on infinite sets

A set is a collection of things.Things can be digits, names, sets and everything. More actually, sets can be given as abstraction from collection treated by our thought. When thinking about things, we may put some of them together because they have something in common. For example, we may put Rome, Paris, Berlin together when we think about the capitals of European countries. In history, there are some specific sets accepted by people though they are fairly somewhat abstract, like a set of natural numbers denoted by $mathbb{N} = {0, 1, 2, …}$. This set can be dated back to counting sheep in ancient times. Mathematically, we can denote a set like this( considering the proceeding capitals ) : ${ Rome, Paris, Berlin }$ . It is exactly a conceptional set.

What more about set? Intuitionally, we can feel that a set should be more than a collection; and it must have more properties.

Given a set $mathbf{A}$ , its members or elements can be anything.

We call an element belongs to a set when we describe the membership. Suppose an element $x$ belongs to set $mathbf{A}$ , we denote the membership with $ x in mathbf{A} $ .

What about a set with nothing in it? It is defined as empty set, $varnothing$ .

Set $mathbf{A}$ is the subset of set $mathbf{B}$ , when

$$forall x in mathbf{A} , x in mathbf{B}$$

Denoted by $mathbf{A} subset mathbf{B}$
If $mathbf{A} subset mathbf{B}$ , but $ mathbf{B} not subset mathbf{A} $ , we call $ mathbf{A} $ is the proper subset of $mathbf{B} $ , id est $ mathbf{A} subseteq mathbf{B} $

Empty set is the proper subset of any nonempty set.

The power set is a set of subsets. Consider set $mathbf{A}$
$$ Power set of mathbf{A} : mathcal{P}(mathbf{A}) = { all subsets of mathbf{A} } $$

Consider set $mathbf{A}$ and $mathbf{B}$ :

$$ Union: { x | x in mathbf{A} or x in mathbf{B} }$$
$$ Intersection: { x | x in mathbf{A} and x in mathbf{B} } $$
$$ Complement : { x | mathbf{A} subseteq mathbf{B} , x in mathbf{B} and x not in mathbf{A} } $$

One thing we can easily think up is the volume of elements in a set. We use cardinality to describe how many elements in one set. For example, set $mathbf{A} = { 1, 2, 3, 4, 5 } $ . Its cardinality is:
$$ Card( mathbf{A} ) = 5 $$
It is easy to determine the cardinality of a finite set. What about an infinite set ? Obviously the cardinality of an infinite set is infinite number. When it comes to different cardinalities of infinite sets, we can not give a explicit description with existing theory. To attain this goal, we need to accept the set theory developed by Georg Cantor. He used aleph number to describe cardinality of infinite set. (I will write on this lately. ) For example, the cardinality of $mathbb{N}$ is $ aleph_{0} $ .