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I don't know if this "question" is apt for posting here. If I can't post this, please, tell me where I can consult this kind of questions.

I would like some tips such as some basics and essential concepts about 'Set theory: cardinal numbers', a possible structure of the exposition, quality references of books (online avaliable if possible), web links, even answered question in this web, axioms or results these concepts need so that it makes sense both defining and using them (like the 'axiom of choice', it's necessary to compare the size of sets), most relevant math (or other sciences) authors in history, latest investigations about, etc.

My first year in college in the degree of Maths they taught some things about cardinals, only a few things, not more than three days. Now in the 4th year my knowledge of cardinals isn't very different.

A good intro might be: "Like mass, time, length, pressure, etc, mathematicians also want to measure sets by the use of cardinals. Even they could measure 'infinite' sets. Specifically different infinite numbers [...]".

I also read some Wikipedia articles respecting of these stuff and I think there are good explanations but asking here is more complementary.

Thanks a lot.

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    Have you searched the site, using the tag "cardinals," or the tag "large-cardinals" ? Even looking to the right of your question, you'll see a column of "Related" questions that can get you started. The wealth of questions and answers in these tags will give you a richer understanding of cardinality and large cardinals than can a few answers here. I don't know how much time you are allotted for the exposition, but I am quite sure you'll need to narrow your focus. Search the site for questions and answers about the continuum hypothesis, for example. – 2017-01-09
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    Cardinals and their characteristics span an incredibly wide range in set theory. One might even dare to say (unless I'm mistaken, I heard Woodin saying that during one of his talks) - as a first approximation - set theory is the study of cardinals and their arithmetic. Therefore, like @amWhy said, I think you should be much more focused in your request. If that's not possible, due to lacking knowledge, you may want to read an introductory textbook on set theory first (e.g. Kunen's 'Set Theory'). – 2017-01-09
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    There is a related exchange/overflow site devoted to math education – 2017-01-10

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Suppose that you are a bushman, some 30,000 years ago, in the dunes of Africa. You are familiar with the concept of one, two, and many. How can you tell that your hands have the same number of fingers (or not?), you put them together and you see that your hands have a matching between the set of fingers in each hand.

Formally, this means that we have a bijection between the set of fingers on the right hand, and the set of fingers on the left hand. And formally, this is how we define the notion of equipotency: the existence of a bijection.

It is important to remark that cardinality ignores "pre-existing structure", so the fact that $\Bbb Q$ or $\Bbb Z$ are countable means that there is a bijection of those sets with $\Bbb N$, but this bijection does not preserve the order or the additive and multiplicative structures. It's just a bijection.

Two important theorems about cardinals which are worth mention:

  1. Cantor's theorem, for every set $A$, the power set of $A$, $\mathcal P(A)$, has a strictly larger cardinality than $A$.

  2. The Cantor–Bernstein theorem, if $A$ is equipotent with a subset of $B$, and $B$ is equipotent with a subset of $A$, then $A$ is equipotent with $B$. So equipotence does in fact define an equivalence relation.