On "trivial" statementsThere are no comments on this post.
May 08 2020, 16:08
Last night someone asked a question on Math.SE regarding a lemma used in proving certain chain conditions hold when iterating forcing with finite support. The exact details are not important. The point is that the authors, almost everywhere, regarded this as a trivial case.
Indeed, in my answer I also viewed this as trivial. It was tantamount to the claim: If \(\cf(\alpha)\neq\cf(\kappa)\), then every subset of \(\alpha\) of size \(\kappa\), contains a subset of size \(\kappa\) which is bounded.
It is clearer in the case where \(\kappa\) is regular, but this also holds for a singular \(\kappa\). It just takes a bit of thought. But let's assume regularity for a moment.
In the case where \(\cf(\alpha)>\kappa\) this is obvious. But what happens if \(\cf(\alpha)<\kappa\)? Well. Let \(A\) be a set of size \(\kappa\), then by definition of a cardinal, \(\kappa\) has an order embedding into \(A\), the image of the embedding \(B\), is a subset of \(A\) of size \(\kappa\). If \(B\) is unbounded in \(\alpha\), then the cofinality of \(B\) is \(\cf(\alpha)\) as well. Oops, so it has to be bounded.
But this did not sit well with the user asking the question. They had to take time to digest the claim, question it, and ended up posting their own proof which is certainly too convoluted, compared to the above.
This is a good thing.
Not the convoluted proof, although this is certainly important for students and learners to try and fill in the gaps by themselves. The above quick proof can be easily quoted from a previous result in the book(s), or proved explicitly. It's not long. Why not include it, one might ask? Well, it's a good thing to force people to reflect back on the material and learn how to pick the observation needed.
This is an important skill for research. For understanding. For developing a mental map of the subject matter, and how different theorems interact with one another. If you sit down to do research, and you have a question, you are not going to systematically throw everything against your question and see what sticks. That's just not how research works, at least initially. What we expect to happen is for some results to make "more sense" than others in different contexts.
The ability to make these connections is something non-trivial that is often, maybe too often, neglected from the discussion about self-learning, or even learning. It is the book equivalent of homework, or "in lecture questions".
Learning is not easy. It is hard. It is meant to be hard, so it will get easier later when applying the knowledge in "real situations" (research, work, whatever). And for this reason, I think it is important to leave "trivial" remarks, even when they are not necessarily obvious to any newcomer to the subject.
(Okay, if you're really curious, this was the post.)