Infinity in Mathematics: What It Really Means, Explained
Infinity is not a real number but a set of distinct mathematical tools, including the extended real line, the projective real line, the Riemann sphere, nonstandard analysis, and cardinal numbers, each built for a different purpose. Treating infinity as an ordinary number produces contradictions such as “2 = 1.” Mathematicians use infinite constructions because they simplify calculus, limits, and the comparison of set sizes, not because physical infinity is known to exist.
Table of Contents
Key Takeaways
- Infinity does not belong to the set of real numbers, so arithmetic rules like division do not automatically apply to it.
- The extended real line adds two points, positive infinity and negative infinity, to the real numbers, but leaves expressions like infinity minus infinity undefined.
- The projective real line adds a single point at infinity to the real numbers and turns the real line into a topological circle.
- The Riemann sphere adjoins one infinite point to the complex numbers.
- Cardinal numbers measure the size of infinite sets, and Georg Cantor was the first mathematician to show that some infinities are strictly larger than others.
- The natural numbers, integers, and rational numbers all share the same infinite size, called countable infinity, while the real numbers form a strictly larger, uncountable infinity.
Why Is Infinity Not a Real Number?
Treating infinity as if it obeyed the same arithmetic rules as ordinary numbers leads directly to contradictions. Consider the equation 2 times infinity equals infinity. Dividing both sides by infinity would appear to give 2 = 1, an absurd result.
The resolution is that infinity is simply not a member of the set of real numbers. Infinite quantities can be adjoined to the real numbers in various frameworks, but they do not follow the same rules as finite numbers once added. Expressions such as infinity divided by infinity are often left undefined in these frameworks. When a problem asks for a solution within the real numbers, working with infinity directly is not permitted unless explicitly stated, and even then the rules governing it must be specified carefully.
Why Do Mathematicians Use Infinity If It Doesn’t Exist Physically?
Whether physical infinity exists is unknown, and it is irrelevant to why mathematicians use it. Infinity is used in mathematics because it is often more convenient than avoiding it, not because of any claim about the physical universe.
Consider measuring the heights of a group of people, producing values such as 1.70 metres, 1.76 metres, and 1.84 metres. Modeling the space of possible heights as the real numbers is convenient even though no measurement will ever equal an irrational value such as the square root of pi metres exactly. Choosing the real numbers as the model makes the underlying mathematics simpler and more powerful, since it allows the tools of calculus, including curve fitting, slopes, and areas, to be applied directly. Without an infinite model, these tools become far harder to use or impossible.
What Does “Infinity” Actually Mean in Mathematics?
There is no single definition of infinity in mathematics. Instead, several distinct notions exist, each serving a different purpose depending on the context.
Sometimes infinity functions purely as a symbol rather than an object with defined arithmetic. Examples include limit notation, such as the limit of f(x) as x approaches positive infinity, and the description of an element’s order in a group as “infinite.”
In other contexts, these symbols are given precise meaning by formally adjoining infinite quantities to an existing set, which allows arithmetic to be performed with them under specific rules. Frameworks that do this include:
- The extended real line, which is the real numbers combined with positive infinity and negative infinity.
- The projective real line, which is the real numbers combined with a single point at infinity.
- The Riemann sphere, which is the complex numbers combined with a single point at infinity.
- Nonstandard analysis, which introduces both infinite numbers and infinitesimal numbers.
Infinity can therefore serve two roles at once: a notion of unbounded size, and, within an appropriate framework, an actual object that can be manipulated with defined rules. Cardinal numbers are the tool used specifically to measure the size of infinite sets.
How Is Infinity Used in Limits?
In limit notation, infinity typically functions as shorthand for unbounded but still finite growth. The statement that the limit of f(x) as x approaches positive infinity equals a means that taking x sufficiently large makes f(x) arbitrarily close to a, and taking x larger still makes f(x) even closer.
For the function f(x) = 1/x, choosing x = 1,000 gives f(x) = 0.001, and choosing x = 100,000 gives f(x) = 0.00001, a value closer to zero. In this case f(x) is said to converge to 0.
This idea can be stated formally: the limit of f(x) as x approaches positive infinity equals a if, by taking x sufficiently large, the distance between f(x) and a can be made as small as desired. In this usage, the infinity symbol indicates values that grow arbitrarily large while remaining finite at every step.
What Is the Extended Real Line?
