How about this example: the natural number 0 is not a member of the integers, from a set theoretic perspective.
In the natural numbers,
0 = {}
However, {} is not an element of the integers.
This is not something I expect to be easy to understand. This is the standard set theoretic definition of integers that mainstream mathematicians use. This is not some esoteric, fringe theory.
Yes, the empty set is a subset of all sets. I think some wires got crossed somewhere because whether zero is a subset of some other set doesn’t fit into the questions we are trying to answer.
You could pick a construction where the natural numbers are a subset of the integers. This is trivial, but this is a poor strategy overall, because you can always find a bigger set of numbers to work with. You can’t take the “biggest” set of numbers and then define all other sets of numbers as subsets of that. It would be kind of like trying to count down from infinity.
The Surreal Numbers manage to be a sort of "biggest" ordered field, in NBG, though they're a proper class not a set. All other ordered fields are subfields of the Surreals. Of course "ordered" is doing a lot, since it excludes the complex numbers, vectors, bivectors, etc. Whether elements of some object that isn't an ordered field should be considered "nubmers" is related but different question.
Yes, {} is a subset(⊂) of all sets, but it is not a member(∊) of all sets. For instance, {} is not a member of {{{}}}. In the Von-Neumann definition, 1:={0}={{}}. So, 0∊1, but 0 is not a member of {1} which is the above set. https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
In the natural numbers,
However, {} is not an element of the integers.This is not something I expect to be easy to understand. This is the standard set theoretic definition of integers that mainstream mathematicians use. This is not some esoteric, fringe theory.