Is infinity a number?

Question

So I've been on a number of math fora, part of learning some calculus (not much of set theory, no). To my surprise I found what I would describe as strong resistance from some folks against (using) infinity in calculus. I find this baffling knowing that with limits, one essentially simplifies the problem (finding a limit) to dividing by infinity to get 0 and some pesky terms, as one author likes to say, vanish.

The assertion in one case was "infinity is not a number" (and so ... no math is possible)

My attempt at a response was to mention the Greeks, who would simply use arbitrarily large numbers when the computation involved infinity (computing pi for example). I guess that means if it ever came to 1/infinity, we could simply compute 1/1,000,000. (From random articles on Wiki)

Is infinity a number?

Answer 1

Infinity is not a real number. All real numbers x have the property x + 1 > x. Infinity does not share this property.

Infinity is an element in the system of extended real numbers. However, this system does not have many of the convenient and intuitive algebraic properties the system of real number has (it doesn’t even form a semigroup, much less a field).

Answer 2

It depends entirely on what you mean by "number." You might be surprised to learn that there is no standard definition of the word "number" in mathematics! Instead, there are many, many different number systems, with varying properties. The most common definitions of the word "number" are probably:

1. "Number" means "nonnegative integer." Infinity is not a nonnegative integer.
2. "Number" means "integer." Infinity is not an integer.
3. "Number" means "real number." Infinity is not a real number.
4. "Number" means "complex number." Infinity is not a complex number.

(It's important to note that the word "real" in the phrase "real number" is only there for historical reasons; there is nothing particularly real about the so-called "real numbers," nor is there anything particularly unreal about numbers which are not "real numbers.")

However, there are lots of other number systems! Here are some other possible meanings of the word "number":

5. "Number" means "element of the projectively extended real line." Infinity is an element of the projectively extended real line.
6. "Number" means "element of the extended real line." Under this definition, there are two infinite numbers: one is called "negative infinity" and the other one is called "positive infinity."
7. "Number" means "ordinal number" or "cardinal number." Under either of these definitions, there are tons and tons of infinite numbers (infinitely many of them, in fact). Since there are so many different infinite numbers, we assign distinct names to them; none of them have the name "infinity."

Answer 3

You have run into a common situation in philosophy, where you asked a question using a word that is weak to that question.

The existing answers are good, but miss an opportunity to get better at philosophy.

We are communicating in English, which is not defined by a table of rules somewhere, but by common usage. As such, each word has variation in meaning from person to person and community to community. For almost any word in common usage, you can construct a flawed question where the "right" answer depends not on some high philosophical truth, but merely which community you happen to be in. Beware if you find yourself confidently answering these questions, and be ready to notice that you are asking this kind of question, and start over with a new question.

"Is a tomato a fruit?" defeats the word fruit. You cannot successfully answer the question until the word fruit is replaced with "culinary fruit" or "botanical fruit."

"Is Guy Fieri an actor" defeats the word actor.

Math is not immune.

"Does 4 ÷ 2 * 2 = 1?" defeats the word ÷.

"Is the previous question actually about math" probably defeats the word math.

Is philosophy a number? No. Is 3 a number? Yes. Is infinity a number? Whoops, "number" crapped out.

Answer 4

The answer to your question is context-dependent and definition-dependent; it varies between different areas of mathematics. In ordinal arithmetic, for example, the first transfinite number, ω, can be involved in addition, multiplication and exponentiation; and the Riemann sphere is a model of the extended complex plane which has a distinguished point "at infinity". On the other hand, non-finite numbers in cardinal arithmetic behave differently than non-finite numbers in ordinal arithmetic. Thus, whether 'infinity' is appropriately regarded as a 'number' depends on the specific area of maths involved.

Answer 5

Many mathematicians (and perhaps authors of some of the other answers to this question) believe, following Cantor, that there are just two types of infinite number in mathematics: ordinal and cardinal. Following Detlef Laugwitz, I would argue that it is more accurate to assert that there are (at least) three kinds: ordinal, cardinal, and ringinal. The latter term is a bit of a neologism though it has already appeared in some publications (published or forthcoming).

What is a ringinal? The reference is to a number that occurs as an element of a ring such as the hyperintegers *Z, or semiring such as *N. These are numbers that are greater than any counting naive (or metalanguage) integer 1,2,3,... (a more precise technical term is unlimited number). Cantor himself was fiercely opposed to numbers of this kind, and went to great length to vilify them in his correspondence with contemporary authors. Some of the epithets he introduced were documented by P. Ehrlich and J. Dauben; one of them is "cholera bacillus of mathematics".

If ω denotes such an infinite ringinal, then naturally ω-1 is smaller than ω while ω+1 is bigger than ω. This does not work for either ordinals or cardinals. Therefore ringinals are the type of infinite number useful in infinitesimal calculus (indeed, 1/ω is an infinitesimal).

