If a set is infinite, no matter how many members of the set are counted, there are still more that have not been counted.^{[1]} Infinity can also be said to be larger than any arbitrary value. Infinity is not a number, so it can not be added, subtracted or multiplied. The symbol for infinity is like an 8 lying on its side: ∞.
A set is finite if it does not go on forever. A set that is finite can be very large, such as the natural numbers from 1 to 1 billion. If it is possible to count all the members of a set, then it is finite. It would take a long time to count from 1 to 1 billion, but it is possible^{1}.
A set is infinite if the end of the set can not be reached by going through all the elements of the set. Try counting all the real numbers between 0 and 1. Here is one way to do it:
In 1874, Georg Cantor (1845 – 1918) published a proof that not all infinities are the same. One class of infinity is denumerable, meaning countable.
The set of integers is an example of a denumerable infinity. If one starts at zero and counts the integers, each integer can be counted. In reality, it would take an infinity of time, but in principle they can be counted.
The set of real numbers is not denumerable. Start with 1. Then add a decimal point: 0.1. Now keep adding a zero after the decimal point: 0.01, 0.001, 0.0001, .... This can go on forever. Then it would need to be repeated for 2, 0.2, 0.02, .... So the size of the set of real numbers is a nondenumerable infinity.
A subset of an infinite set may be either finite or infinite. Take the set of integers. Since integer go on forever, the set of integers is an infinite set. The set { 1 } is a subset of the set of integers. It is a finite set, as it has exactly one member. The set of all even numbers is also a subset of the integers. However, the set of all even numbers is infinite, since it is impossible to find the highest even integer.
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