User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180201150715/@comment-30118230-20180204124023

"The correct answer to the strength of both TON and BMS is "we have no idea". " I actually agree with that completely. "Do you have any evidence that TON reaches Z2 let alone ZFC?"

For ZFC,no. For Z2,I'm currently working on it,so please be patient.

"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."

Except Dmytro Taranovsky is a professional mathematician at MIT,so I wouldn't compare him to the thirteen-year-olds that claim to have found the largest number ever.

"If we have no evidence that a system reaches 2nd order arithmetic, then it probably doesn't."

Assuming a statement false because you don't have proof for it is just as wrong as assuming it right. We have no proof for the continuum hypothesis,and we might never have one,but you don't see mathematicians assume it to be false. At this point things like the strength of TON have become the continuum hypothesis of googology and ordinal analysis.