Mathematical definition of infinity [duplicate]

I've heard from mathematicians, and not mathematicians, that the infinity is not a number. Is a concept, an idea.

Good! But we are in the world of mathematics, a world in where it's supposed to be able to define any mathematical object (I have doubts about this...) with precision and rigour.

I've been dealing with this concept in many areas but, SINCE THEN, I don't have the precise meaning. I don't like the idea of talking without definitions in math...

So I was wondering if the math definition of this concept exists? And where I can find it?

• Since there are many distinct infinities, you might want to be more precise. – user251257 Aug 12 '16 at 22:26
• And for the record, number is also just a human concept... – user251257 Aug 12 '16 at 22:27
• There are many different concepts of infinity. I think the definition in terms of cardinality of sets is one of the most useful ideas. E.g. $\aleph_0$ is the cardinality of the integers (countable infinity) and $\mathfrak c$ is the cardinality of the real numbers (uncountable infinity). See: en.wikipedia.org/wiki/Aleph_number – jdods Aug 12 '16 at 22:27
• You may also be interested to read about the extended real number line which may be the type of "infinity" you are wishing to talk about and is the type of infinity we use when we say something like $\lim\limits_{x\to\infty} x^2=\infty$. – JMoravitz Aug 12 '16 at 22:31
• @JasonM There are infinite hyperreal numbers, but $\infty$ is bigger than all of them. – user14972 Aug 13 '16 at 18:12

As others have noted, there are many different kinds of "infinity" used in mathematics. It's sometimes easy to forget that the notion of a "number" is also somewhat ambiguous. So, before you can even try to answer the question of whether infinity is a number, we need to be clear about what we mean by each term---"infinity" and also "number."

In my experience, usually people who ask about "infinity" have in mind the quantity $\infty$ used in evaluating limits. Mathematicians sometimes call this an extended real number. The same goes for its negative counterpart $-\infty$.

But just because some people call it a number doesn't mean it is a number in every sense of the word. In some contexts, a "number" refers to a natural number, in which case $\infty$ clearly does not qualify. Other times we are talking about real numbers, which, again, excludes $\infty$.

By the way, the term "real numbers" is somewhat lamentable, as all numbers are abstractions, failing to exist in the physical world. So, for example, the number 2 is no more "real" than the number $\infty$---both are abstract concepts used in math, for measurement and other purposes. Two eggs sitting on the counter do not form the number 2. Instead, the number 2 is an abstraction we use to think about those two eggs.

That said, some abstractions are weirder than others. The numbers $\infty$ and $-\infty$, as part of the extended real number system, don't behave as nicely as the real numbers themselves. So, for instance, it's impossible to define the subtraction operation $\infty-\infty$ in any reasonable way. The same goes for the division operation $\infty\div\infty$, which is also impossible to reasonably define.

As noted above, there are many other kinds of infinite numbers, perhaps most famously the infinite cardinal numbers introduced by Cantor. Instead of extending the real numbers, Cantor's cardinals extend the natural numbers. Cantor showed using a famous diagonalization argument that some infinite cardinals are "larger" than others (so to speak). In fact---in the context of cardinal numbers---there are infinitely many different sizes of infinity!

Infinite cardinal numbers also fail to behave as nicely as finite ones, and this has led some philosophers to deny the existence of infinite collections of objects. For example, William Lane Craig, as part of his project to prove the existence of God, has argued extensively against the existence of what he calls "actual" infinites. In my opinion, Craig's arguments are deeply flawed, as explained, for instance, by critics like Wes Morriston in his excellent paper "Craig on the actual infinite" (2002).

So as you can see, the issues here can get very complicated very quickly. So, in my opinion, it is a bad idea to say "infinity is a number" or "infinity is not a number." Even if you know what you mean by saying such things, your audience probably doesn't. Instead, I believe we should be more precise and say something like this: "$\infty$ is not a real number, but it is an extended real number."

There are quite a lot (I almost wrote "an infinite number") of different concepts that all use the word "infinity". Each does have a precise mathematical definition. They are not (ordinary real) numbers, though some can be called "numbers" in certain senses. You might look at Wikipedia.

I don't like the idea of talking without definitions in math...

Even in math that is completely rigorous, there's things that are never defined, and this is not just due to laziness. Here's a simple example: let $X$ denote a set with two distinct elements (and no others), call them $a$ and $b$. Question: what is the definition of $b$?

