4 ms·
You can't enumerate the real numbers, but you can grab them all in one go - just draw a line! The more I learn about this stuff, the more I come to understand
by gloftus 10mo ago
You can't enumerate the real numbers, but you can grab them all in one go - just draw a line!
The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.
- veltas 10mo agoSorry I apologise, I didn't realise I wasn't allowed to care about countability.
- gloftus 10mo agoCountability is the whole point, there's no need to apologize. I was merely offering the perspective that "towers of infinity" is possibly the least useful consequence that comes from defining the notion of countability. To my mind, what we really reap from Cantor's work is a better understanding of the topology of the real numbers. But you have to define countability first in order to understand what uncountability really implies.