User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-40154718-20190917091450/@comment-35470197-20190917230804

> Do you mean that we are not sure that TON expressions really represent ordinals ?

Right. It is an open question, although many googologists regard the well-foundedness as a fact.

> If they do not, may we call it an "ordinal notation" ? Or should we say "Taranovsky Conjectured Ordinal Notation" instead of "Taranovsky Ordinal Notation" ?

If TON does not express ordinals, i.e. is not well-founded, then it is not an ordinal notation. For the precise definition of the notion of an ordinal notation, see this blog post. Then you will find that an ordinal notation should be well-founded by definition.

The reason why we call it TON (shorthand of Taranovsky's ordinal notation) is just because many googologists who understand the definition of TON believe the well-foundedness.

For example, many googologists believed that BM2 (a previous version of Bashicu matrix system) could be regarded as an ordinal notation, and talked as if the well-foundedness had been obvious or doubtless, just because they did not know an example of an infinite loop. Since it is known not to be well-founded, there is no reasonable way to regard it as an ordinal notation.

The current version, i.e. BM4, plays the same role now. It is still unknown to be well-founded, but many googologists talk as if the well-foundedness were obvious or doubtless saying "because it expands in a very natural way". Namely, the term "ordinal notation" frequently means "notation whose infinite loops are unknown" in googology.

> 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 ?

The well-foundedness of TON is equivalent to the existence of an order-preserving injective map from TON to ordinals.