The extended real line was introduced to make certain limit statements mathematically meaningful. Within the ordinary real numbers, an expression like “x equals positive infinity” makes no sense, since infinity is not a real number. Adjoining two new points, positive infinity and negative infinity, to the real numbers produces the extended real line, in which expressions such as “f(x) equals positive infinity” become well defined.
Certain arithmetic expressions are defined within the extended real line, including positive infinity plus positive infinity equals positive infinity, one divided by positive infinity equals zero, and 2 is less than positive infinity. However, not every expression is defined: positive infinity minus infinity remains undefined even within this framework. Further detail is available from Wikipedia’s entry on the extended real number line.
What Is the Projective Real Line?
The projective real line takes a different approach than the extended real line. Rather than adding two separate infinite points, it adjoins a single element, infinity, to the real numbers, representing both directions of unboundedness at once. This turns the real line into a shape that is topologically a circle.
Within this model, one may write that 1 divided by 0 equals infinity, but infinity minus infinity remains undefined, just as it is on the extended real line. Ordinary real numbers generally cannot be compared to the projective infinity, so a statement such as “2 is less than infinity” is not meaningful in this framework. Further detail is available from Wikipedia’s entry on the real projective line.
What Are Cardinal Numbers and How Do They Measure Infinite Sets?
Cardinal numbers are the mathematical tool used to distinguish different sizes of infinity, showing that some infinite sets are, in a precise sense, larger than others.
The underlying idea starts with finite collections. Imagine a toddler who cannot yet count being given two sets of marbles and asked whether the sets are equal in size. The toddler can still determine this by pairing marbles one-to-one, picking one marble from each set and matching them until no further pairings are possible. If one set has marbles left unpaired, that set is larger.
This same pairing idea extends to sets of any size, including infinite ones. Two sets A and B are defined to have the same cardinality if there exists a one-to-one correspondence, or bijection, between every element of A and every element of B.
This definition produces a genuinely counterintuitive result for infinite sets. Consider the set of all natural numbers, A = {0, 1, 2, 3, 4, 5, …}, and the set of even natural numbers, B = {0, 2, 4, 6, …}. Even though B is a strict subset of A, the two sets have the same cardinality, established through the bijection that maps each number n in A to 2n in B. Every element of A pairs with exactly one element of B and vice versa, so the two sets are the same size despite one appearing to be “smaller.”
Using this same reasoning, the natural numbers, the integers, and the rational numbers can all be shown to be countably infinite, meaning they share the same cardinality. The set of real numbers, by contrast, is uncountable and strictly larger than any of these countably infinite sets. Georg Cantor was the first mathematician to demonstrate this difference in the sizes of infinite sets.
Glossary
- Extended real line: the real numbers with two additional points, positive infinity and negative infinity, adjoined to allow certain limit expressions to be well defined.
- Projective real line: the real numbers with a single point at infinity adjoined, forming a structure that is topologically a circle.
- Riemann sphere: the complex numbers with a single point at infinity adjoined.
- Nonstandard analysis: a framework that introduces infinite numbers and infinitesimal numbers alongside the real numbers.
- Cardinal numbers: numbers used to measure and compare the sizes of sets, including infinite sets.
- Bijection: a one-to-one correspondence between two sets in which every element of one set pairs with exactly one element of the other, and vice versa.
- Countably infinite: a set, such as the natural numbers or the integers, whose elements can be placed in a one-to-one correspondence with the natural numbers.
- Uncountable: an infinite set, such as the real numbers, that is strictly larger than any countably infinite set.
Frequently Asked Questions
Is infinity a number?
No. Infinity is not a member of the set of real numbers. It can be adjoined to the real numbers or other sets within specific mathematical frameworks, such as the extended real line, but it does not follow the same arithmetic rules as ordinary finite numbers once added.
Why does dividing by infinity seem to create contradictions like 2 equals 1?
These contradictions arise from treating infinity as if it obeyed ordinary arithmetic rules, which it does not. Starting from 2 times infinity equals infinity and dividing both sides by infinity is not a valid operation, since expressions like infinity divided by infinity are typically left undefined.
What is the difference between the extended real line and the projective real line?
The extended real line adjoins two separate points, positive infinity and negative infinity, to the real numbers. The projective real line adjoins only a single point at infinity, representing both directions at once, which turns the real line into a shape that is topologically a circle.
Are all infinite sets the same size?
