5 ms·
I contend no one understands real numbers if they can’t explain (without resorting to essentially purely symbolic rigor) why you can cover the rationals with in
by l33t2328 5y ago
I contend no one understands real numbers if they can’t explain (without resorting to essentially purely symbolic rigor) why you can cover the rationals with intervals of arbitrarily small total length, but you can’t do the same with R.
- Koshkin 5y agoWell, to understand an explanation one must already know something, otherwise first you have to explain those other things, e.g. the simple fact that unlike the reals the set of rational numbers is countable…
- l33t2328 5y agoCardinality hasn’t much to do with it since there are uncountable sets which you may cover like that.
- Koshkin 5y agoSure, but for countable subsets (such as the rationals) it is easy to show.
- l33t2328 5y agoSure. Explaining what’s different about the reals is the hard part
- Koshkin 5y agoHow so? Even a real interval of a finite length cannot be covered by any set of intervals of a smaller total length. (Unless the person you are trying to explain this to starts raising questions about the meaning of 'interval' or 'length', in which case the meaning of the original question becomes just as uncertain in the first place.)
- ravi-delia 5y agoI feel like that's downstream of seeing the reals as the rationals with the holes plugged, no? Obviously you can start from either end but everything special about the reals comes from being the complete version of the rationals.
- deleted 5y ago[deleted]
- shkkmo 5y agoI've been thinking about the reals a lot...beyond the rationals are all the real numbers like pi that have finite definitions, (even if those definitions, like pi's, require infinite computation.) But there is also this vast set of reals that are simply undefineable, non-repeating sequences. These numbers are unmentionable and unknowable. Does it really even make sense to say that this subset of the reals exists in the same way that definable numbers do?
- l33t2328 5y agoYes, of course. Because definable (really, computable) numbers don’t really “exist” either.
- shkkmo 5y agoWe can can use definable numbers. Pi is used with such massive frequency that is seems silly to say it has the same nonexistence as numbers that we can literally never even mention, let alone use. My assertion is that there is something wonky with these u definable numbers and that wonkiness is directly related to how absolutely massive the infinity of reals is.
- auggierose 5y agoWell, first you would have to explain what you mean by your statement.
- l33t2328 5y agoI mean that for any epsilon > 0, you can have a set of intervals of the form (a_i, b_i) where every rational number is in some interval and the sum over all i of b_i - a_i < epsilon. That is, you can cover the rationals with intervals of arbitrarily small total length.
- auggierose 5y agoI see. But is that not just a simple conclusion of the fact that the rationals are countable, and that there exists a converging series of positive numbers?
- l33t2328 5y agoRight but explaining why the reals are different is the tricky part