Answer 6

Despite some historical doubts and temporary issues, infinity is studied rigorously in mathematics. That doesn't mean every single mathematician (or even aspiring student to give them some credit) is comfortable with the wide open gates into mathematical rigor. The gate guard essentially lets any object into mathematics that is consistent in some theory we have strong reason believe is itself consistent (inconsistent mathematics notwithstanding). That is too permissive for some, probably a small minority, but see relevance logic as an example of consistency alone not being sufficient. And to be even more blunt, some mathematicians simply don't think every concept in math will turn out to be consistent, they think there are errors deep down (Ed Nelson, Poincare, etc), and infinity usually has something to do with that.

Then calculus had a period of actual inconsistency regarding infinitesimals. Graham Priest regards this as a feature, not a bug; that historically math let in inconsistent objects and handled them rigorously, just not under classical logic.

Plus a sufficient definition of limit can't really take place until mid-university where you learn either enough logic or analysis to actually prove it. Given some historical tumult, not having a proof or not being trained in proofs yet while in early calculus courses means the doubts have room to grab hold. Most early courses don't teach a formal definition of limits, and most don't teach students how to understand proofs until after calculus.

In summary, we can highlight three large issues leading to doubts of treating infinity as we do in calculus: 1) we didn't understand infinity rigorously until recently so past doubts are still circulating, 2) even with the rigor, infinity being consistent is still doubted by a few and consistency isn't the only bar to mathematics, 3) and the pedagogical glossing over too much and while not treating infinity as rigorously as we should in these earlier courses.

For "infinity is not a number" specifically: as an aside it is a number in certain areas of mathematics. But to me it just reiterates we don't do a good job telling/showing students we can handle infinity rigorously enough for mathematics, especially in calculus.

Answer 7

A common reason we differentiate between infinity and real numbers in (introductory) calculus is because we define limits differently between them.

We say lim x->0 f(x) = L if and only if, for all δ > 0, there exists an ϵ > 0 such that for all h with |x - h| < δ, |f(h) - L| = ϵ. For limits at infinity, however, we replace ϵ with c, such that for all h with h > c, |f(h) - L| < ϵ. (This is what you're saying with "using a sufficiently large number")

When you're doing math, it pays to be rigorous, and so in introductory calculus, you can't simply treat infinity as a number and use it in the same ways you'd use a real number. Hence your strong resistance.

If you keep doing math, you'll encounter the topological definition of a limit, which is much more suited to taking limits of infinities and other weird topologies, as long as you can define what "near infinity" means in your topological space.

Answer 8

Mathematics is a game of definitions.

In most fields of modern mathematics, you start with some things that you define to be true, which you call axioms, and ask what conclusions you can draw by reasoning starting from these axioms.

Of course an obvious question then is "which axioms should I start with?" And whilst this is a subtle question, one thing you probably want is for your axioms to lead to useful results.

So in particular, in calculus there is a fairly standard set of axioms that almost everyone works with, and in these axioms infinity is not defined to be a number. There are alternate formulations of calculus that do permit infinity as a number, but I'm not aware of any useful results that these formulations have lead to, which may explain why they are not widely used.

Other fields deal with infinite numbers more frequently (for example set theory), but in a lot of them you won't see "the number infinity", because their axioms lead to multiple infinite numbers. So mathematicians working in these fields have more specific names for some of these infinite numbers.

Answer 9

Your title question is: Is infinity a number?

In the body you say that you found a strong resistance from some folks on math fora against using infinity in math.

Let me suggest that the title question is a red herring. After all, one can keep wiggling definitions ad infinitum(!) to obviate embarrassments — as many of the answers here demonstrate.

The real point of your question (I feel) is about the "resistance" — I'd say rather, chariness — that some math folks show toward infinity.

It actually gets worse!

Gauss stated: I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking.

This was directly contradicted by Cantor's Axiom of Infinity, which beyond the jargon baldly asserts:

There exists an infinite set!

So the direct answer to your (real) question is that mathematicians on calculus/analysis fora are typically more Gaussian than Cantorian.

The real question you are unearthing is more disconcerting: Since both Real Analysis and Set Theory are taken to canonically be math, either mathematicians are confused or schizophrenic — surely it does not require a PhD in math or philosophy to see that both Cantor and Gauss cannot be right (Nagarjuna aside).

This brings us to one of the central disputes in philosophy of math for the last 150 years: Should math be exclusively constructive or are non-constructive constructions(!!) allowed?

One important point of the dispute is about infinity: Gauss quoted above was in the camp of almost all mathematicians until Cantor. His statement can also be restated: Infinity is fine as an unending process but never a completed infinity.

The difficulty of jumping off the Cantor's infinite number-of-infinities-all-the-way-to- nonsense into the constructive camp is that Cantor's diagonalization is central to much 20th century math.