No. Cardinal numbers show that infinite sets can differ in size. The natural numbers, integers, and rational numbers are all countably infinite and share the same cardinality, but the real numbers form a strictly larger, uncountable set, a difference first demonstrated by Georg Cantor.
How can a subset be the same size as the set it belongs to?
For infinite sets, a subset can have the same cardinality as the full set if a bijection, or one-to-one correspondence, exists between them. For example, the even natural numbers have the same cardinality as all natural numbers, via the pairing that maps each number n to 2n.
Why do mathematicians use infinity if it may not exist physically?
Whether physical infinity exists is unknown and irrelevant to mathematical practice. Modeling quantities such as physical measurements using the real numbers, which include infinite precision, makes tools like calculus far simpler and more powerful to apply than avoiding infinite models would.
Advanced education and experience with mathematics
infinity is 0
You can experience infinity in real life. Try to imagine NOT existing at all. The closer you get to an authentic visualization, the farther you get also. Try it. Another example is counting. They say "Count as high as you can and then I'll just add one." and "If you find the edge of the universe and then stand on it and shoot an arrow…" The most famous argument against infinity is actually the most sound proof that infinity does exist in the real world. Infinity is an Anti-number of sorts. The fact that you can "add one" or "shoot another arrow" means that there is actually and anti-force which accommodates possibility. And you want infinitesimal? Add a drop of water to a swimming pool at rest. Does it rise?
Even elementary introductory physics requires at least simple calculus which necessarily involves the concept of infinity.
The observable universe is a sphere, centered on us, where the radius is the horizon distance, which is the distance light could have traveled since the Big Bang. The observable universe is finite, but is only a subset of the entire universe. Is the entire universe finite or infinite? If the universe has positive curvature, like a sphere, it could be finite. If it has zero curvature, like a plane, or negative curvature, like a hyperboloid, it is infinite. According to all our measurements, the universe is flat, and thus infinite. There is still the possibility that the universe could have positive curvature, and be finite, but where the radius of curvature is so large, that the deviation of flatness, from our point of view, that it would appear flat to us.It used to be thought that the universe might have enough mass to recollapse into a Big Crunch. This has been disproven. In fact, the expansion of the universe is accelerating. That means, we know that it will exist for an infinite length of time in the future.The Big Bang has been confirmed by the CMB, so we think of the Big Bang theory as having won the Big Bang versus Steady State debate. However, we still don't know whether the Big Bang was the fundamental beginning of time, which is the traditional view, was instead only a local Big Bang, which created this specific part of the universe, which we think of as the universe. According to chaotic inflation, at any time, a given patch of space might suddenly undergo inflation. According to this view, time would extend infinitely backwards.So does infinity exist in the real universe? According to our recent theories, the universe is very probably infinite in space, definitely infinite in future time, and possibly infinite in past time. There are other occurrences of infinity in physics, such as having to sum over an infinite number of Feynman diagrams.Someone here acted like if you can't count to infinity, then infinity doesn't exist. That is a misunderstanding of infinity. When you ask, "Can you count to such and such number?", what you are asking is, does number X appear in the set of integers, Z = 1, 2, 3, …? Well, the number 1/2 also does not appear in the set of integers. Does that mean 1/2 does not exist? Why single out the integers as your set of comparison? Why not choose some other set, such as the prime numbers? Why not say the number 9 does not exist because it does not appear among the prime numbers? You can't count to infinity. You also can't count all the numbers that appear between 0 and 1. Does that imply that these numbers don't exist? You can't write down all of the digits of pi. Does that imply pi doesn't exist?
Mathematics is not physics so whether or not something in mathematics is "in our universe" is irrelevant to its existence and importance in mathematic. There are methods of making "infinity" as well as "infinitesmals" rigorous. One of them is in what is called "non-standard analysis":
http://mathworld.wolfram.com/NonstandardAnalysis.html
I get frustrated when anyone talks about infinity as if it is something that actually exists rather than something that a variable tends towards. Same with the infinitesimal. Indeed when most mathematicians write the symbol for infinity into an equation they are using it to describe the different rates at which a variable tends towards infinity/-infinity. In the real world nothing will ever reach infinity. You could argue with me that "Hey! I'll just start counting," (maybe you'll even start from a really large number), "I'll surely get to it one day," you say. Then I say to whatever number you have arrived at "Okay, now add 1 to that number." Well wait a minute. "If I counted forever, though?" you suggest. Then I say, "how do you intend to do that? Will you just go on living forever?" Then, you say: "Well I'll build a computer that is tough enough to process this problem forever, and continues to create its own storage space to allow the counting." Then I will just say: "Well if we assume there is enough matter in the universe to store all that information, then my only concerns are this: What does the computer do when it has lived until the end of this universe? How will it process if this universe has reached its natural end?" Then you suggest: "Maybe the universe goes on forever!". I will smile and just say: "Okay, tell me what it's age in seconds is when this universe reaches "forever", and I will ask you to add 1 second to that number. Then, come back to me in a second and tell me if the previous forever was truly "forever".OK, this thought experiment might show that there is no infinity in our universe (at least, we won't be able to count it). But why does that mean that infinity cannot be a useful model and approximation for reality?
