Nature of Mathematics: Infinity as a Number

Our class period on Tuesday, June 21 brought us into a fairly heated argument regarding the existence of infinity as a number. Naturally, the discussion at one point veered towards using the Merriam-Webster-approved definition of the terms “number” and “infinity,” trying to pinpoint exactly what it is we are asking when we hear the question: “Is infinity a number?”

Infinity is seen all throughout mathematics, particularly in calculus when mathematicians pose questions using phrases such as “…as x approaches infinity.” We interpret this to mean “…as the independent variable gets either impossibly large, or impossibly small,” to a degree that we do not know, but we know that it is impossibly large or small. So what we can tell is that infinity is used to put a limit to a value that is absurdly huge, or absurdly small. Merriam-Webster helps us out quite a bit here with their definition of infinity (which coincidentally has also been used to disprove infinity’s existence as a number):

the limit of the value of a function or variable when it tends to become numerically larger than any preassigned finite number

Within the definition of infinity, we see that infinity represents both a limit and a value. In order to claim infinity as a number we must also know what a number is defined as, and we see two definitions that can either help or harm infinity’s role as a number:

  1. a word or symbol (such as “five” or “16”) that represents a specific amount or quantity
  2. a number or a set of numbers and other symbols that is used to identify a person or thing

What I can tell you from the first definition is that infinity is a symbol, however it does not represent a specific amount or quantity, since it is not physically possible in the real world to have an exact infinite amount of anything, and that was one of the strongest argument against infinity as a number. However, within the definition of infinity, we see the phrase “…preassigned finite number.” To me, this suggests that numbers are not inherently finite, or else why would a the numbers that infinities exist outside of have to be be described as such? If numbers were inherently finite, there would be no need for the phrase “finite numbers.” The second definition of number can help clean things up a little, so let’s get some background on what a set of numbers is.

A set of numbers is a group of numbers that share the same qualities, and are grouped according to those qualities. The following is a graphic from that gives a simple representation of different sets of numbers, and how they relate to each other.


During our discussion in class, we also decided to provide a set of numbers for infinity, as well, declaring that the term infinity refers to many different infinities, some bigger than others, but all representing some unknown value. We called these the Infintegers.  This is where the strongest argument supporting infinity’s existence as a number comes into play. Looking at this diagram we could ask the question: “Are Whole Numbers, numbers?” (We could do the same for Integers, naturals, etc.) The answer would be “Yes.” Slotting the Infintegers into this chart, we could ask the same question, except about infintegers. Would the Infintegers, existing in this same chart of number systems, not warrant the same response? The values within the Whole Numbers all represent different values, and likewise the infinities within the Infintegers all represent different values. Just as each value within the Whole Numbers qualifies as a number, so should all of the values within the Infintegers.



3 thoughts on “Nature of Mathematics: Infinity as a Number

  1. Coming from a fellow supporter of infinity as a number, I love this post. You really do bring up a good point about how there are different infinite numbers of certain sets of numbers so why would infinities be any different? I would be interested to see how a non-supporter of infinity as a number would respond to this argument. Thanks for sharing!


  2. Like to see you give Cantor some love for the transfinite number idea, that infinity is also a quantity. You could get into aleph_null and c &/or aleph_1. (A bit more content would be here.)

    But I like the argument you’ve got here a lot, too.

    C’s: 4/5


  3. Well first, let me state I whole heartedly agree that infinity is a number. However, even when stated in class I believe there was a “misspeak”, and even in your post. We actually don’t refer to things as approaching infinity when they are infinitely small. “Infinitely” is different than “infinity” and infinity is only spoken of as positive or negative… which is very normal for a number. “Infinitely” is actually an adjective to describe something that is very very very “something” and needs to be modified by using big or small. Infinity the number can stand alone. In the instance that we say something is infinitely small is when it is approaching zero! So, boom. That argument is null and void. 🙂 I agree with Heather, your finite vs. infinite argument is on point.


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