Intuition for countable vs. uncountable

[redrzewski]

Is there any way to visualize what is happening here, or do we just have to rely on the definitions/theorems?

1. Every open segment of reals (a,b) is uncountable
2. Every open segment of reals contains a rational
3. cardinality(R) = cardinality(PowerSet(N)). So an uncountable set is "much" bigger.
4. The function f(x): = 1 for x rational, 0 for x irrational is discontinous everywhere.

I follow the proofs/definitions justifying all the above. But intuitively, it doesn't make sense.

There are so many more uncountable numbers, it seems like there must be a bunch of them adjacent without a rational in between. But there can't be that many adjacent, because then we could actually create an open segment without a rational. But as soon as you have more than 1 point in a connected metric space, then we're back to an uncountable set, and rationals creep back in.

Does anyone have an intuitive explanation for this?

 

[Dragonfall]

You must remember that there is no "adjacent" real number to any other real number, so don't try to visualize it that way, or at all. Instead just remember that:

1. the rationals are countable and dense in the reals;
2. the irrationals are also uncountable and dense in the reals;

and know that these two facts do not contradict each other.

 

[Hurkyl]

As Dragonfall said, you went wrong as soon as you thought about "adjacent"; you aren't looking at anything resembling a sequence or a graph. Your error of intuition has nothing to do with cardinality!

You'll notice, of course, that between any two distinct real numbers, there are more irrationals than there are rationals, just as our intuition demands. (both in the sense of cardinality and in the sense of measure)

 

[Coelum]

There are so many more uncountable numbers, it seems like there must be a bunch of them adjacent without a rational in between. But there can't be that many adjacent, because then we could actually create an open segment without a rational. But as soon as you have more than 1 point in a connected metric space, then we're back to an uncountable set, and rationals creep back in.

Does anyone have an intuitive explanation for this?

Imagine you have a segment (a,b) whose endpoints are real (not rational). Both a and b admit a decimal representation, for example:

a = 0.123487498436621123...

b = 0.123487498436621987...

Then a position i exists in their decimal representations where they must differ (if it does not exist, a=b counter our hypothesis). In our example, i=16 and the last three positions are different. You can easily build the decimal representation of a number x such that a<b<x by picking some arbitrary sequence of numbers whose value is intermediate between a and b, e.g:

a = 0.123487498436621123...

x = 0.123487498436621500

b = 0.123487498436621987...

If you truncate the sequence representing x at any term in a position j>i, then x is a rational number. In other words, whatever two real numbers you pick, there is always a rational number in between.

Hope it helps,

Francesco

 

[mathwonk]

my intuition is on continuity versus non continuity. i.e. continuous intervals are uncountable. but rationals merely appear repeatedly, and never fill up continuous intervals.

 

[Blouge]

I'm not sure if it makes things more or less clear, but the surreal numbers and the hyperreals are infinitely more dense than the real numbers. Conceptually, what seems like a point in these number systems is actually a monad (a small interval full of points) when viewed at a finer scale.

 

[mathwonk]

think pointilism versus smooth brush strokes.

 

[benorin]

I'm going to give some food for thought: Recall that two sets have the same cardinality iff their elements can be put into a one-to-one correspondence.

Consider these sets: R=the real number line, C:=the unit circle. Now superimpose the C on top of R with the zero of R at the center of C. Establish a one-to-one correspondence thus: from the point at the "top" of C, call it N, (N is the point (0,1) in Cartesian coordinates) construct a ray from N through any point P on the line R and map it to the point Q were the ray intersects C. For example: if P is zero, then Q is at (0,-1). Draw it. What does Q approach as P goes to infinity?

Notice that C may as well be a line segment of length 2\pi (or any length by adjusting the radius of C,) hence the set of real numbers and the set of points in a line segment have the same cardinality.

Try the same thing with a plane and a sphere.

 

[CRGreathouse]

There are so many more uncountable numbers, it seems like there must be a bunch of them adjacent without a rational in between. But there can't be that many adjacent, because then we could actually create an open segment without a rational.

As the others have pointed out, don't think adjacent. You can't even have two adjacent rationals -- between any two distinct rationals there is a third, distinct from the other two.

 

[cks]

It's quite obvious from your example that P cannot go to infinity because (1,0) will be the limit where P can goes and still connected with Q. So, what are you trying to tell us from the question "What does Q approach as P goes to infinity?"

Sorry, couldn't catch it.

I'm going to give some food for thought: Recall that two sets have the same cardinality iff their elements can be put into a one-to-one correspondence.

Consider these sets: R=the real number line, C:=the unit circle. Now superimpose the C on top of R with the zero of R at the center of C. Establish a one-to-one correspondence thus: from the point at the "top" of C, call it N, (N is the point (0,1) in Cartesian coordinates) construct a ray from N through any point P on the line R and map it to the point Q were the ray intersects C. For example: if P is zero, then Q is at (0,-1). Draw it. What does Q approach as P goes to infinity?

Notice that C may as well be a line segment of length 2\pi (or any length by adjusting the radius of C,) hence the set of real numbers and the set of points in a line segment have the same cardinality.

Try the same thing with a plane and a sphere.