Consider about "sets in ordinals" instead of
"ordinals in sets". There's a usual idea then
that ordinals besides reflecting an inductive set,
and a well-ordering in any mapping from them,
also that the whole numbers or "number theory's
numbers", so sit with them.
So, consider prime numbers with unique
factorization, and also a subset of prime
numbers that only have at most one power
of each prime factor. This is for the "fundamental
theorem of arithmetic", numbers (natural integers,
with regards to the cases for 0 and 1), that
numbers have unique prime factorizations,
and, of those, some have unique instances
of factors.
numbers: 1 2 3 4 5 6 ...
primes: 2 3 5 7 11 13 ...
composite: 4 6 8 9 10 12 14 15 ...
"uniposite": 2 3 5 6 7 10 11 13 ...
So, when modeling "sets", in "numbers",
there is a default model giving each element
in the universe of sets or the domain of discourse,
a prime number assignment, then a given set,
is just the multiple of those in these "uniposites",
where for example prime numbers generally
model "multi-sets", quite directly.
Now, I don't know too much who talks about
"primes and composites with only unique
factors in their decomposition, 'uniposites'",
but it would be interesting to really that a
very natural model of this sort _arithmetization_,
of sets, is exhibited _naturally_ because the
usual operation of union is taking the product
of the numbers and the usual operation of
membership is a divisibility test, and these kinds
of things.
union: product (least common multiple)
membership: divisibility test
intersection: greatest common denominator
disjoint: dividing out members
So, this introduces the usual notion of
"arithmetization", that then the operations
of arithmetic implement the set-theoretic
operations, then for such notions as
"geometrization", the sort of continuous analog,
of "arithmetization".
This sort of "composite numbers are a natural model
of a multi-set", can go a long way helping show that
the fundamental relations model and model and
model each other again, helping show why and how
it's simple that "resources in relations" establish
their orders, ..., of complexity in relation in type.
Here it's introduced the utility of
"uniposites: a sub-class of numbers
whose unique prime factorization has
unique elements".
2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47,...
https
oeis.org/search?q=2%2C3%2C5%2C6%2C7%2C10%2C11%2C13%2C14%2C15%2C17%2C19%2C21%2C23%2C26%2C29%2C31%2C33%2C34%2C35%2C37%2C39%2C41%2C42%2C43%2C46%2C47&language=english&go=Search
https
oeis.org/search?q=2,3,5,6,7,10,11,13,14,15,17,19,21,23,26,29,31,33,34,35,37,39,41,42,43,46,47&language=english&go=Search
Hmm, these aren't in the Online Encyclopedia of Integer Sequences?
https
oeis.org/A000040
https
oeis.org/A002808
The idea is simple that multi-sets are modeled by numbers ,
and, sets modeled by these "uniposite", numbers. Surely,
or rather, Shirley almost certainly, these already are,
"known".
A most usual sort of modeling a set as a number
is a "bit-map", for a word as wide as
the domain-of-discourse.