Weyl Gordan Wittgenstein

Personally I find Weyl's take on Set Theory persuasive: He said in effect:

Cantor proved via diagonalization that the real numbers cannot be ‘lined up’.
That much is fine.
To go from there to saying that the set of real numbers exists and is larger than the set of natural numbers is a gross form of reification fallacy. theory

Gordan: This is not mathematics; this is theology.
[regarding cavalier use of infinite sets]

Wittgenstein responding to Hilbert's No one can drive us from the paradise that Cantor has created for us.

To this Wittgenstein responded: If one person can see it as a paradise of mathematicians, why should not another see it as a joke?

Answer 10

Mathematicians sometimes use the symbol ∞ in contexts where they do not consider it a “number.” When they read notation such as lim n→∞ out loud, they often say, “infinity,” but this is just a shorthand for the definition of limit, convergence or asymptote that they learned in introductory classes, for example:

The limit as n goes to infinity of one over n is zero.

The sum over i from one to infinity of two to the power of minus i is one.

The tangent of x goes to infinity as x approaches ninety degrees from below, or to negative infinity as x approaches ninety degrees from above.

Even though many of these notations allow you to use a (real, natural, rational, etc.) number in the same place you write ∞, the meaning is often very different (improper rather than proper integration, a series that diverges instead of converging, and so on).

Mathematicians also sometimes use the word “infinity” to refer to something they also call a “number,” such as the cardinality of an infinite set. In this context, there is more than one kind of “infinity.” For example,

The number of even numbers is the same as the number of whole numbers or rational numbers: a countable infinity.

You could probably find someone who thinks this is as much an abuse of notation as the first three examples I gave. A constructivist might even say that “infinity” is meaningless because no infinite object exists in their mathematics.

Answer 11

The practical reason to avoid using infinity in computations.

If you're taking calculus you've probably seen (or will soon see) that quantities like (infinity) - (infinity), (infinity)/(infinity), etc. are indeterminate forms, meaning that you may encounter one limit having the form (infinity)/(infinity) and correctly compute it to be 7, while another limit of the form (infinity)/(infinity) may correctly be found to be 0. So this is a good reason to avoid calling infinity a number: dividing a nonzero number by itself should always result in 1, but that doesn't appear to be the case with infinity. This is a good practical reason to say "infinity is not a number," although (as you observed) it is not quite correct to conclude "... so no math can be done."

Real numbers and infinity.

Many of the responses here indicate that the word "infinity" does not specify a single concept, in the way expressions like "1", "pi", "the square root of 2", and "the square root of -1" do. So there are two interesting aspects to your question:

1. What does it mean for a number to be finite?

2. Are there any numbers that are not finite?

It may be useful to analogize these to the questions "What does it mean for a number to be rational?" and "Are there any irrational numbers?"

The first question is apparently easy to answer if you're working in the real number system:

Definition 1. A real number x is finite if there is a natural number n such that -n < x < n. A real number x is infinite if it is not finite.

If one is comfortable defining the set of natural numbers to be the smallest set which contains 1 and is closed under addition, Definition 1 is a satisfactory definition of "infinite." If one insists on defining "natural number" in a more rigorous way, this opens up some other mathematical and philosophical issues that I won't pursue here.

With the first question out of the way, the second question has many interesting answers. One can prove from the standard axioms of the real number system that there are no infinite real numbers. This is a famous consequence of the Completeness Axiom, known as the Archimedean property.

The Archimedean property of the real numbers (and some other considerations that I'll digress toward if asked) might be a good philosophical reason to say that "infinity is not a number" when one is doing calculus.

If one is willing to consider number systems that do not have the completeness property, then infinite numbers are available. A well-developed theory called nonstandard analysis allows one to form ordered fields containing the integers, and containing elements strictly greater than every natural number. In other words, this number system has a number x satisfying n < x for every natural number n. Sometimes this number system is called the hyperreal numbers. However, there are many infinite numbers in this system, not just one, so the word "infinity" does not refer to a single number. (Nonstandard analysis is not the only way of constructing ordered fields with infinite numbers, but I don't want to digress into a list of examples here.)

Cardinal numbers and infinity.

In mathematics, the cardinality of a set is the number of elements of a set. For example, the set {1,7, 500} has cardinality 3. The empty set, {}, has cardinality 0. Defining the concept of cardinality for infinite sets is an interesting development: https://en.wikipedia.org/wiki/Cardinality. So if one believes that there is such a thing as "the set of natural numbers", and that every set has a "number of elements", then one would believe that there is such a thing as an infinite number: namely, the number of elements of the set of natural numbers is an infinite number. This number is called the "cardinality of the set of natural numbers." However, this definition of "number" is quite different, and not quite comparable to, the numbers in the hyperreals mentioned above. Cardinal numbers do have their own kind of arithmetic (including addition, multiplication, and exponentiation), which has been the subject of intense study by set theorists.

