User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-40154718-20190917091450/@comment-40154718-20190917220840

Do you mean that we are not sure that TON expressions really represent ordinals ? If they do not, may we call it an "ordinal notation" ? Or should we call it "Taranovsky Conjectured Ordinal Notation" instead of "Taranovsky Ordinal Notation" ? Reciprocally, if the well-foundedness of TON is proved, would it be sufficient to prove that there is an order-preserving map from TON to ordinals ?