From Goldblatt, since {0..H} is internal.

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?

I used compactness in recent comment reply. Hypernaturals are uncountable because they are bigger than all the nats and so can’t be counted. Whether cardinality of continuum is equivalent to continuum hypothesis

I thought about this since. Bigger is not the right word. Complicated maybe? Like how the unit interval contains non-measurable sub intervals, or a compact set contains non-compact subsets.

Each number gets infinitesimal weight. Which infinitesimal is basically arbitrary.

P v NP: https://en.wikipedia.org/wiki/Generic-case_complexity

iLate reply, but the slicker bit is going in more fully. The appeal of the NSA approach here is axiomatizing it which helps people understand because people already know what numbers are, so 'inf big' is much less of a stretch than going the usual crazy inference depth math has.

This really benefits from a picture. Calling something “a nonstandard number” doesn’t really convey anything about them and a better name I’ll use is “infinitely big”, because they are.

< makes sense because the 2 chains are finite numbers and infinitely big numbers and an infinitely big number is bigger than any finite one because it’s , well, infinite. I can elaborate more technically, but I think trying to develop some numeracy for infinite numbers is a lot like learning about negatives and rationals and complex numbers. Just play with some expressions and get used to them. then look at the more technical treatment even if you have the ability to read it. Someone gave that example with a flat list but I think and feel that tapping into one’s existing NUMBER (and not list) sense is very powerful since it’s the first math we learn and the only one people use every single day.

"This" is the broken duality phenomenon?

Through stone duality. What about them in particular?