As mentioned in other answers, "real number" and "cardinal number" are not the only way of precisely defining the concept of "number," and the concept of "infinite number" can be considered in many of these other settings.

By the way, there are people who don't believe in the existence of infinite sets: https://en.wikipedia.org/wiki/Finitism. This belief comes in several flavors.

Answer 12

There are many different sorts of "infinity" in mathematics.

In calculus, one speaks of the limit of a function as its argument approaches plus infinity or minus infinity, or of a limit of a function being equal to plus infinity or minus infinity as its argument approaches some finite number.

This "infinity" is not considered a number.

However, there are also various different infinite cardinalities in set theory. These are often called "numbers." But they are not numbers called "infinity", but have various different names.

There are also the infinite nonstandard real numbers of Robinson's "nonstandard analysis."

Then there are the points at infinity in projective geometry.

And things like the point at infinity on the Riemann sphere. With that one, one can to some extent actually do arithmetic. But I've never heard anyone call that a number.

And there are the infinite values of things like Dirac's delta function and it multiples and its derivatives. I don't think anyone calls such things numbers. You can't do arithmetic or anything else with them except insofar as one does arithmetic or other things with the generalized functions of which they are values.

So "infinity" is not a number, but there are some objects used in mathematics that are in some sense infinite that are conventionally called numbers.

Answer 13

Infinity is a concept not a number. It refers to the natural phenomenon that no matter what value of n, there is always n+1. When it is said, "as x approaches infinity, y converges," it refers to the fact that if you continue moving x in the up direction, y continues to converge, and if you reverse x back towards the beginning, y will become less accurate. Following the pattern of division, as you decrease the denominator, the result will always increase in size, therefore if you keep decreasing the denominator, the result will approach infinity, but there is never a value that such a result can settle on, because division by zero is not valid, and there is no smallest number to use as a denominator so the result will simply keep going up relative to the smallness of the denominator, this is an example of infinity, the concept that it will ceaselessly increase to any arbitrarily large value but never settle upon a final value. Thus, infinity is just the concept that any pattern can continue incrementing forever without ever reaching a final value.

When infinity is referred to typically it is under the context that a pattern can be continued forever.

You can keep infinitely shrinking the denominator, and the numerator will keep infinitely increasing, there is no number it will ever settle on, and infinity refers to the direction the number is headed in, which never ends or reaches a value, nor approaches a specific value.

The name infinity literally means "the concept of that which has no end." Fin meaning end, in meaning no. It relates to the concept of that which cannot be stopped and has no ending point, like the value of numbers, which can keep increasing forever, etc.

Answer 14

Mathematics are close to phylosophia on one point: definition is essential.

And neither numbers nor infinity have a unique definition.

For the common definition of numbers, we start with the well known sets N for the natural numbers (ie positive integers), Z for relative numbers (ie positive or negative integers), Q for rational numbers (numbers in the form p/q where p is a relative integer and q a non null positive integers). Then we reach R which is the smallest topologically complete set containing Q, and C which is a superset of R (seen as a field) where every polynomial of degree n has exactly n roots, distinct or not. For all of those sets, the infinity is never a number.

But nothing is ever definitive. Some mathematicians have defined a concept of surreal number. The set of those surreal numbers is defined to be a superset of R (with the same field structure) containing also the infinite cardinals. Because infinite sets can also be ordered by their cardinality. For exemple, it is possible to have a bijection from Q to a subset of N and a bijection from N to a subset of Q, so those sets have the same cardinal (noted aleph_0). But the set of the subsets of N have a strictly greater cardinality (which is also the cardinality of R and noted aleph 1). And so on and so forth...

As the set of those surreal numbers is indeed a field (you can do additions and multiplications with its elements), its elements can be called numbers. And it does contain an infinity of infinite numbers... So in this definition you can say that the number elements of the N set (the usual positive integers) is both infinite and also a (surreal) number.

So depending on what set you considere, you can either say that the infinity cannot be a number of that some numbers can be infinite. Definition matters...

Answer 15

As mentioned in all the answers, it depends on definition. Rather than try to formally define "number" and "infinity", let's compare based on operations:

1. Addition: numbers are additive, Infinity is not: Infinity + Infinity = Infinity or Infinity + 5 = Infinity
2. Subtraction: numbers can be subtracted. Infinity - Infinity = Undefined or Infinity - 5 = Infinity
3. Mutiplication: Infinity x Infinity = Infinity
4. Division: infinity/Infinity = Undefined but, importantly, 1/Infinity = Undefined but approaches 0.

In nearly all of the simplest mathematical operations, Infinity cannot be used as a number. The important exception is 1/Infinity. Since this approaches a number, zero, it can be treated as a number.

Answer 16

There are sets, which include infinite quantities. For instance, the Hardy fields include the germs of functions at infinity.

