User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180201150715/@comment-30754445-20180203204350

The correct answer to the strength of both TON and BMS is "we have no idea". We have lower bounds which aren't anywhere near Z2 which were explicitly calculated, but that's it.

Do you have any evidence that TON reaches Z2 let alone ZFC? Links to Taranovsky himself asserting things without proof do not count, as he is obviously biased.

Keep in mind that the burden of proof lays on those who claim the system is stronger. If we have no evidence that a system reaches 2nd order arithmetic, then it probably doesn't.

It is primarily for this reason that I, personally, highly doubt TON reach 2nd order arithmetic. Such claims remind me of all those newbies that declare they've found "the biggest number evar!!!".and then proceed to post an (ω+2)-level number. It's just silly.