This kind of thing actually happens all over the place. Let $X$ denote a monoid with underlying set $\{1,a,b\}$, subject to the laws: $$aa = a, \;ab = b, \;ba = b, \;bb = b$$

Question: what is the definition of $b$? Of course, this has no good answer: what we're defining is $X$, not $b$. Now let $[-\infty,\infty]$ denote the set $\mathbb{R} \cup \{-\infty,\infty\}$, and make all the obvious definitions like $$\infty+x = \infty.$$

So, what is the definition of $\infty$? Again, this has no good answer. A better question would be: what's the definition of $[-\infty,\infty]$? Try searching the phrase "extended real number line" if you want to learn more.

This is an informal description of how "infinity" is used in math.

Mathematicians use "infinity" differently in different contexts. One of the more common context is limits. In general, limits are evaluated as so: $$\lim_{x\to a} f(x)$$ Informally, this expression asks what $f(x)$ gets closer and closer to as $x$ gets closer and closer to $a$. For "$a=\infty$" we have: $$\lim_{x\to \infty} f(x)$$ Which evaluates to whatever $f(x)$ gets closer and closer to as $x$ gets larger and larger. This is kind of a notation shorthand though, since it is NOT correct to say that larger numbers are closer to infinity than smaller numbers. In most cases where infinity is written into an expression, e.g. $$\sum_{i=1}^{\infty}\frac{1}{i^2}$$ or $$\int_{1}^{\infty}\frac{1}{i^2} dx$$ what is really meant is $$\lim_{t\to \infty}\sum_{i=1}^{t}\frac{1}{i^2}$$ and $$\lim_{t\to \infty}\int_{1}^{t}\frac{1}{x^2}dx$$

Another way "infinity" is used is to describe the size of sets. There are an infinite number of integers, and also an infinite number of even integers, and also an infinite number of prime integers, not to mention rational numbers (fractions), or even the set of all polynomials. The size of these sets can be described using Cardinal numbers. For example, the number of integers is $\aleph_0$, the number of real numbers is aleph_1, [edit:commenters are disputing my claim about $\aleph_1$, nonetheless $|\mathbb{R}|>\aleph_0$]. You can read about the differences between those and the reasons behind that difference here: https://en.wikipedia.org/wiki/Cardinality

• Your second to last sentence is false: it is consistent with the axioms of set theory (ZFC) that the size of the reals is not $\aleph_1$, but rather something larger. The statement "$\vert\mathbb{R}\vert=\aleph_1$" is called the Continuum Hypothesis. – Noah Schweber Aug 12 '16 at 23:08
• The number of real numbers is at least $\aleph_1$. The inequality is backwards (Though, more usefully, it is $2^{\aleph_0}$) – Milo Brandt Aug 12 '16 at 23:17
• Thanks, I always interpreted CH as claiming there is not a set S such that $\aleph_0<|S|<aleph_1$, given that $\aleph_0=|\mathbb{Z}|$ and $\aleph_1=|\mathbb{R}|$ – rikhavshah Aug 12 '16 at 23:17
• @user125261 See en.wikipedia.org/wiki/Aleph_number#Aleph-one. – Noah Schweber Aug 13 '16 at 0:32
• @user125261: Yes, that is what CH says. But CH is not proven. And it can't be proven using the currently fashionable axioms of set theory. In fact there is not even broad agreement on whether or not it "would" be true if only we could find the right axioms. (And in fact there is not even broad agreement on whether that last sentence makes sense.) – TonyK Aug 13 '16 at 14:11

(for the short attention span, it may be worth skipping the main content and just looking at the bottom to see the projective numbers)

It may be interesting to see one of the first fully rigorous accounts of the geometric sense of infinity.

In Euclidean geometry, it was realized that there was a duality between points and lines — it was realized that many theorems in the field of projective geometry were duals of one another: you could take one theorem, swap the notion of "point" and "line", fix up some details, and the result would be the other theorem.

As a simple example of this idea, "through every two distinct points there is exactly one line" becomes "on every two distinct lines there is exactly one point", which fixes up to "every pair of distinct lines is either parallel or has exactly one point of intersection".