For instance, the germ of function f(x)=x is greater than the germ of functions x-5 and x/2. But all 3 are infinite, greater than any real number.

Hardy fields can be canonically embedded into surreal numbers, so one can see each such germ as as surreal number.

It should be noted that the word "number" is a bit arbitrary, for instance, germs or formal series are usually not called numbers even though they are fields and can be seen as subsets of surreal numbers.

Answer 17

As others have pointed out, in mathematics, you really need to define your terms and the axioms they follow before you can reason about them. Thus when you bring such an "overloaded" concept to people knowledgeable in mathematics to some degree, either they expect you to use the "common" language (what is taught in schools), or show that your "alternate" approach is sound and that you know what you are doing (as there are many people who don't, yet they think they do).

However, since this is philosophy, we don't need to be so precise and rigorous and invalidate the question (and point you to ordinals, hyperreals, surreals, projective geometry, p-adics, and other fun stuff) ‒ we can allow ourselves speak about necessarily fuzzy objects (such as "infinity" and "numbers") in fuzzy terms, and stick more to our intuition than strict definitions. This way, you might find the answer yourself.

Using calculus

How do you find infinity? The usual calculus gives you the real number line and nails two direction signs to it, labelled "+∞" and "−∞", going in opposite directions. What are your options in such a space?

• You can get closer and closer to any arbitrary real number, i.e. for an arbitrarily small bound, you can always get closer than that to it.
• You can get closer and closer "to infinity", i.e. for an arbitrarily small bound, you can always find a number whose reciprocal is within that bound to 0.

The infinity in this case is obviously not a real number, because you cannot use the first definition to arrive at the second (which is why we call such limits improper). These two definitions could however be unified by defining ε-neighbourhood (U_ε(x)) on the extended real number line, where +∞ and −∞ are actual elements (but just as named special objects). This is actually a very common practice in mathematics ‒ even negative numbers came to existence just to get rid of exceptions and corner cases!

In this regard, infinity is number-like because it has a neighbourhood in the real numbers. However, when you use limit arithmetic, you don't even use the real numbers! It is either a blatant misunderstanding or gross oversimplification to think that finding a limit at a is somehow achieved by setting x = a ‒ limit is an operator that turns a function operating on real numbers into a function operating on neighbourhoods, so when you ask about the limit at a, what you inspect is the neighbourhood created by y as x is in the neighbourhood of a, and the result is the element of the extended real number line that gives you that neighbourhood (if there is any).

At this point, you need to ask yourself what are numbers ‒ assuming that real numbers are numbers, are neighbourhoods in real numbers also numbers?

It is reasonable to say yes, since you can do the usual arithmetic on neighbourhoods of real numbers and it translates to operations on real numbers (e.g. adding numbers from two neighbourhoods gives you the neighbourhood of the sum of those two numbers). It is not too alien to define specific numbers as sets, after all, this is also how the real numbers are defined.

By extension, you can also treat the neighbourhoods of +∞ and −∞ as numbers themselves, to complete this space. This also gives you some basis for computing limits arithmetically, but you have to be aware of the limitations ‒ indeterminate forms like ∞ − ∞ or 0⁰ occur precisely when this fails, as you can no longer assign the result to a concrete neighbourhood. This is however still fine ‒ you cannot divide by 0 normally, and natural numbers cannot be even subtracted without exceptions!

To sum up the results so far:

• Infinity is number-like because it has a neighbourhood in the real numbers like any other real number. That alone does not make it a number, however.
• Neighbourhoods of real numbers can be treated as numbers, so it is not too alien to simply call them numbers and work with them that way (like mathematicians always do).
• In this point of view, the neighbourhoods of +∞ and −∞ are numbers. The complete set does not have some "nice" properties (such as being a field), but not all number sets have those anyway.

Now, you might wonder why you don't see neighbourhoods nearly as often as I mention them here, and the reason is notation. Using the limit notation, you can just write numbers as usual, and you can even use infinities, but it is all symbolical ‒ the underlying objects are still neighbourhoods (possibly one-sided).

This is as far as calculus can get you, but it is still quite far ‒ if by ∞ you mean U_ε(∞) (as one often does), then yes, it is a number. Of course, U_ε(∞) + 1 = U_ε(∞), so it does not behave like real numbers, but not all numbers do (after all, ℵ₀ + 1 = ℵ₀ too, where ℵ₀ is a cardinal number).

In more metaphorical terms, infinity, like all real numbers, is a destination, but you cannot reach it. Like all real numbers, you can approach it, and the way you approach it (the direction towards it, so to say), is actually a number.

Using hyperreals

Even if you could use ∞ as a number, you risk running into the same issues as before with indeterminate forms, but now without the notation saving you from the gritty details and those pesky terms ‒ for example notice how the neighbourhood of +∞ is necessarily one-sided, while the neighbourhood of 0 is two-sided ‒ that means that 1/∞ is not simply 0, it is 0⁺ (if you use the projective real number line, you actually get just one ∞ which you can approach from both directions, but then you can't compare it meaningfully to other numbers, so it, arguably, becomes less of a number)!

