Can infinity be defined as rational or irrational - and why?

Answer 1

Infinity is not just really big number - and consequently the concepts of rational vs irrational cannot be applied to it. It is a marvelously useful concept with great utility in mathematics but don't confuse it for being the same as a number that we could write out and categorize just because we have a symbol for representing it. When you stick infinity into an equation you get things like "limits" rather than a fixed answer; for example - for the function f(x) = (x-1)/x, if x = ∞ you don't actually get a value for the function- rather you get a limit that it approaches as x goes off to infinity; in this case the limit as x approaches infinity is 1. For the function f(x) = (x-2)/x, the limit as x approaches infinity is ALSO 1, and for the function f(x) = x/(x-1) the limit as x approaches infinity is .... 1. Obviously for any finite number they will not have the same value, but conceptually they all converge to the same value as you go to infinity. Hopefully this illustrates why you cannot apply the concept of rational vs irrational to "infinity".

Answer 2

There is a short answer and a long answer. The short answer is that infinity is not a specific number and so it cannot be defined as rational or irrational.

If you are interested, the longer answer is as follows.

As mentioned above, infinity is not a specific number but a cardinality. The cardinality of a finite set is simply 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 of this set is an infinity, called aleph-null.

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! The classic exposition for this is Hilbert’s Grand Hotel. See https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel for details.

It is possible to devise a diagonal scheme which gives a one-to-one correspondence between all rational numbers and all natural numbers - even though the natural numbers comprise a proper subset of the rational numbers. Thus there are aleph-null rational numbers. Since each of these sets can be put into 1-to-1 correspondence with the counting numbers, each set is said to have countably infinite elements.

You may have noticed that, when introducing aleph-null, I used the phrase “an infinity”. This is because there is another, higher cardinality: the uncountably infinite. This is the cardinality of the power set of a set with countably infinite elements. To illustrate this, the set {1, 2, 3} has cardinality 3 and its power set consists of the sets {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} : all the subsets of that set. So the power set has cardinality 2^3. Therefore the power set of a countably infinite set is "2-to-the-power-aleph-null". This infinity is also known as the continuum. Georg Cantor proved that the cardinality of Irrational Numbers (and therefore of real numbers) is the continuum and also that there are no orders of infinity between aleph-null and continuum.

It has been proved that the “number” of rationals between any two rational numbers – no matter how close - is the same as the “number” of rationals in total. Similarly, the “number” of reals between any two reals is the same as their “number” in total.