its strength against Taravonsky and, naturally, if it fails at that level it's reasonably likely to fail later too.) of the system presented in the section you mentioned, which also claims to have a proof of consistency in ZFC.