The above is sort of ambiguous. A set, each of whose members is true in some model, may not have a model itself (i.e. if it (i.e. each of its members) is not simultaneously satisfiable). To ask whether a set of sentences has a model is equivalent to asking whether it is consistent. I don’t see that there is a difference between (i) the (extended notion of the) truth of a set of sentences S, and (ii) the consistency of S. Of a set of sentences, the property ‘being simultaneously true’ and ‘being consistent’ are coextensional.

Also, in order to know that a set is consistent one must know that each of its members forms a consistent singleton. So the case for consistency is not any different from that of truth, in any sense relevant to the quotation above.

“…although I might know that “the set of all mathematical truths that I know is a consistent set”, it seems plausible that if I was presented with this same set under a different mode of presentation, I might not recognize it as a consistent set.”

Does this matter? It isn’t important that it is not *known* that it is a consistent set under a different mode of presentation, but that it is not *knowable* that it is consistent under that mode of presentation. The argument relies on knowability.

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