As a freshly-minted CS doctoral student who plans on working on the more formal side of things, I've been studying up on my math in detail. While reading through a proof of Zorn's Lemma, I was struck by the complexity and creativity of it. (The proof was in Paul Halmos's Naive Set Theory.) By contrast, whenever I try to do practice proofs of anything nontrivial, I feel like I spend hours banging my head against the problem and worrying whether I've made any small mistakes that cause the whole tower to come crashing down.
My advisor is relatively math-savvy for the field, but even he doesn't do as math-intensive work as I'm planning on doing. My question is： Would a math professor be able to look at a novel problem like Zorn's Lemma and prove the thing in one sitting with no mistakes? Is that the level that will be expected of someone doing serious math work? Or is a nontrivial proof something that takes a lot of time, where small flaws in the proof is an understood part of the practice?
Sorry if this is a bad question; I suppose I'm partly just wondering how insecure I should be about my math skills.
It will all depend on your research question/s, topic, your supervisor, your ability and creativity and the amount of time you spend working on this.
You may be lucky and solve some question quickly (matter of hours), but then this will likely be one minor result. You will likely attempt many questions/problems without getting anywhere (especially if these are not proposed to you by your supervisor).
If you chose un unsolvable problem, you will obviously never get a direct result for this. You may on the other hand write up a a mostly trivial result. You want to spend minimum possible time doing these two things.
Ideally your advisor will have appropriate questions for you to work on. They will have a good idea on how much time it should take.