# Infinity: Countable, Uncountable

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Is infinity a number? The simple answer is no. Infinity is a concept, an idea that is related to numbers and counting systems; but it is not a number.

The infinity symbol (∞) is a lazy eight curve— an eight lying on its side. An easy way to associate this mentally with the concept of infinity is to realize that you can travel endlessly around the curve— there is no beginning and no end. That’s what the concept of infinity means— something without bound, larger than any natural numbers.

## If Infinity it isn’t a number, what is it?

A common definition for infinity is that it’s a quantity bigger than any number. That isn’t quite correct, but it’s good enough for most purposes.

Which exact definition you use depends on which specific area of math you’re working with. For example, if you’re looking at a number line, positive infinity is the last possible “number” to the left; negative infinity is always the last possible “number” to the far right.

In set theory, things get a little more complicated, because sets can have countable infinity, or uncountable infinity. Countable infinity is when it’s easy to label the numbers (like if you’re counting whole numbers 1, 2, 3…. Uncountable infinity is where the set contains all the numbers and everything in between, like 1, 1.22, 1.23, 1.256…. Set theory is beyond the scope of this article but if you’re interested in reading more on countable and uncountable sets, read Countable and Uncountable Sets.

## Is Infinity a Number if It’s Transfinite?

Although infinity is not a number, there are a special class of numbers sometimes referred to as “infinite numbers” which are bigger than all finite numbers. More properly, these are called transfinite numbers. So, if a number falls into the “transfinite infinity” class, then yes— it’s a number.

Although infinite numbers— i.e, the transfinites— are sometimes called numbers, it’s important to realize this does not mean they are natural numbers, real numbers, or any other type of number that we are used to working with and manipulating. The Peano axioms, for instance, the basic rules which guide the way we add, subtract, and understand real numbers, do not apply in any way to the transfinites.

The infinite (transfinite) numbers each have their own designation. The first transfinite cardinal number is written as Ν0, which we read as Aleph-naught. This represents the cardinality of the infinite set of natural numbers— the number of elements in that set. The next cardinal number is Ν1.

## References for Is Infinity a Number?

Rucker, Rudy. The Science and Philosophy of the Infinite. Bantum Books. Retrieved from