# What is (the nature of) infinity?

#### phoenixthoth

Perhaps some consensus can be arrived at in regard to what infinity is. After that, perhaps its nature can then be discussed.

One approach to defining infinity is to first define what finite means and then say something is infinite if it is not finite. Rather than define infinity by what it isn't, let's try to define it by what it is.

This definition is intentionally vague, in part because it has precise definitions in math. The sort of thing I want to discuss is what isn't covered in math. A definition of infinity that isn't in terms of what infinity isn't will be motivated by what we think its nature is, will dictate what its nature is, or both.

So let's give it a go...

Infinity is a quality or quantity for which it is possible to be reduced in a way that the reduction is, in some sense, equivalent to the original.

What a reduction is and what it means to be equivalent is, of course, crucial. A particular example of infinity would be an infinite set which is infinite if reduction means removing a single element of the set and two sets are equivalent if there is a one-to-one correspondence between them (i.e., there is a bijection between them).

What it means to be finite could then be a quality of quantity that is not infinite. For example, if reduction means subtraction and equivalence is taken to be equality, no counting number has the quality of infinity since, when reduced, no counting number is equal to the original.

Then perhaps we can answer some basic questions such as is there anything in the universe (or is the universe itself) infinite?

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#### daveb

This definition is intentionally vague, in part because it has precise definitions in math. The sort of thing I want to discuss is what isn't covered in math....A particular example of infinity would be an infinite set which is infinite if reduction means removing a single element of the set and two sets are equivalent if there is a one-to-one correspondence between them (i.e., there is a bijection between them).
Talking about sets means you're talking about math. I suppose, in the end, it's hard to discuss infinity without resorting to math since math is how we model the universe.

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#### wuliheron

Like any mathematical concept, infinity can be described in words.

dictionary.com said:
infinity
1. The size of something infinite.
Using the word in the context of sets is sloppy, since different infinite sets aren't necessarily the same size cardinality as each other.
2. The largest value that can be represented in a particular type of variable (register, memory location, data type, whatever).

in·fin·i·ty /ɪnˈfɪnɪti/ Pronunciation Key - Show Spelled Pronunciation[in-fin-i-tee] Pronunciation Key - Show IPA Pronunciation
–noun, plural -ties. 1. the quality or state of being infinite.
2. something that is infinite.
3. infinite space, time, or quantity.
4. an infinite extent, amount, or number.
5. an indefinitely great amount or number.
6. Mathematics. a. the assumed limit of a sequence, series, etc., that increases without bound.
b. infinite distance or an infinitely distant part of space.

7. Photography. a. a distance between a subject and the camera so great that rays of light reflected from the subject may be regarded as parallel.
b. a distance setting of the camera lens beyond which everything is in focus.
What none of these definitions touches upon is that no one has ever proved infinity exists in the real world. Therefore, infinity is a speculative quantity.

Lao Tzu said,

Tho it has no limit, I call it infinite.

This is a recognition that to say something "has no limits" is, in itself, to impose a limit. The limit that it has no limit.

To Phoenixthoth:

I would suggest you begin your inquiry about the topic of the infinite with Aristotle, 'Physica", Book III (B), Chapter 6. Here is his definition:

"A quantity is infinite if it is such that we can always take a part outside what has been already taken"

That is, it is not a "set" that has nothing outside it and can be reduced that is infinite, it is the "set" that always has some"thing" (some part) that we can take outside it that is infinite, such that the next part which is taken out is never the same as the previous.

#### Hurkyl

Staff Emeritus
Gold Member
The sort of thing I want to discuss is what isn't covered in math.
This seems like a good place to start -- what do you think is lacking about the mathematical treatment of the infinite, and why would it require a nonmathematical treatment?

#### sd01g

This is a recognition that to say something "has no limits" is, in itself, to impose a limit. The limit that it has no limit.
If no limits means no limits, then no limits does not impose limits.

