This perhaps is not a math problem. I donot know if it fits in here. But it confuses me a lot these two days. I have wiki it. But it does not work for me. I will give some easy "proofs" to explain my confusion.
Prove that every set has the same cardinality with itself.
I prove it by contradiction. Suppose this statement is not true, then we pick one from what?... The elements fail in this statement cannot form a set?(How do I test whether it is a set? Is "pick" not a map which defined by set-theory ?). Even this is not a set, we still can pick one, right?
Confusion comes from induction.
General speaking, I donot encounter troubles use induction. But many authors like the following manner which I think is equivalent to induction: Suppose statement $S$ is not true for all natural numbers. By well ordered property of natural numbers, we can find a smallest number $n$ such that $S(n)$ is not true. Then.... Until here no problems. But what if the elements in $S(n)$ is not a set. For example, we want to show statement about all finitely generated $R$-module is true. When $R$ runs, module structure runs, can I say that "since $S(n)$ is not true, say $(M,T)$ is false in $S$,(where $M$ is a $T$ module generated by $n$ elements)" ? But here we can fix a ring $R$ first, then continue.
I think there are several inconsistencies in my argument.
Next is a wrong proof but not wrong too much.
Show every field has an algebraic closure.
proof. Say $k$ a field. Consider $\Sigma=\{L|L \text{ is an algebraic extension of }k\}$. Apply zorn's lemma, easily find a maximal element in $\Sigma$, then this element is a algebraic closure of $k$. But be carefully, we do not test whether $\Sigma$ is a set!
All right. I am sorry, I cannot figure out what my confusions.
I am not familiar with set-theory.
What exactly is a statement? How am I convinced myself that I have gave a right proof?
What is a fast way to convince a thing is a set for people having only naive set-theory foundation? Or how do we avoid this(not need to judge whether a class is a set) ?(This will be a primary question in this post).
I donot know what should be tagged to this. Thanks for reading.
Sorry for being unclear.