Gonna sleep bc 3 am but will respond later. Also the remark that hyperfinite can mean smaller than a nonstandard natural just seems false, where did you get that idea from?
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 n. If the set is in bijection with the numbers up to n, then it would seem to have cardinality less than n+1[1].
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.
Gonna sleep bc 3 am but will respond later. 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 n. If the set is in bijection with the numbers up to n, then it would seem to have cardinality less than n+1[1].
Obviously this doesn’t hold for transfinite sizes, but we’re merely considering hyperfinite sizes, so it should hold there.
From Goldblatt, since {0..H} is internal.
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.