#### sd01g

Perhaps some consensus can be arrived at in regard to what infinity is. After that, perhaps its nature can then be discussed.

?
The nature of infinity is that it has no natural nature. It has no empirical component. Infinity is a rational construct for a mental process that has no limit or end. Its main use is in mathematics. No one has observed or experienced infinity and no one ever will. No matter how large a number is, it is not infinity.

#### Hurkyl

Staff Emeritus
Gold Member
According to classical mechanics and GR, my lifespan encomasses an infinite number of points, and at this very moment, I occupy an infinite number of points of space. QM agrees too, in so far as things like "my lifespan" make sense.

"Infinity", in its lay usage, is a very poorly conceived word. It is a noun, but it is typically used when one really wants an adjective. e.g. a layperson tends to says "infinity" when he really means "an infinite number". Even worse, the word is used as an identifier; as if there was only one thing called "infinity". And, unfortunately, language has a tendancy to shape one's thought. Last edited:

#### wuliheron

If no limits means no limits, then no limits does not impose limits.
It is a paradoxical statement. For example, if I were to say I have no brothers this would impose the limit that I have no brothers, a perfectly reasonable statement. However, if infinity has no limits than it is limited because it does not include limits which, contradicts the fact that it is infinite.

Now in mathematics there are several distinct types of infinity. One is the all encompassing type of infinity, while the rest have verious types limitations. For example, an infinite series of numbers is limited in that the only thing infinite about it are the numbers. It does not include horses or whatever.

Nonetheless, all of these types of infinity also display this same central paradox. To say that a number series is infinite is still a contradiction in terms because it excludes limited numbers such 5.

#### Hurkyl

Staff Emeritus
Gold Member
It is a paradoxical statement.
Word games do not a paradox make.

Now in mathematics there are several distinct types of infinity. One is the all encompassing type of infinity,

#### russ_watters

Mentor
What none of these definitions touches upon is that no one has ever proved infinity exists in the real world. Therefore, infinity is a speculative quantity.
Not true at all. Infinity is an integral (pun intended) part of mathematical descriptions of real-world object and scenarios.

#### Sauwelios

One approach to defining infinity is to first define what finite means and then say something is infinite if it is not finite. Rather than define infinity by what it isn't, let's try to define it by what it is.
There is no difference. To say that infinity is not finiteness is to say that it is {not finiteness}. Indeed, "in-finite" means "not finite". So by definition, infinity is a negative concept.

I adhere to Nietzsche's Zarathustra's maxim:

"Could ye conceive a God?—But let this mean Will to Truth unto you, that everything be transformed into the humanly conceivable [denkbar, "thinkable"], the humanly visible, the humanly sensible!"
[Thus Spake Zarathustra, In the Happy Isles.]

Infinity is not humanly conceivable and therefore nonsensical.

This does not mean that what is humanly conceivable is true. To the contrary, what is humanly conceivable must be false! For to conceive means to grasp, and to grasp means there is something that can be grasped, some thing, some unity, some oneness. Humans can only think in particles. Even the "quantum" is a kind of particle. Instead of "particles", quantum mechanics uses "quanta" (amounts); instead of unities, it uses units. But an amount is needs a definite amount. An amount of "1" is an amount of exactly ...000,001.000... -- that is, it presupposes "an infinite amount" of decimals. But an amount must needs be definite, i.e., it cannot be infinite. Therefore, there are no amounts.

But if there are no particles and no amounts, if there are no units and no unities, then all ideas thereof must be illusions, namely simplifications (literally "single-makings", "one-makings"). In order to conceive anything at all, in order for his consciousness to have its necessary object, without which consciousness is impossible, man must falsify the world by simplifying it. He must de-fine the world, impose boundaries on the chaos, ordering it. Quanta are indefinite (as they cannot be exact to an "infinite amount" of decimals), but man must regard them as definite in order to regard them at all.