The following also gives a good discussion about infinity.http://www.quora.com/What-is-the-difference-between-2-infty-and-infty-2
I get frustrated when anyone talks about infinity as if it is something that actually exists rather than something that a variable tends towards.I think you are voicing the opinion found in [https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From]. Many of us echo your sentiments about the importance of understanding how the theory of computation relates to the mathematical concept of infinity, and your frustration, as best as I can tell, comes with those who advocate the Platonic philosophy of mathematics (which misses the big picture pretending that mathematical ideas (like numbers and algorithms) are floating around outside of information systems like mathematical angels and demons in the ether). Most of the responders are right about infinity being understood quite rigorously within mathematics, and I think the problem is not so much the concept of infinity, but the poor way in which those who practice elementary and undergraduate mathematics are taught it. It's easy to see when the symbol is treated both as a quantity and not a quantity in notation. Normally, we use the lemniscate just as we would a real number notationally (e.g., ## -β < r < β : r in β ## , and looking at it, you can see why someone would get the impression it is a "number". That's exactly the point of why we extend the reals. (See [https://en.wikipedia.org/wiki/Extended_real_number_line].) What I think you're frustrated with is mathematicians who don't understand the nature of computation in working with infinity, and those same people usually treat irrational numbers the same way. "All" numbers come from computation, because all numbers are data in an information system made of physical media.
But because this is all true doesn't make infinity and the infinitesimal any less important or valid. In fact, it's well understood both in the philosophies of mathematics and logic AND the theory of computation.
You said, "I would suggest this is true in any reality, no matter how finely a variable is allowed to be describes, it has to stop somewhere." In practice, yes, but besides having a computational value, it also has a conceptual value. It is the symbolic notation for the concept of infinite regress which is EXTREMELY important when considering models of computation. If we say ## f(x) := g(x) + 1 ## and ## g(x) := f(x) + 1 ##, then for any ## x in β ## computing either ## g(x) ## or ## f(x) ## will result in ## β ## as an answer, and the importance of this is not whether we can continue calculating forever (the science seems stacked against that as you point out), but AVOIDING creating mathematical systems that create the situation to begin with! (Anyone who has written an infinte loop or a recursive computer function with no base case can attest to the dangers of that.) In fact, Zen philosophy which has amused mathematicians and computer hackers alike, is a worldview that intentionally likes to point the importance of understanding the importance of the infinite not just within mathematics, but personal ontologies! (See [https://en.wikipedia.org/wiki/Infinite_regress].)
In the real world nothing will ever reach infinity.Nothing has to 'reach' infinity for infinity to be a logical and rigorous concept. If I divide space into infinitely small or infinitely many sections, I am taking no physical action. I'm not taking a space-time knife and slicing it into small slices. I'm merely performing a very useful, and very well supported mathematical process.
So, despite my rude way of arguing my point, you can see why I don't want mathematicians to waste their time with logically inconsistent concepts such as the infinite or infinitesimal.No, I can't. Calculus is built on those concepts and is itself a fundamental mathematical subject that has countless applications. If there was something wrong with infinities or infinitesimals then calculus wouldn't work and wouldn't be used.
I get frustrated when anyone talks about infinity as if it is something that actually exists rather than something that a variable tends towards.What do you think actually exists?
So, despite my rude way of arguing my point, you can see why I don't want mathematicians to waste their time with logically inconsistent concepts such as the infinite or infinitesimal.The ideas of the infinite are rigorous.
Another argument against the infinite (or in this case infinitesimal) is a slight variation on Zeno's paradox. "If a person walks from one side of a room to another, they must first move halfway there, and then they must move halfway between that point and the other side of the room. They must keep making this division in space and if they do (if they could) they would never reach the wall on the other side of the room.Why can't Achilles traverse infinitely many intervals and overtake the tortoise?