While this allows you to differentiate 1/∞ from −1/∞ = −0⁺ = 0⁻, there is still no advantage to using this instead of the usual limits. What you need is not only the direction from which you approach a number, but also the speed: when x⁴ grows faster than x², you lose important information if you make them equal to ∞ at x = ∞.

What you can do instead is to identify all growing sequences and use them to compare the rate of growth of functions. Even this has its flaws, as functions like x + sin x and x − sin x are incomparable, but you can use some set-theoretic trickery to assign a definite (but essentially arbitrary) interpretation to them. This gives you a set of hyperreal numbers, where each number is an equivalence class of sequences that agree on some (infinitely many yet infinitely scarce) key indices, and all just works (any first-order logic statement holds).

This construction allows you to identify numbers beyond the reals, such as ω as the equivalence class of (0, 1, 2, 3, …), which is bigger than any real number. Now you can easily do any operations on ω and they all just get applied element-wise on the sequence. These numbers are called transfinite, as they are not necessarily bigger in size, but they are definitely beyond all finite numbers.

If you wish to evaluate a function f(x) at infinity, what you want is f(ω). However, this is, strictly speaking, not the infinity ‒ you can take f(ω + 1) or f(2ω) equally well and they all give you the same information about the behaviour.

Arguably, you might even say that ω overshoots infinity by a huge degree! If the sequence 1, 2, 3, 4, … approaches ∞, you should get closer and closer to it, but now you have numbers like ω, ω/2, ω/3, √ω, ln ω, which are all bigger than any element in the original sequence, yet can get arbitrarily small! If infinity is a concrete point that borders the real numbers, we have got past it, not to it!

Nothing stops us from repeating this process again within the hyperreals, which leads us to a number ω₋₁ equivalent to ((0, 0, …), (1, 1, …), (2, 2, …), (3, 3, …), …). This number is still larger than any real number, but smaller than ω and any transfinite number constructed from ω. We can repeat this process again to get ω₋₂, ω₋₃, ω₋₄, in another infinite sequence forming increasingly smaller transfinite domains. And even then, we can imagine (but not construct this way) an even smaller number ω_−ω below all of them! We are not even close!

Using surreals

There is a similarly-looking number system called the surreal numbers, №, which is also known as the largest (proper class-sized) ordered field. Numbers in this system are formed by pairs of sets of previously-created numbers, identifying in some way the simplest number that lies between all the numbers in those sets. Thus { | } is 0, { 0 | } is 1, { 0, 1 | } is 2, { 0 | 1 } is ½, and so on. Eventually, you arrive at { 0, 1, 2, 3, … | } = ω, { ω | } is ω + 1, etc. Using another set-theoretic magic, you can outreach numbers we have seen before like ω₋₁, or go in the opposite direction with ω₁ and get as far and as wide as set theory permits you (for every ordinal number, you get new surreals).

Alternatively, we can define these numbers as ordinal-length sequences of ↑ or ↓, such as ↑↑↑ = ↑³ = 3, ↑↑↓↑ = 1.75, ↑^ω = ω, ↑^ω↑ = ω + 1, ↑^ω↓ = ω − 1, and so on (the superscript is the ordinal ω, while the result is the surreal ω).

There are more surreal numbers than elements in any set, since you can embed the cardinal numbers in them, thus they cannot be a set ‒ indeed they are a proper class. From a set-theoretic perspective, we can only safely speak about them when we bound their construction by a sufficiently large ordinal. This is actually very similar to what we did before in calculus, only this time we get even more insane than ∞. Even though set theory breaks at this point, we can still imagine an entity { № | } = On = ↑^On, which corresponds to the class of ordinal numbers, bigger than any surreal number. This is not a number, but a gap ‒ we know its position in relation to the other numbers, but we lack the foundation to treat it as one.

With this notion in mind, we don't necessarily need to go hunting for ordinal-level infinities, as we already know where the original one should be ‒ down there, bordering the finite and the transfinite. Between the real numbers and all transfinite surreal numbers (a proper class thereof), we finally find it: ∞ = ↑^ω↓^On.

Infinity is not a number, it is an abyss.

---

^{There is also infinitesimality, 1/∞ = ↑↓ω↑On, smaller than any positive real number, but bigger than any infinitesimal surreal number.}

Answer 18

Infinity is not a number, but moreover a direction. When something is infinite, we're saying it has no end, which is difficult to understand: all the objects we know have physical limits (time and space are not objects, but contexts where objects exist).