As there are no definite quanta, no exact quanta, i.e., as there is not even one self-same quantum, there can definitely not be multiple equal quanta. But a finite number is a definite number. Therefore, there is no finiteness, either. But man can only conceive of finiteness, definiteness. So infinity is whatever cannot be conceived.

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#### Hurkyl

Staff Emeritus
Gold Member
But man can only conceive of finiteness, definiteness. So infinity is whatever cannot be conceived.
And since this contradicts the fact that people do conceive of the infinite, it necessarily follows that you have erred in your post.

#### Math Is Hard

Staff Emeritus
Gold Member
Infinity is not humanly conceivable and therefore nonsensical.

This does not mean that what is humanly conceivable is true. To the contrary, what is humanly conceivable must be false! For to conceive means to grasp, and to grasp means there is something that can be grasped, some thing, some unity, some oneness. Humans can only think in particles. Even the "quantum" is a kind of particle. Instead of "particles", quantum mechanics uses "quanta" (amounts); instead of unities, it uses units. But an amount is needs a definite amount. An amount of "1" is an amount of exactly ...000,001.000... -- that is, it presupposes "an infinite amount" of decimals. But an amount must needs be definite, i.e., it cannot be infinite. Therefore, there are no amounts.

But if there are no particles and no amounts, if there are no units and no unities, then all ideas thereof must be illusions, namely simplifications (literally "single-makings", "one-makings"). In order to conceive anything at all, in order for his consciousness to have its necessary object, without which consciousness is impossible, man must falsify the world by simplifying it. He must de-fine the world, impose boundaries on the chaos, ordering it. Quanta are indefinite (as they cannot be exact to an "infinite amount" of decimals), but man must regard them as definite in order to regard them at all.

As there are no definite quanta, no exact quanta, i.e., as there is not even one self-same quantum, there can definitely not be multiple equal quanta. But a finite number is a definite number. Therefore, there is no finiteness, either. But man can only conceive of finiteness, definiteness. So infinity is whatever cannot be conceived.
Sounds a little like Leopold Kronecker bellowing, "God made the integers; all else is the work of man!" That didn't go over so well.

This thread is on thin ice. Some of the comments are getting not only philosophically scattershot, but irritating.

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#### wuliheron

http://en.wikipedia.org/wiki/Infinity

Infinity is a valid and useful concept, but that does not mean it exists in the real world. The map is simply not the territory. In fact, as Wikipedia confirms, infinite results in physics are considered as meaningless and useless as paradoxical results. The only practical use for a paradoxical/infinite result is as a shortcut for understanding where our reasoning went wrong.

#### ZapperZ

Staff Emeritus
2018 Award
http://en.wikipedia.org/wiki/Infinity

Infinity is a valid and useful concept, but that does not mean it exists in the real world. The map is simply not the territory. In fact, as Wikipedia confirms, infinite results in physics are considered as meaningless and useless as paradoxical results. The only practical use for a paradoxical/infinite result is as a shortcut for understanding where our reasoning went wrong.
Oh really?

If that Wikipedia is correct, then did it explain away why, in condensed matter physics, there is such a thing as critical point, as in classical and quantum critical point? This is where several state functions and parameters change abruptly and therefore, quantities that depends on the variations of these parameters are infinite that that point.

Open up a book on phase transitions. Are these books less valid than a silly Wikipedia article that could have been written by someone whose credentials you know nothing about?

Zz.

 http://xxx.lanl.gov/abs/cond-mat/0011011
 http://arxiv.org/abs/cond-mat/0503002

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#### russ_watters

Mentor
http://en.wikipedia.org/wiki/Infinity

Infinity is a valid and useful concept, but that does not mean it exists in the real world. The map is simply not the territory.
Saying it over and over again does not make it true. Does Pi exist in the real world? How far will a photon of light travel if it never hits anything? Hurkyl gave other examples.
In fact, as Wikipedia confirms, infinite results in physics are considered as meaningless and useless as paradoxical results. The only practical use for a paradoxical/infinite result is as a shortcut for understanding where our reasoning went wrong.
The first part is often true, but not universally true. There are plenty of useful examples of infinity and useful infinite solutions in the real world. And the second part is a misunderstanding of the first part! The Wik article does not say that infinites don't exist in the real world!

