Question

What the difference between cardinality and dimension of a set? for example for R^3 and R^n.

Answer 1

See explanation...

Explanation:

Cardinality compares the sizes of sets by considering one to one functions between them...

If there is a one to one function from a set `A` into a set `B` then:

`abs(A) <= abs(B)`

If there is also a one to one function from `B` into `A` then:

`abs(B) <= abs(A)`

and we can deduce:

`abs(A) = abs(B)`

If you include the axiom of choice in your axioms of set theory then there will also be a bijection between `A` and `B`.

In the case of `RR` and `RR^n`, we do not need the axiom of choice in order to construct bijections between `RR` and `RR^n`. We can do it constructively.

For example, in https://socratic.org/s/aFEfkfWA I constructed a bijection between `RR` and `CC`. As sets we can consider that as a bijection between `RR` and `RR^2`.

If you have such a bijection, then you can use it to construct bijections between `RR` and `RR^n` for any finite `n`, by applying repeatedly to pairs of coordinates.

So:

`abs(RR^n) = abs(RR)`

for any finite positive integer `n`.

Dimension is a very different concept. `RR^n` is the set of `n`-tuples of real numbers. It is also a vector space over the reals. That is, it has operations of scalar multiplication and vector addition. With those algebraic structures, it has dimension `n` over the reals.

That does not mean that it is any larger as a set - it isn't.

By the way, you may think of the real numbers `RR` as one dimensional, but they actually form an infinite dimensional vector space over the rational numbers `QQ`.