4 ms·
Funny you picked those examples, because I recall them both seeming quite obvious to me when I learned them. And I'm no great mathematician. It's, um, obvious
by pash 15y ago
Funny you picked those examples, because I recall them both seeming quite obvious to me when I learned them. And I'm no great mathematician.
It's, um, obviously quite hard to explain why you find something obvious, but I think in the first case it's because you can just think about rescaling the number line (using a bigger unit) or shifting the origin, which to me makes it clear that bigness is completely arbitrary and implies that there can't possibly be any biggest number in this sense; in the second case, there clearly is no "next" number in the reals as there is in the naturals, and that insight gets right at the heart of the concept of countability.
- xyzzyz 15y agoin the second case, there clearly is no "next" number in the reals as there is in the naturals, and that insight gets right at the heart of the concept of countability. I think you missed the point. There is no "next number" concept in the rationals either, and yet they're countable. There is "next number" concept in the first uncountable ordinal, and, as the name suggests, it is uncountable.
- pash 15y agoYou're right. The "next number" test (under the natural ordering) makes the concept of density obvious, not countability. So I think I've given credence to the notion that those who claim to find something obvious might not fully understand it. Or at least not at four in the morning. (Oh, and a point that Stephen Abbott made in his wonderful intro analysis book now springs to mind: proofs are useful also as a check on our intuition.)