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#### CRGreathouse

Homework Helper
Nonetheless, all of these types of infinity also display this same central paradox. To say that a number series is infinite is still a contradiction in terms because it excludes limited numbers such 5.
Interesting. When you use the term "infinite", you mean what I might call "coempty" (c.f. cofinite).

Infinity is a valid and useful concept, but that does not mean it exists in the real world. The map is simply not the territory. In fact, as Wikipedia confirms, infinite results in physics are considered as meaningless and useless as paradoxical results. The only practical use for a paradoxical/infinite result is as a shortcut for understanding where our reasoning went wrong.
So the class of all things, i.e. the universe, does not exist? I'm taking this directly from your indirect definition above of the infinite as that which fails to exclude any thing.

#### Sauwelios

And since this contradicts the fact that people do conceive of the infinite, it necessarily follows that you have erred in your post.
Please tell me how one may conceive of the infinite.

#### Sauwelios

Does Pi exist in the real world?
No, it does not.

There are plenty of useful examples of infinity and useful infinite solutions in the real world.
Could you give an example of infinity in the "real world"?

#### ZapperZ

Staff Emeritus
2018 Award
Could you give an example of infinity in the "real world"?
Oy vey. Considering that I gave to references to phase transition in which there are critical points that represent these "infinities", that is a rather strange question.

You want another example? The BCS density of states has the form of

$$N = \frac{eV}{\sqrt{(eV)^2 - \Delta^2}}$$

where eV is the energy of that state and $\Delta$ is the superconducting energy gap. So guess what happen at $eV = \Delta$?

Zz.

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#### ZapperZ

Staff Emeritus
2018 Award
And oh, since people have such affinity to believing what's in Wikipedia, why don't you look up "van Hove Singularity" on there and figure out what kind of system such a thing is occuring? [Hint: phonons occur in practically ALL solids structure, including the insulators, conductors, and semiconductors that you are using in your modern electronics. Would that qualify as "real world" enough?]

Zz.

#### russ_watters

Mentor
No, it does not.
It doesn't? Does a circle have a circumference and a diameter in the real world?
Could you give an example of infinity in the "real world"?
Besides the several already given, to find the total strength of a field (say, when computing the escape velocity at a point in a planet's gravitational field) you need to integrate out to an infinite distance from the object.

Since we like wik: http://en.wikipedia.org/wiki/Escape_velocity#Derivation_using_only_g_and_r

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#### Sauwelios

Oy vey. Considering that I gave to references to phase transition in which there are critical points that represent these "infinities", that is a rather strange question.
Considering that this is the philosophy board, it is rather strange to assume that every philosopher knows the details of such matters.

You want another example? The BCS density of states has the form of

$$N = \frac{eV}{\sqrt{(eV)^2 - \Delta^2}}$$

where eV is the energy of that state and $\Delta$ is the superconducting energy gap. So guess what happen at $eV = \Delta$?
I have no idea, as I don't speak that language. Please provide a theoretical account, in English, of how infinity occurs in the real world.

#### Sauwelios

It doesn't? Does a circle have a circumference and a diameter in the real world?
Does a perfect circle ever occur in the real world?

Besides the several already given, to find the total strength of a field (say, when computing the escape velocity at a point in a planet's gravitational field) you need to integrate out to an infinite distance from the object.
"Infinite distance" is a self-contradiction. A distance is by definition a definite distance. Describe to me how you think you can ever have an "infinite distance" to an object. Hint: "distance" is a relative concept (there can never be a question of "distant" or "not distant", but only of more or less distant).

I don't necessarily like wik.

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