|
 |
Orchid XP v8 <voi### [at] dev null> wrote:
> Kevin Wampler wrote:
> > I suggest you start by proving that there's the same number of rationals
> > as natural numbers.
> But there are surely *more* rational numbers than natural numbers?
There's an infinite amount of rational numbers. There's an infinite
amount of natural numbers. Which set of numbers is larger?
The answer is that the sets have the same size. That's because a unique
mapping from each rational number to a distinct natural number can be
constructed. This means that for each rational number there exists a
distinct natural number. Hence the amount of both is the same.
Now, why this is even relevant? It's relevant because real numbers,
quite surprisingly, do *not* have such a mapping to natural numbers.
It's impossible to construct a unique mapping from each real number to
a distinct natural number. There are "too many" reals for this. For this
reason the set of real numbers is "larger" than the set of natural numbers
(and consequently the set of rational numbers).
Where this gets unintuitive is when you think about the relation between
rational numbers and real numbers: Take any two rational numbers, and
between them there will be an infinite amount of real numbers. Take any
two real numbers, and between them there will be an infinite amount of
rational numbers. No matter which pair of numbers you choose, there will
always be numbers from the other set between them. Yet the set of real
numbers is larger than the set of rational numbers.
> Actually, let's try something easier: Common sense tells you that the
> number of 2D coordinates is obviously [vastly] greater than the number
> of 1D points. Yet set theory asserts that both sets are exactly the same
> size. How can this be?
You may be confusing the density of the numbers with their total amount.
Even though more rational numbers can appear in a smaller place than
natural numbers doesn't mean their total amount is larger.
--
- Warp
Post a reply to this message
|
|
