Because the rationals aren’t complete. The closed set could be empty in the case of the rationals.
and you guys laugh at people who believe in God!
Exactly. I perhaps should have spelt it out more completely, but letting L be the supremum of the lower bounds and R be the infimum of the upper bounds, we have that either A) there are finitely many constraints, L < R, and the solutions comprise the interval (L, R), or B) there are infinitely many constraints, L <= R, and the solutions comprise the interval [L, R].
The trouble with the rationals is that the supremum L and infimum R may fail to exist as rationals in the infinitely many constraints case; indeed, what would happen if you took an enumeration of the rationals is that you would end up in the infinitely many constraints case with L and R being the same irrational, so that [L, R] was inhabited by just one irrational point.
(Also, to be very nitpicky, instead of alternating between lower bound and upper bound constraints, Cantor apparently just waited for two things to come along satisfying the existing constraints, rather than taking them one at a time… It’s all the same, though)
Unlike the religious, mathematicians demand their results be logically consistent, and can change their axioms without cutting anyone’s head off.
Martin Luther survived nailing his axioms to the door.
Although he did say “Ouch” very loudly. (And in German, which required fourteen syllables.)
See, I don’t see how there can be a 1-1 mapping between reals and rationals, since reals include the subset of rationals. So there’s already a built-in 1:1 mapping, yet there are the irrationals which are also real. In other words, rationals are a subset of reals.
That’s what “hurts my head” about these. It’s no problem thinking a mile is greater than an inch, although both contain an infinite number of points. What I can’t do is somehow see that a mile and an inch contain the same number of points.
As it so happens there isn’t, but I’ll treat this as if you said “rationals and integers”, and address the subset issue.
For any infinite set, there is a 1-1 mapping between that set and some proper subset of that set. That is one way infinite sets differ from finite sets.
Here is a one example of a injective mapping between a set and one of its proper subset. (And showing there is an injective function shows there are at least as many elements in the proper subset.) Take I to be the integers {0,1,…}, N to be the natural numbers {1,2,…}, and f(x) = x +1. It seems obvious (maybe it isn’t) that f(x), given x is an integer, is a natural number, and a different natural number for each x; so there are at least as many integers as natural numbers (which is the un-intuitive direction).
There’s a one to one mapping between the counting numbers and the even counting numbers. (The relationship is obvious: a = 2*b.) And yet the even counting numbers are a subset of the counting numbers…
Screwy, innit? Infinity goes all wonky.
Martin Gardner had a cute one. Suppose, every single day, a person or group of persons throws out ten pieces of trash as street litter. But, every single day, you, a good citizen, pick up one piece of trash and dispose of it responsibly. At the “end of forever,” how many pieces of trash are littering the street?
If you use ordinary counting numbers as your metric…none! You picked up piece 1 on day 1. You picked up piece 10 on day 10. For any given piece of trash, you picked it up on day n.
To make the problem fit our “common sense,” you have to go with a different relationship. On day 1, people threw away pieces of trash 1a, 1b, 1c, 1d…1j. You picked up piece 1a. On day 2, people threw away 2a through 2j, and you picked up piece 2a. Using this relationship, which far more accurately models what we naturally think of, the trash piles up as high as the skies.
The lesson is: choose your relationships wisely. (All my old girlfriends say this.)
Well, the sense in which a mile and an inch contain the same number of points is this:
Imagine an inch-long stick very close to you and a mile-long stick very far away… if you set it up just right, the inch-long stick just covers the mile-long stick in your field of view. The left end of the inch long stick goes over the left end of the mile-long stick, the midpoint of the two sticks line up, the 30% from the left point of both sticks line up, etc. Each point on one corresponds to precisely one point on the other.
You could also think of a rectangle with an inch-long base and a mile-long diagonal (each point on the diagonal lying above a unique point on the base, and vice versa), or other such things. You could even forget pictures and just think purely linguistically “Ok, I can stretch an inch into a mile, and I can squeeze a mile into an inch, and if I do these one after the other, they cancel each other out”.
If you like, you can call this “there are the same number of points”, and indeed, that’s the sense in which mathematicians often use that phrase. You don’t have to use that phrase, if you don’t like, but the important thing is recognizing that simple stretching and squeezing functions take the inch to the mile and back in an inverse fashion. Surely you can wrap your head around that?