- #1
redrzewski
- 117
- 0
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?
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?