Intuition for countable vs. uncountable

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In summary: 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.
  • #1
redrzewski
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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?
 
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  • #2
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.
 
  • #3
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! :smile:

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)
 
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  • #4
redrzewski said:
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
 
  • #5
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.
 
  • #6
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.
 
  • #7
think pointilism versus smooth brush strokes.
 
  • #8
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 [tex]2\pi[/tex] (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.
 
  • #9
redrzewski said:
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.
 
  • #10
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.



benorin said:
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 [tex]2\pi[/tex] (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.
 

1. What is the difference between countable and uncountable?

Countable and uncountable refer to the way in which we classify nouns in English. Countable nouns are objects or ideas that can be counted and have a singular and plural form (e.g. books, apples). Uncountable nouns, on the other hand, cannot be counted and do not have a plural form (e.g. water, knowledge).

2. How can I identify if a noun is countable or uncountable?

One way to identify if a noun is countable or uncountable is to see if it has a plural form. If it does, then it is a countable noun. Another way is to look at the meaning of the noun - if it represents something that can be divided into smaller parts, then it is countable. If it represents something that is abstract or cannot be divided, then it is uncountable.

3. Can a noun be both countable and uncountable?

Yes, some nouns can be both countable and uncountable depending on their usage. For example, "paper" can be countable when referring to individual sheets, but it can also be uncountable when referring to the material in general. In these cases, the meaning and context of the noun will determine if it is countable or uncountable.

4. What are some examples of uncountable nouns?

Some examples of uncountable nouns include abstract concepts (e.g. love, happiness), substances (e.g. water, sand), and collective nouns (e.g. furniture, equipment). These nouns cannot be counted and do not have a plural form.

5. Why is it important to understand the difference between countable and uncountable nouns?

Understanding the difference between countable and uncountable nouns is important for proper sentence structure and noun agreement. Countable nouns require a singular or plural verb depending on their number, while uncountable nouns always take a singular verb. Additionally, knowing the difference can also help with vocabulary acquisition and using the correct determiners (e.g. "a" or "an" for countable nouns, "some" or "any" for uncountable nouns).

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