# Aedating null set rar

*20-Nov-2019 05:32*

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?