The breakthrough came up algebraically. Recall that you can define points in the plane as a pair $(x,y)$ of coordinates, and you can define lines as triples $(a:b:c)$ of coefficients (where at least one of $a$ or $b$ is nonzero), and a point lies on a line if: $$ax + by + c = 0$$ (I use colons (:) instead of commans (,) to denote that multiplying through by a nonzero constant gives the same line)

It was realized that you could improve the symmetry between points and lines by representing a point as three coordinates $(x:y:z)$ with $z \neq 0$ (corresponding to the two-coordinate form $(\frac{x}{z}, \frac{y}{z})$), and then a point lies on a line if $$ax + by + cz = 0$$

At this point, it is an easy step to relax the restriction on nonzero coordinates: the projective plane consists of points with coordinates $(x:y:z)$ with at least one nonzero coordinate, and lines with coefficients $(a:b:c)$ with at least one nonzero coefficient.

From the Euclidean perspective of the $(x,y)$ plane, all of those points with $z=0$ are said to be "at infinity". Now, every pair of Euclidean parallel lines does meet at a unique point at infinity. There is one additional line, the "line at infinity", which passes through all of the points at infinity.

But algebraically, it's clear that infinity is not special; it behaves just like any other place on the projective plane. In fact, if we decide to fix $y \neq 0$ and work with the $(x/y,z/y)$ coordinates, we get another Euclidean plane and most of those points that were at infinity are now ordinary Euclidean points.

And this isn't just an esoteric thing; projective coordinates are very important for doing geometry, particularly when working with perspective (e.g. computer graphics) or when doing geometry algebraically. Also when doing algebra geometrically!

If you repeat this construction with the line rather than the plane, you get the projective line. You can naturally extend arithmetic; i.e.

• $(a:b) + (c:d) = (ad+bc : bd)$
• $(a:b) - (c:d) = (ad-bc : bd)$
• $(a:b) \cdot (c:d) = (ac : bd)$
• $(a:b) / (c:d) = (ad:bc)$

where the operations are only defined if the right hand side is not $(0:0)$. As before, the colon means that $(a:b)$ and $(ac:bc)$ mean the same thing if $c \neq 0$.

Each ordinary number $x$ corresponds to the point $(x:1)$. Of particular note is that $1/0$ exists, and is $(1:0)$. It is convenient to name $\infty = (1:0)$, and so we have $1/0 = \infty$. And then we have other things like $\infty + 1 = \infty$, $1/\infty = 0$, and $\infty + \infty$ is undefined.

(in calculus, we tend to do things differently, so that we get two points at infinity, rather than just the one)

There are incredibly many infinities in mathematics. When we talk about infinity in mathematics, we almost always talk about a size (cardinality) of a set. Sets can have finite size, but infinite sizes exist as well. The smallest infinite size is the size of the set of natural numbers $\mathbb{N} = \{1, 2, 3, 4, \ldots \}$. There are other, larger infinite sizes. For example the set of real numbers $\mathbb{R}$ can clearly be said to contain a larger amount of elements than the set of natural numbers. The set of rational numbers $\mathbb{Q}$, however, has the same size as the set of natural numbers. We write $|\mathbb{Q}| = |\mathbb{N}|$.

Other concepts of infinity arise when we are looking at limits. For example, suppose we have a sequence $(2,~ 4,~ 8,~ 16,~ \ldots)$. Precisely we can say that the $n$th element of this sequence is $2^n$. We can ask ourselves the question where this sequence goes to (spoilers, it's infinity). Intuitively, we want to know about the infinitieth "element" of the sequence. Formally, we call that number the limit of the sequence. We write this as $$\lim_{n\to\infty}2^n = \infty.$$

Now in this case $\infty$ isn't really a real number, it's just another name for the positive end of the real number line (or the upper end of the natural numbers). The other end of the real number line, of course, is called $-\infty$.

Note that only the ends of the real number line are named using the $\infty$ symbol. Cardinalities (sizes) don't use the $\infty$ symbol, after all there are many different infinite sizes. We do have some specific symbols for particular sizes, for example $\aleph_0$ (aleph zero) is the symbol for the size of $\mathbb{N}$.

There are more concepts of infinity in mathematics, but these two are by far the most common.

