/sci/ - Science & Math

I guess with the insight from yesterday, that with CH as in
i : R -> w1 a bijection
we can order all of R via
x <_{w1} y := i(x) < I(y)
so that

for all (y in R).
for all (x <_{w1} y).
exists (b: N -> {u <_{w1} x)

i.e. while R is not countable, there's an order for every y such that everything up to it is actually countable.

If we e.g. say R is like w_2, then there's elements Y so late in R that things below Y can't be counted.

This actually strikes me as a quintessential character of the hypothesis.

Are there any natural objects that are by construction of the cardinality of an uncountable ordinal?
I suppose the Neumann Hierarchy isn't, since their objects (while indexed by ordinals) have sizes soley determined by the power set operation