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.

I'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?

Yes, of course. Because definable (really, computable) numbers don’t really “exist” either.

We 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.

I 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.

Well, 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…

Cardinality hasn’t much to do with it since there are uncountable sets which you may cover like that.

Sure, but for countable subsets (such as the rationals) it is easy to show.

Sure. Explaining what’s different about the reals is the hard part

How 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.)

> Even a real interval of a finite length cannot be covered by any set of intervals of a smaller total length.

You’ve only restated the problem without saying _why_ covering the reals is different.

Because the 'intervals' in question are already 'real'?

Well, first you would have to explain what you mean by your statement.

I 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.

I 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?

Right but explaining why the reals are different is the tricky part

The way I explain this super simply to people is absolute value.

To most people, it just means that when you take the absolute value of a negative number it becomes positive, and the absolute value of a positive number stays positive.

Now there is more to it, but how you might think of absolute value instead is as a distance function, particularly how far away from zero you are on a number line.

This is way over simplified, but an example of how there can be a little more buried beneath the surface.