# Aedating null set rar

Complex numbers are, well, tuples of real and imaginary part. However, besides proper classes (and proper classes of proper classes, etc), everything based on a set theory operates on sets.

And each of these sets originates from an empty set wrapped in different ways.

For comparison, you can subtract one sequence from the other and check the sign. So far everything was more or less convoluted way of wrapping up empty set and sets of the empty set, and sets of sets of the empty set, etc. Vectors and matrices could be defined as functions from elements coordinates to the element (sets). Algebras are sets of values with operations on these values (functions, thus also sets). They did it because sets have to be constructible from bottom-up (one way or the other), which excludes e.g.

We build this way natural numbers, integers, rationals and reals, tuples and functions. Or tuples (vectors) and tuples of tuples with matching sizes (matrices). ability to build the set of all sets, which might be limiting. Also, an example of a proper class - a class which is not a set.

If we define it like: What would be the meaning of that?

(I saw a definition of a subtraction in lambda calculus, that would return 0 - because it has to return something, but we would like something better than that). At this point, I’m skipping showing how comparison work, or things like neutral elements, but the curious reader has all the tools to figure them out.

In mathematical sets, the null set, also called the empty set, is the set that does not contain anything. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable.

This is because there is logically only one way that a set can contain nothing.

The axiom of choice is sometimes replaced by well-ordering theorem or Kuratowski–Zorn lemma. Minimal requirements for building up natural numbers were defined in the 19th century by Giuseppe Peano.Axiom schema of replacement is a way of stating that if way of creating function can contain more elements, than required above (von Neuman set is just a minimal example).Having one infinite set, we can create other infinite sets just by applying other axioms on it.So, let’s Now, let’s say from now on we’ll identify a natural number with the group to in which it is a result of subtraction. So far everything could be solved with tricks like wrapping things up in a specific way. In a way, as for something based in eponymous reality, they are the first thing we cannot easily construct from smaller things.As a matter of the fact, what we did here is creation of an equivalent class - we partitioned the whole set of pairs to smaller disjoints subsets, that altogether cover the whole original set , where each pair is identical to the other in some regard (here: same result of subtraction). This indirectly became an issue for Pythagoreans, when they found out that numbers like cannot be constructed from (finite amount of) rational numbers, no matter how you arrange them. Even millenniums later we call such numbers irrationals.