Aedating null set rar
Let’s say we have a set ) of cardinalities of multiplied sets.
So, if all of the sets are non-empty the cardinality of a product should also be non-zero and the set itself non-empty.
If we said, that set can contain itself, or some other set that would contain such set, things would get pretty hairy, pretty soon - it would easily lead to the paradox of set of all sets (Russell’s paradox).
And not sets, that contains some special ingredient somewhere deep inside - but by basically wrapping up empty sets and merging it with more wrapped empty sets in various degree of wrapping? If you are curious how to do it, let us take a small walk through the foundations of the set theory! Therefore we say, that two sets are equal (and they are the same set) if they contain the same elements.
If we want to build up the whole universe with sets, we need to set up some ground rules to follow. If we want to build set(s) up, we need to start up with something. The classical set theory does it in another way - uses axiom schema of specification, which tells that for each set you can define a subset.
Because they are the same (in a way), we might not distinguish between them, and even identify the whole set with them. Yet, we need them to be able to describe positions and sizes in (among others) Euclidean spaces.
And (as long as we won’t violate the property used to partition them), we can translate the operation on elements of the set to the operation on a whole set. if we What we might have not paid attention so far, is that from the very beginning such definition defines classes of equivalence for all possible pairs - even those, that were not handled by the original definition of subtraction. There are 2 popular ways to define real numbers using rationals: Dedekind’s cuts and Cauchy’s sequences.
If we define it like: What would be the meaning of that?