There is no answer to the question unless you come up with a way of comparing infinite lists. Cantor proposed one that most mathematicians find very useful, so it has become the standard.

He said that two infinite sets are "equal" in size if you could put them in 1:1 correspondence. To compare the the positive integers, you make a list and see if you can include all the numbers in that list. So, for example, you can make a list of primes:1. 22. 33. 54. 75. 11etc

The list will be complete in the sense that every prime will have a definite place in that list.

More remarkably, you can't do that for the real numbers! The proof is remarkably simple. Let's assume that someone claims to have a list. It doesn't matter how that list has been created; let's just say he claims to have it. The beginning of the list might look like this:

1. 3.1415926535 ...2. 1.000000000 ...3. 0.5772156649 ...4. 2.718288183 ...etc.

Note that some of the real numbers are represented by an infinitely long expansion, such as pi = 3.14159... with the ... representing the rest of the digits.

Well, I can find a number that is not on that list. I make up a number that is different from every number on the list in at least one digit. I make a decimal fraction that differs from the first number in the list by the first digit after the decimal point; that differs from the 2nd number by the second digit after the decimal point; differs from the third by the third digit; etc.

My number might be 1.0185 ...

The person who created the list says, "No -- your number is on the list!" I ask, where? He says it is number 326,543. I say, no it isn't. It differs from that number in at least one digit, in fact, in the 326,543rd digit. Of course, I am right and he is wrong. His list did not include all real numbers.

That proves that no such list exists. According to this approach, developed by Cantor, there are "more" real numbers in their infinity than there are positive integers. Cantor would say that both sets of numbers are infinite, but that the "order" of the infinities are different.

As an exercise, see if you can show the following facts in the Cantor approach:The number of positive and negative integers is equal to the number of positive integers.The number of even numbers is equal to the number of integers.The number of fractions is equal to the number of integers.

I learned all this when I was a teenager by reading the wonderful little book "One, two, three, ... Infinity" by George Gamow. The title of the book was based on this very problem of counting infinities.