• Are the infinite "limits" of $n$ and $2^n$ really the same thing? I'm not trying to be silly. A divergent sequence doesn't really go anywhere, per se, it just grows without bound. So for $\lim a_n=\infty$, the infinity doesn't represent an object, but just the unbounded growth behavior. Just thinking out loud here... – jdods Aug 13 '16 at 0:31
• In terms of the concept of limit, those of $n$ and $2^n$ are the same. You can of course distinguish between these forms of growth by other means, e.g. using $o$ and $\mathcal O$. – Robert Israel Aug 13 '16 at 0:39
• @jdods: When I write $\lim a_n = \infty$, the $\infty$ does represent an object. Also, in this case, $\lim a_n$ doesn't have "unbounded growth behavior" -- it's $a_n$ that has "unbounded growth behavior". That the limits of $n$ and $2^n$ are the same is basically the same thing as the fact the limits of $1 - \frac{1}{n}$ and $1 - \frac{1}{2^n}$ are the same. – user14972 Aug 13 '16 at 18:08
• No, really. As mathematical concepts, those two versions of infinity don't relate at all. Of course you could always relate them by intuitive analogy, but there is absolutely no canonical relationship, and you can relate anything to anything by intuitive analogy. – Anon Aug 13 '16 at 22:04
• @jdods If you want an intuitive counterargument, then $2^\infty = \infty$, since $\lim_{n \to\infty} 2^n = \lim_{n \to\infty} n$, but $2^{\aleph_0} = \beth_1 > \aleph_0$, where $2^{\aleph_0}$ is the size of the power set of any set of size $\aleph_0$. Note that we have two completely different definitions of $2^\cdots$ here. – Anon Aug 13 '16 at 22:40

Well, it depends on what domain you are working in. Infinity has multiple meanings, I will try to answer it from a set theoretic perspective.

Let $f : A \to B$. $f$ is said to be bijective if every element in $A$ is paired with one, and only one, element in $B$ by the map. In more formal language, we a bijective function is defined as a function being both injective (a function that maps distinct elements to distinct elements) and surjective (a function in which all elements in the codomain has exactly one element in the domain that maps to that particular element).

Bijective functions are very important for understanding how and why infinities are equivalent.

Two sets are said to have the same cardinality ("size"), if there exists such an injective function between them. Formally, we say that $|A| = |B|$ iff. there exists some bijection $f : A \to B$. In a sense you can view such a bijection as a function that uniquely pairs every element in the domain with another element in the codomain, which is why we use this definition.

Now, when is a set infinite?

We define an infinite set as a set, $A$, such that there is a strict subsets $B \subset A$ with $|A| = |B|$. For example, to show that $\mathbb N$ is infinite, we consider the subset of even naturals, $2 \mathbb N$, and notice the bijection $f : n \mapsto 2n$, which you can verify to be a bijection:

1. Injectivity: If $f(x) = 2x = f(y) = 2y$, then, by the cancelation property, $x = y$.

2. Surjectivity: We can construct an right inverse: $f^{-1}(x) = \frac{x}{2}$.

We usually classify infinities into two categories:

1. Countable infinite set: If we can "enumerate" the elements (i.e. they have same cardinality as $\mathbb N$), we call the infinity countable.

2. Uncountable infinite set: Any infinite set which is not countable.

A natural question is: Does uncountable infinities exists? This is equivalent to asking "Are there different infinities?".

The answer can be quite intuitive to freshmen: Yes, there are different infinities.

$\mathbb R$ can be shown to be of another cardinality than $\mathbb N$.

Let $f : \mathbb N \to \mathbb R$ be a bijection from the naturals to the reals (an enumeration of the reals). There is some sequence of bijective functions, $B_n(x) : \mathbb R \to \{0, 1\}$ such that for any arbitrary $r \in \mathbb R$, $r = \sum_{n \in \mathbb N} 2^n B_n(r)$.

To find a contradiction, all we need to is to construct a real which cannot be in the codomain of a $f$. Take $k = \sum_{n \in \mathbb N} 2^n (1 - B_n(f(n)))$. To prove that $k \notin \operatorname{dom} f$, show that no element $n \in \mathbb N$ can have $k = f(n)$.

To show such a thing, observe how $B_n(k) = 1 - B_n(f(n)) \neq B_n(f(n))$. That's a contradiction! $1 - 0 \neq 0$ and $1 - 1 \neq 1$.

In fact, we can generalize this, to say that there exist no bijection from a set to its power set (the set of all its subset). The above is a special case of that: $|\mathbb R| = |\wp(\mathbb N)|$, and thus $|\mathbb R| \neq |\mathbb N|$.

Notice something? This allows us to generate an infinite number of distinct infinite cardinals: $|\wp^n(\mathbb N)|$.

• Also your definition of infinite set isn't right in the absence of the axiom of choice (see en.wikipedia.org/wiki/Dedekind-infinite_set), but that's a minor point. – Noah Schweber Aug 13 '16 at 16:11
• Apparently I suck more at set theory than I thought. It should be fixed now. – user347499 Aug 13 '16 at 16:58