This suggests, that whether it is a result of space being discrete at some level, or a result of the motion of the object and how its kinetic energy is defined being discrete, that at some point our reality gives up on the infinite, and just allows the object to move to the next position. I would suggest this is true in any reality, no matter how finely a variable is allowed to be describes, it has to stop somewhere.Current physical theory demands the idea of the infinite. E.G. the Shroedinger equation for a free particle assumes an infinite domain and the size of that infinity is way beyond any idea of a limit of a counting process. General relativity assumes a continuous space/time so that is also an uncountable infinite domain, Electricity and Magnetism is modeled on a trivial U(1) bundle, again uncountable. String theory – which models everything – takes place on a manifold – uncountably infinite again.
– The theory of the infinite is profound, stunning, important, and beautiful. Instead of disparaging it as "inconsistent" why not learn something about it?
I get frustrated when anyone talks about infinity as if it is something that actually exists rather than something that a variable tends towards. Same with the infinitesimal. Indeed when most mathematicians write the symbol for infinity into an equation they are using it to describe the different rates at which a variable tends towards infinity/-infinity. In the real world nothing will ever reach infinity. You could argue with me that "Hey! I'll just start counting," (maybe you'll even start from a really large number), "I'll surely get to it one day," you say. Then I say to whatever number you have arrived at "Okay, now add 1 to that number." Well wait a minute. "If I counted forever, though?" you suggest. Then I say, "how do you intend to do that? Will you just go on living forever?" Then, you say: "Well I'll build a computer that is tough enough to process this problem forever, and continues to create its own storage space to allow the counting." Then I will just say: "Well if we assume there is enough matter in the universe to store all that information, then my only concerns are this: What does the computer do when it has lived until the end of this universe? How will it process if this universe has reached its natural end?" Then you suggest: "Maybe the universe goes on forever!". I will smile and just say: "Okay, tell me what it's age in seconds is when this universe reaches "forever", and I will ask you to add 1 second to that number. Then, come back to me in a second and tell me if the previous forever was truly "forever".
So, despite my rude way of arguing my point, you can see why I don't want mathematicians to waste their time with logically inconsistent concepts such as the infinite or infinitesimal. In 3D gaming engines they often have Euclidian coordinate systems using floating point numbers that have corrections for the cases where a repeating decimal is the true answer to where an object should be moved (if moving at a certain velocity as defined by their spatial coordinate system and the computer clock) and techniques for making sure these corrections don't cause additive errors. I would suggest this to be akin to our universes quantum mechanical effects.
Another argument against the infinite (or in this case infinitesimal) is a slight variation on Zeno's paradox. "If a person walks from one side of a room to another, they must first move halfway there, and then they must move halfway between that point and the other side of the room. They must keep making this division in space and if they do (if they could) they would never reach the wall on the other side of the room. This suggests, that whether it is a result of space being discrete at some level, or a result of the motion of the object and how its kinetic energy is defined being discrete, that at some point our reality gives up on the infinite, and just allows the object to move to the next position. I would suggest this is true in any reality, no matter how finely a variable is allowed to be describes, it has to stop somewhere.
(Y) Wonderful !! I have a question if infinity doesn't fall in the set of real number than ever do it stand ?? Umm complex no?No. It's not a number at all, either real or complex. Infinity is more of a concept than a number.
Semi-relevant and interesting problem I stumbled across the other day: If you have an infinite number of infinitely small objects, would it take up near-zero volume or infinite volume? You will have to define "infinitely small" objects. I suspect you mean "infinitesimal" but the answer can be either of those (or a non-zero finite number) depending on exactly how those infinitesimal objects are defined.
Semi-relevant and interesting problem I stumbled across the other day: If you have an infinite number of infinitely small objects, would it take up near-zero volume or infinite volume?It depends. It could be anything in between, depending on the situation.
Simple examples:
All real numbers between 0 and 1 – length (measure) =1
All rational numbers – length (measure) = 0
All real numbers – length (measure) is infinite
(Y) Wonderful !! I have a question if infinity doesn't fall in the set of real number than ever do it stand ?? Umm complex no?
Semi-relevant and interesting problem I stumbled across the other day: If you have an infinite number of infinitely small objects, would it take up near-zero volume or infinite volume?
Nice work guys!