In the next section of his paper, Hellman expounds a version of the Dummett indefinite extensibility argument. You know the sort of thing! ‘Take some ordinals; then, whatever we start with, there’s an operation which gives us a new ordinal (take the successor of the greatest, or if there is no greatest take the limit ordinal) … Hence there can be no determinate domain containing, once and for all, all ordinals’.

What Dummett actually says (in a passage Hellman quotes) is “Given any precise specification of a totality of ordinal numbers, we can always form a conception of an ordinal number which is the upper bound of that totality, and hence of a more extensive totality.” But, as the version above shows, it seems we don’t need to lean heavily on the notion of a ‘totality’ to get the argument going — a point that Hellman also makes. Oddly, however, Hellman seems to think that the argument presupposes a plenitudinous platonism that involves the thought that “the very possibility of mathematical objects suffices for their actuality”. But that doesn’t seem right. You could surely be a selective platonist (a sort of Quinean platonist?), who thought that there were kosher mathematical entities that really exist and which are to be contrasted with the mere fictions of mathematical game-playing, and who thought that — among the kosher entities — are ordinals, indispensable for theorizing about the order structures we find in the world. But you could still be struck by the thought that, once we countenance do ordinals and the standard ways of getting from old ordinals to new ordinals, then there is no non-arbitrary way of calling a halt which is true to the very concept of an ordinal. And being struck by that thought doesn’t require being in thrall to a plenitudinous platonism.

Could it be, though, that we could accept the argument that there is no determinate domain of ordinals (for example) but still countenance quantification over absolutely everything? The thought would be that we can talk determinately about everything that there is, even if we cannot determinately corral off just that portion of what there is that ought to count as an ordinal (any attempt will leave outside it entities with just as much right to be called ordinals). Hellman thinks that this is unpromising for the reason that the “mathematical operations appealed to in connection with pure mathematicalia [as in forming new ordinals] can also be applied to mathematicalia-cum-non-mathematicalia”. But this goes too fast. Suppose we have some ordinals plus some other things (making up everything there is!); then we can apply the familiar operation to the given ordinals to give us another thing. But this operation need not extend the tally of everything there is: it could just be that one of those “other things” that made up everything there is has turned out to have as much right to be deemed an ordinal as the ordinals we started with.

Re your last claim (“…has turned out to have…”), that seems to mean that it

was alreadyan ordinal, unless ‘ordinal’ names a vague concept (?)