Is infinity a real number

Answer 1

Infinity is not a specific number but a cardinality. The cardinality of a finite set is the number of elements in the set. For example, the cardinality of the set {1, 2, 3, 4, 5} is 5. Simple enough, but what about the cardinality of all natural numbers? There is no end to natural numbers so the cardinality cannot be a number in the normal sense. The cardinality is an infinity, called aleph-null. [As an aside, aleph is the first letter of the Hebrew language – which, along with the next letter, beth, gives us the word alphabet.]

The cardinality of any set which can be put into one-to-one correspondence with the set of natural numbers (or conversely) is also aleph-null. You then have the curious result that, using the mappings x-> 2x-1 and x -> 2x the cardinality of positive odd number is also aleph-null as is the cardinality of positive even numbers. Comparing cardinalities, you get the aleph-null + aleph-null = aleph-null or 2* aleph-null = aleph-null.

This result can be extended to all integers so that n * aleph-null = aleph-null for all integers n. This leads to the counter-intuitive result that aleph-null * aleph-null = aleph-null! It is possible to devise a diagonal scheme which gives a one-to-one correspondence between all rational numbers and all natural numbers. So there are aleph-null rational numbers. The classic exposition for this is Hilbert’s Grand Hotel. See https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

The above sets are said to have countably infinite elements (since they can be put into 1-1 correspondence with the counting numbers.

You may have noted that, when introducing aleph-null, I used the phrase “an infinity”. This is because there is another, higher cardinality: the uncountably infinite. The cardinality of the set of all subsets of a set with countably infinite elements, or 2-to-the-power-aleph-null. This infinity is also known as the continuum. Cantor proved that the cardinality of Irrational Numbers (and therefore the real numbers) is the continuum and also that there are no orders of infinity between aleph-null and the continuum.

Answer 2

yes!