Remark that infinity is normally associated with a process, which implies a chronological progression. This simplifies the understanding of infinite, but introduces issues. Upon this understanding, you will hear that infinity is a growing number or that infinity continues to grow. This is useful to get a notion of infinity, but introduces a problematic time dimension in the idea. Infinity has no associated time, it is just a direction.

If we remove the chronological element from the understanding of infinity, we get something like the directional component of a vector.

You will wonder "if infinity is not a number, why does Cantor define types of infinity, where some are larger than others?"

In such case, infinity is just a mathematical gadget. Mathematics needs of objects, not undetermined elements. So, mathematically, infinite objects have sizes, which can be compared (e.g. using series) and assessed (e.g. calculating limits over infinite curves). But remember that mathematically, we can't assess the behavior of infinity, we can just assess the tendency of finite segments towards infinity. When you calculate the surface of the normal distribution curve, you are not really measuring infinity, you are just observing its behavior tendency towards infinity.

Therefore, infinity as the mathematical gadget means usually addressing infinity as a number. But conceptually, infinity is not a number. It is the opposite of a limit, it is a direction, it is a system without boundaries, which is quite difficult to get notion of.

Answer 19

In mathematics, infinity is not considered a number in the conventional sense. It is considered an idea or concept.

Once you have defined numbers as finite specific quantity then obviously infinity wont fit in. Once you have defined numbers as quantities adhering to specific arithmetic operations then obviously infinity wont fit in.

It is a matter of definition. But practically infinity is very much part of mathematics and number system. I am more inclined to call infinity a number by defining it as some something larger than the largest number known(defining positive infinity).

Infinity is used in limits , integration , counting etc. There are infinite “numbers” of integers. Infinite is uncountable but it is definitely a number. Arithmetic operations can be applied on it in the limiting sense.

Therefore , in my opinion, infinity is a number and it can be defined using numbers. Mathematical community needs to accommodate infinity as number by defining it as uncountable number but having a practical definition of it as I have given above or can be defined using limits( for example - a number X is as large as infinity if 1/X equals 0.)

Answer 20

There are a number of good answers here which explore the natural language issues with the terms "infinity" as well as "number".

One "infinity" you are explicitly mentioning though is the one often used in calculus. An example could be "lim_x->0 1/x = ∞": We are tempted to read this as "1/x becomes infinity when x becomes 0".

But as a computer guy I understand this (and all similar statements) as the description of a procedure, an algorithm. There is no "infinity value"; this is a statement about what happens when x shrinks in an unbounded fashion. And what happens, when we perform this gedankenexperiment, on the right side of the equation is that 1/x grows correspondingly without an upper bound. This is denoted by the somewhat misleading shortcut "∞". It would be more consistent and less misleading if the right side notation was,
corresponding to the left "x->0", a "->∞" as well.

The beauty, as with all math, is that this is very well defined. This limes consideration for x and 1/x says "if we cannot provide a minimum for x we cannot provide a maximum for 1/x". Of course, physicists, let alone engineers, simply insert 0 for that in many contexts. But the rules that allow them to do that have been carefully and safely chiseled out by mathematicians.

Answer 21

Is forever a time? It has temporal characteristics. But it is non-specific.

Infinity is not "a" number.

It is numerical in nature, has numerical characteristics. But it is non-specific. And that prevents it from being described as "a" number

ADDITION July 10th, 2:00am (EST)

I think it also important, when contemplating "infinity" , to remember that numbers are kinda "imaginary" until they are describing "somethings"... like dollars, oranges, tractors, degrees, satellites, dominoes, or whatever.

In the case of "numbers", are there an "infinite" amount of numbers? Well... could the "largest number" ever be reached, by writing it down, including all its digits?

No. Not really. So, we could never present document evidence in court to prove an "infinite number of numbers exist"

I can't imagine any witnesses being able to convey the largest number (and all digits), verbally, either.

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I also think it probable, that physically, reality is infinitely vast, and eternally old.

And that whatever is "fundamental" to reality, is infinite in quantity, and non-aging (meaning it doesn't get older)... resulting in the reality that is infinitely vast and eternally old.

And if it is so... that whatever is fundamental, there are an infinite number of them...

... then also it is so that... all the things that emerge into existence within reality, are also infinite in quantity. Like stars, planets, galaxies, etc.

If the fundamental quantum are infinite in quantity... then all emergent phenomenon are infinite in quantity as well.

HOWEVER... each "thing" when considered as a specific type of "thing" different from other "things", such as "stars", or "planets"... they each have a "percent of existence by volume ratio".

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What I mean is this... let's say "fundamental" turns out to be "strings"... and it turns out "strings" have always been, do not age or evolve or change nature...and these fundamental "strings" fill reality, make the quantum fields, which make all else...

... AND we determine "strings" to be non-aging...

... THEN we will conclude string fill reality extending forever in every direction, and there was no big-bang or expansion.

