How can you compare the number of real numbers in the interval [0,1] and [0,10]?

Question

There are infinite number of real numbers between 0 and 1,i.e in the interval [0,1]. So definitely there should be more numbers in the interval [0,10] because it includes the numbers in the first case and also the numbers in the interval (1,10]. But we can map(one to one) every number in [0,1] to every number in [0,10]. This means that there are equal numbers in [0,1] and [0,10]. What does this really mean though??

Answer 1

The fact that the set $A$ is a proper subset of a set $B$ does NOT mean that the cardinality of the set $A$ is smaller than the set $B$. In fact, a set is infinite if and only if it contains a proper subset with the same cardinality as itself.

For example, the set $\{2,3,4,\dots\}$ is a proper subset of $\mathbb N$, but has the same cardinality as $\mathbb N$, as the bijection $n\mapsto n+1$ clearly shows.

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I admit, this can, at first, be slightly confuzing, but eventually, you get used to it. The main point is this: the ONLY proper way to determine whether sets have the same cardinality is to show whether there exists a bijection from one to the other. A bijection can be viewed as a pairing. By that I mean that a bijection $f:A\to B$ produces a set of pairs $\{(a,f(a)|a\in A\}$ where you provided each element of $A$ with an element of $B$ to pair up with. Since all the elements are now paired up, it makes sense to say that yes, indeed, there is the same "amount" of them in both sets.

Answer 2

People are used to work with finite numbers. In case of infinity, our intuition fails. In set theory, the 'measure' of the set is its cardinal number. It tells us how many elements does the set have. In the finite set it is easy to find its cardinal number: count the elements. Counting means finding a bijection between the given set and the set $N_k=\{1,2,...,k\}$ for any $k \in N$.

For sets with infinitely many elements there are two problems with our intuition:

i) all sets seem to have same number of elements: infinitely, so their cardinals are equal;

ii) as in your problem, the 'bigger' set must have more elements.

But non of above is correct.

What we do here, is just the same: take a bijection between two sets. Because we've used to work mostly with natural or real numbers, in the infinite case we search for bijections between that set and natural or real numbers. First person to handled these problems was Georg Cantor. He proved that we cannot find any bijection between natural and real numbers, so it means that cardinals of natural and real numbers are not equal!

But, we can always find a bijection between any interval [a,b] and real numbers, and a bijection between any two intervals, so we can also find a bijection between [0,1] and [0,10].

So, the explanation why it is so is related to the nature of infinity (or infinities). Imagine if you add a finite elements to a set with infinitely many elements. How many elements will the set have? What if we add infinitely many elements to a set with infinitely many elements?!

Also remember that 'counting the number of elements' means finding a bijection!