Right, on the standard account mathematicians have chosen for how to technically formalize “length”.
But that is a very technical beast, very far from ordinary-language “length”, and never forget that you are free to find some other way to formalize the notion of “length”, should you like.
Or actually, I should say, on one standard account mathematicians have chosen for how to formalize “length” (Lebesgue measure of a subset of a one-dimensional space).
It’s not uncommon even in math to consider other notions of “length” which are only defined for, say, continuous curves, such that the question of the length of the irrationals doesn’t even come up…
[All this is in reply to post #2[sup]10[/sup]]
On the Science Friday program yesterday, there was a brief mention of .999~ Steven Strogatz: The Joy Of X : NPR
See, that’s what I was wondering about when I wrote that post. The interval [0, 1], minus all the rationals, still has uncountably infinitely many points – yet no two are adjacent. So for all those points, you still have no actual continuous sub-interval there, however short. So I was kinda wondering how you define any kind of “length” for the sum total of the points you have left.
I have a two-year degree in Math (A.S.) so I’ve had all the Calculus/DiffEq (and Stat and Finite Math), but none of the upper-division advanced math. So it’s interesting to see occasional discussions like this, described at a level that I can grok.
The idea is that you adopt the principle “A countable disjoint union of sets has length equal to the sum of their individual lengths” (so, e.g., [0, 1] u [2, 3] has length 2) and the principle “A point has length 0” (or other principles leading to these, though these are straightforward in themselves). If you put those two principles together, the rationals, being a countable collection of points, have total length 0. And then the remainder of [0, 1] has to contribute the rest of its length; the irrationals will be assigned length 1.
The word “length” may not seem like the right word for this; perhaps “measure” is better. Think of it as a special case of integration: the “length” or “measure” of a subset is the same as the integral of its indicator function (the function which is 1 on that subset and 0 elsewhere).
(Now, it may not be obvious why we should consider the integral of the indicator function of the irrationals in [0, 1] to be 1, either. For example, this function is not Riemann integrable; it’s often used as an example to motivate more general notions of integration. Again, it comes down to rule-choices like the ones I mentioned above: e.g., the integral of a countable sum of functions is the sum of their individual integrals, etc.)
It’s strange and weird to think of it that way – decidedly “non-intuitive”. So a segment of the number line derives its “length” entirely from its irrational points, while the rational points – all of them together – contribute nothing. Weird. So much for “intuition”.
ETA: For a real headache, just contemplate (go ahead! Try it!) Thomae’s function, a.k.a. the “Popcorn Function”! IIRC, wasn’t there a thread on this not long ago. I’m pretty sure I first read of it here. It’s discontinuous at all of those rational numbers, but continuous at all the irrational numbers! :eek: Somehow, this is integrable.
(Missed edit window for above post.)
Yeah, there was a thread on the Popcorn Function, started by Frylock, 10/02/2011.
So what happens is we divide the irrationals up? Say into the algebraic and non-algebraic irrationals?
The algebraic rationals also contribute a total measure of 0. The non-algebraic irrationals in [0, 1] have total measure 1.
Any countable set of points has total measure 0. Each particular point has measure 0, and measure is countably additive.
[All on the conventional account of measure, of course. Feel free to devise another.]
Yeah, I was really wondering if we could divide the irrationals into a couple of uncountable sets easily, in a fashion that still intermingled them in a similar manner to the intermingling we get with the rationals. But it doesn’t really get us anywhere.
I guess it all comes back to - rationals are countable, irrationals aren’t. We have Aleph null rationals, Aleph 1 irrationals, and the continuum hypothesis says there are no intermediate counts. This seems to say that for points to be dense on a line we need an uncountable number of them, and that is about it. (It has been decades since I studied set theory, and it shows. I pulled out the textbook a while ago, and it still sort of made sense, but so much has leaked out of my ears it is depressing. Whether my musings are coherent is mostly a matter of luck. 
)
It does strike me that a lot of the arguments about 0.999… are in some sense related to the continuum hypothesis, and the difficultly of getting one’s head around it.
Actually, we have Aleph null rationals, and we don’t know how many irrationals we have. The continuum hypothesis says they’re as small as Aleph 1, but the set could be much larger.
And a countable set can be dense in the real line: The rationals are! (Meaning every nonempty open set of real numbers must contain rationals).
I said it had been a long time 
I’m misremembering a few things here. Time to read that old textbook again.
Just to clarify for laypeople, since very many are misinformed on this point: aleph_1 is NOT defined as 2^(aleph_0) [i.e., the cardinality of the powerset of the naturals; i.e., the cardinality of the reals; i.e., the cardinality of the irrationals]. That’s the definition of beth_1. aleph_1 is defined as the smallest (well-orderable) cardinality greater than aleph_0. Both aleph_1 = beth_1 and aleph_1 < beth_1 are consistent with the ordinary setup of set theory (in the same way that both “There is a square root of 2” and “There is not a square root of 2” are consistent with the ordered field axioms).
Indeed, in general, there are very loose constraints forced by the ordinary axioms of set theory for the relationship between “The smallest cardinality greater than x” and “The cardinality of the powerset of x”, beyond that the former is <= the latter; all kinds of possibilities abound in various models of set theory.
But anyway, the key point is, to reiterate:
aleph_1 is NOT defined as 2^(aleph_0) [i.e., the cardinality of the powerset of the naturals; i.e., the cardinality of the reals; i.e., the cardinality of the irrationals]. That’s the definition of beth_1. aleph_1 is defined as the smallest (well-orderable) cardinality greater than aleph_0.
Sorry Indistinguishable. You unclarified that for me.
My memory was that
aleph0 is the cardinality of the naturals
aleph1 is 2^aleph0
c, the cardinality of the reals may or may not be equal to aleph1 depending on which axioms one adopts.
I don’t recall much about beth and i know I have read about omega but forgotten everything.
So, have I read some bad sources or is my recall faulty?
Well, one or the other. aleph_1 is not defined as 2^{aleph_0). aleph_1 is defined as the smallest cardinality greater than aleph_0. c may or may not be equal to aleph_1, but c is definitely equal to 2^{aleph_0}.
And another question that is probably more related to the actual topic of this thread.
Is the cardinality of the hyperreals the same as the reals?
It depends on exactly which construction of hyperreals you have in mind. There are models of the hyperreals of every cardinality from c on up. But the usual construction is the minimal one, of cardinality c.
Yeah, slowly comes back to me. I had the steps in the wrong order. 
Ok then. I was under the impression that the hyperreals had a standard definition. I guess I have a lot to learn.
I do have a bunch of questions related to hyperreals, infintesimals surreals and infinities that this thread has brought up. But they are probably worth their own thread. And they are definitely worthwhile pursuing at another time when I am not supposed to be doing something constructive.
But I will ask now…
Is this statement
(a) we don’t know at the moment but it may be possible to figure out
(b) the answer is actually unknowable
(c) the answer depends on the axioms one is adopting
(d) we can choose whether or not to treat them as equal
If (d), then what effect does this have on the mathematics we do and the conclusions we reach?