Then, consider from any point in reality/the universe... you could start an "Inventory Page", and begin an "expanding sphere of content counting"... an imaginary growing circle that gets bigger and bigger and bigger until it has encompassed say, 1000 super-galactic groups.

As it went, the inventory would grow.

If you started the "inventory sphere" at our Sun for example...

Each of the numbers would go to infinity, but each will grow at different rates. Different infinities, but statistically relative to one another.

Making "units" matter significantly.

(Its a tough thing to explain, I hope I did it justice)

Answer 22

The Definition of "Number" is Ever Changing.

The definition of a Real Number has changed so many times throughout history, that most of what we consider real numbers today were once not considered to be numbers at all. But there is one definition of a number that seems to be very close to consistent throughout history: A number is any meaningful value in an equation. I say this is the best and most consistent definition not because it fits perfectly with modern math, but because history is scattered by numbers being declared NOT numbers when the philosophy of a civilization has decided that a thing is not meaningful when used.

A Brief History of What Is A Number

• Before about 2700 years ago, real numbers were things you could count. If you could not count it, it was not a number: so this is what we now call Natural Numbers: 1,2,3...
• Irrational Numbers (things like pi and the square root of 2) were the first type of number to challenge the countability of numbers somewhere around 700BCE. They were too useful and necessary to not be numbers, but thier lack of countability made them controversial until about 350BCE when they were formally declared as Magnitudes, and NOT Numbers, and it was not until about 900CE that irrational numbers came back to being considered numbers again.
• Rational Numbers (fractions and decimals) were used widely in math starting in about 300BCE, but were not formally recognized as numbers until 600 years later. Like Magnitudes, fractions were considered different than numbers, but unlike magnitudes, they could be proven through counting.
• Zero has a really checkered past having been declared a number and not a number many times throughout history since Ancient Babylon. Many systems of Math since the invention of Zero have even gone beyond saying it's not a number, but have excluded it entirely considering it a logical error that breaks math itself.
• Negative numbers were first seen used around 200BCE as a way of documenting debt, but were not seen used in any formal system of math for nearly another 500 years. Because most Classical math is based on Geometry, negative numbers were not considered numbers because the length of any line is an absolute value. Instead negative numbers were handled as opposing vectors with all results being a positive number. It was not until the 1600s that negative numbers really started to be seen as numbers unto themselves.
• Imaginary Numbers were not even documented until the 1500s and when they were discovered, they were widely considered Mathematical errors akin to dividing by zero until thier usefulness in mathematical proofs and 4 dimensional calculations became apparent in the 1800s, and they became seen as actual numbers.

So why is any of this important?

In modern mathematics, we often say that ∞ is meaningless and therefore Not a Number (NaN). But this is a really old song and dance as far as human thinking is concerned that has played out over and over again for as long as humans have done math.

History tells us this: as we explore new branches of science we will eventually find meaningingful applications for every discovered NaN, and when we do that, those who call a thing not a number always lose out to those who say it is. Where infinity is concerned, it is already gaining popularity as a number since the confirmed discovery of black holes and the existence of infinitely dense mass. Now that we know that infinity exists in nature, and we have reasons to mathematically model ∞, there is no reason left to consider infinity NaN, it's just a matter of time before ∞ becomes universally acknowledged as a number.

But is it a Real Number or is it a Complex Number akin to Imaginary Numbers?

Even though Imaginary numbers are considered numbers now, they are not considered "Real" Numbers because modern math defines Real Numbers as a specific category of numbers. The currently accepted definition of a Real Number is a "quantity that can be expressed as an infinite decimal expansion". So, just as pi is a Real Number even though its fractional part extends to infinity, it is equally logical to say that infinity is a Real Number even though its whole part extends into infinity. So, even though many sources will still tell you that infinity is not a number, it is by definition a Real Number and an Irrational Number.

Answer 23

"Infinity is not a number. It just means that numbers go on forever."

Actual quote by my six-year-old daughter (I am even prouder than you think.)

I quote her to my 20-year-old Calculus students every term, but to be fair, I then show them what you can do with the symbol ∞ in Calculus. I also do show them how careful you have to be with that symbol, i.e. what you cannot do with it. And that all boils down to what my daughter says -- plus the fact that you can treat that symbol ∞ like a number in some contexts, but not in others, and your learning outcome is to tell those contexts apart.

Answer 24

See Finitism. The idea here (as best I understand it) is that infinities should be treated merely as abstract ideals: i.e., as linguistic shorthand for an endless series, not as a thing in itself. Thus, something like lim_x→∞ (the limit as x approaches infinity) could just as well be written as lim_x→ (the limit as x increases). Casting infinity as a 'thing' — and in particular, positing multiple levels of infinities — is confusing and misleading.

Mainly this has to do with the theoretical foundations of mathematics. In most practical uses of mathematics infinity is used in a purely formal, symbolic sense the way finitists suggest. Infinity only become a problematic issue on deep dies into fundamental principles.