where is my fault in my proof? problem is denumerable set

Solution 1:

Other people already suggested great comments and answers to your question, even under your severely ill-formulated question. Honestly, I would not be surprised even if your question had been closed. I believe it had not happened because you still tried to provide lots of details and explanations in your attempts, which is really a great attitude. Let me answer your question from the perspective of mathematical- and non-mathematical sides.

First, your definition of infinite sets is not correct. Yeah, it depends on the textbook you are using. I believe it is either You-Feng Lin or Pinter, and the former is awful in the set-theoretic perspective. The latter is better because it was written by a logician (but I bet most instructors in Korea have not been carefully using Pinter's book.) I would be surprised if your textbook is Jech/Hrbacek.

The proper definition of infinite sets is a set that is not equipotent to any natural numbers. Period. You do not need to put any other sentences after this definition.

Then what is the remaining part of your definition? Well, it has a separate name: Dedekind-infinite sets:

Definition. A set $X$ is Dedekind-infinite if one of the equivalent conditions holds:

• $X$ has a proper subset that is equipotent with $X$,
• $X$ has a subset that is equipotent with the set of all natural numbers.

(You can prove that the above conditions are equivalent. That is, the former implies the latter and vice versa.)

Also, you stick to abstract and non-constructive proofs. There is no need for this unless it is necessary. You have to get familiar with the non-constructive nature of set theory, but you should defer it before learning the axiom of choice and cardinals. Abstract proofs are unnecessary, especially if you are working with countable objects on your level.

[Most set theorists would not agree that cardinals are "non-constructive." However, in a pedagogical view, they are unprecedented objects that students had not met before, and their constructions (say, aleph numbers) are bizarre for newbies. Thus students (and possibly, you) need time to be accustomed to them.]

@StinkingBishop gave a clear and nice concrete proof, so let me provide a proof based on Cantor-Bernstein-Schröder theorem.

Let $A$ be the set of all finite subsets of $\omega$, Clearly, there is an injection from $\omega$ to $A$. (That is, $|\omega|\le|A|$.) For the reverse inequality, let $\{p_n\mid n\in\omega\}$ be the enumeration of all prime numbers, and consider the map $$\{a_0,a_1,\cdots,a_k\}\mapsto p_0^{a_0} p_1^{a_1}\cdots p_k^{a_k}.$$

You can see that the above map is one-to-one, so $|A|\le|\omega|$. Hence by Cantor-Bernstein-Schröder theorem, $|A|=|\omega|$.

Let me suggest some tips on how to write a good question:

• Do not be shy with using a translator, dictionary, and Grammarly. It is better than writing down an incomprehensible question. Broken English is unavoidable as a non-native English speaker, but you should try to minimize it.
• Read previous questions and answers, and examine the structure of them. Also, click the "edit" button and see which mathematical symbols (LaTeX codes) they are using. I learned \mid, \mathbb and \operatorname in that way. A Wikipedia page for the list of mathematical symbols is also very helpful. (You can find this post by googling wp:formula.)
• Learn standard terminologies and clarify your definition. Not every mathematical texts use an identical definition, and there is a gap between terminologies from (usually, old) textbooks and working mathematicians.
• Keep practicing. Writing much is the best way of improving your writing skills. (It is very important if you plan to continue studying mathematics.)

Feel free to ask if there is any other question.