@DanielFischer thanks for your help but this argumentas are new for me, and I have a bad time. I'm better ask the question the truth no longer want to bother you with my stupid questions
@MikeMiller I just feel that most of the things I consider reasonable to think about can be approximated to an arbitrary order of magnitude using finite or countable sets
once we start believing in infinite sets we start getting into stuff where I read theorems but i don't care whether they're right or wrong, because the exceptions are goofy. the whole AC thing with well-ordering is a good example.
my secret goal has been to develop a framework in which one can analysis at only reasonable things. An analysis theory where you can't have ridiculous things like cantor sets or Weierstrass functions
Species are supposed to categorify combinatorial proofs, but they only do so for "natural" proofs. For instance, the species Lin (linear orders on a set) and Perm (permutations of a set) are not naturally isomorphic, but obviously |Lin X|=|Perm X| for all X via an "unnatural" proof: pick a "basepoint" in Lin X, etc. Notably, Lin x Lin and Perm x Lin are naturally isomorphic. One recalls the notion of an affine space as an originless vector space.
Wonder if the (categorical or model) nature of unnatural combinatorial proofs and "arbitrary choices" has been studied in more detail.
@anon Since $a\in\mathbb{Z}$, my proof is not resolved since $2^k$ may not represent an integer and my factorization may not represent an integer factorization?
@Anthony so assuming A doesn't contain (0) and is finite, the closure of A is just A. what happens if A contains (0) or has an infinite number of prime ideals?
I was thinking these days that if I managed to publish a book, then I need to do it in such a way it shouldn't be taken as a contest between book. Ovidiu already has a very nice book, and it will remain like that no matter how many books will be published out there.
$X$ book is better than $Y$ book or $Y$ book is better than $X$ book is already a dangerous area, one might remain isolated, alone. I prefer to cooperate with people not to be in a competition with them.
@IceBoy Behind each book there is a crazy effort, and like the clothes, $X$ book might fit better for some and less for others especially if we consider the difficulty level of the problems in the book, the knowledge required.
For instance, I'm thinking now at Au-Yeung series, it's not an easy series, it's actually a difficult one, but the good news is that some research might reveal ways incredibly easy. Then, even if you have harder stuff in your book, adding there some revolutionary solutions is just a very welcome thing in my opinion. You know, the reader might say at first sight "This is a crazy hard question" and then, when he sees the solution "Wow, that's such an easy way!".
I think each question & answer must be as a good lesson to learn, and this is the hard part of the book. Repetitive ways of solving problems (in your book) is not that helpful for a reader that wants to learn a lot of techniques.
Each question should have something special from which you can learn something, that "wow" element one shouldn't think of at first sight.
"In general method is the general or specific way in which an activity is conducted, while techniques are the various methods and processes developed through knowledge, skill, and experience."
I'm focused on techniques that I got through experience. Moreover, the problems I wanna add I doubt they can be approached by some well-known methods. If we talked about a textbook, well, then we could talk more about methods.
@IceBoy Bringing in front of the reader new problems, very nice ones, that are solved by new techniques. Of course, I'll also have some known problems that are solved by new techniques.
This is also my dream when I buy a book, to see something I've never ever seen before, a book with amazing problems, amazing solutions, full of that "wow" element.
@IceBoy Not yet, but I have these things very clear in mind.
@IceBoy At the moment I only work on some problems I wanna add to my book, so I didn't establish yet other details of the book. This will happen later on.
@rehband I think I'll also have some open problems. On the other hand, where is possible, I can refer to results alread obtained from other problems, like some generalizations.
If you have a bunch of groups of different things, and you group the groups into a bigger group, what do you call the bigger group?
