Also the remark that hyperfinite can mean smaller than a nonstandard natural just seems false, where did you get that idea from?
When I look up the definition of hyperfinite, it’s usually defined as being in bijection with the hypernaturals up to a (sometimes either standard or nonstandard, but given the context of your OP I assumed you mean only nonstandard) natural . If the set is in bijection with the numbers up to , then it would seem to have cardinality less than [1].
- ^
Obviously this doesn’t hold for transfinite sizes, but we’re merely considering hyperfinite sizes, so it should hold there.
Ok, so this sounds like it talks about cardinality in the sense of 1 or 3, rather than in the sense of 2. I guess I default to 2 because it’s more intuitive due to the transfer property, but maybe 1 or 3 are more desirable due to being mathematically richer.