What is (the nature of) infinity?

In summary, 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. Its nature is up for debate, but one suggestion is that it is the largest value that can be represented in a particular type of variable.
  • #1
phoenixthoth
1,605
2
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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  • #2
phoenixthoth said:
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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  • #3
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.
See also aleph 0
2. The largest value that can be represented in a particular type of variable (register, memory location, data type, whatever).
See also minus infinity.

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.
 
  • #4
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.
 
  • #5
phoenixthoth said:
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?
 
  • #6
wuliheron said:
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.
 
  • #7
phoenixthoth said:
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.
 
  • #8
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 tendency to shape one's thought. :frown:
 
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  • #9
sd01g said:
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.
 
  • #10
wuliheron said:
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,
Please elaborate.
 
  • #11
wuliheron said:
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.
 
  • #12
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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  • #13
Sauwelios said:
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.
 
  • #14
Sauwelios said:
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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  • #15
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.
 
  • #16
wuliheron said:
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[1] and quantum critical point[2]? 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.

[1] http://xxx.lanl.gov/abs/cond-mat/0011011
[2] http://arxiv.org/abs/cond-mat/0503002
 
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  • #17
wuliheron said:
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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  • #18
wuliheron said:
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).

wuliheron said:
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.
 
  • #19
Hurkyl said:
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.
 
  • #20
russ_watters said:
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"?
 
  • #21
Sauwelios said:
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

[tex]N = \frac{eV}{\sqrt{(eV)^2 - \Delta^2}}[/tex]

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

Zz.
 
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  • #22
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 occurring? [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.
 
  • #23
Sauwelios said:
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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  • #24
ZapperZ said:
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

[tex]N = \frac{eV}{\sqrt{(eV)^2 - \Delta^2}}[/tex]

where eV is the energy of that state and [itex]\Delta[/itex] is the superconducting energy gap. So guess what happen at [itex]eV = \Delta[/itex]?
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.
 
  • #25
russ_watters said:
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.
 
  • #26
Sauwelios said:
Considering that this is the philosophy board, it is rather strange to assume that every philosopher knows the details of such matters.

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.

I don't have too. Even when this is a philosophy subforum, the issue of PHYSICS was invoked. When one claims that "... infinite results in physics are considered as meaningless and useless as paradoxical results..", then it is a valid response to cite EXACT examples of where such a claim is false. I have produced just a few of these examples in which such singularities are not only part of the theory and description of these real-world phenomena, but also have been detected. It is a physics issue, and there's no dumbing it down. It makes zero sense to me to make claims about something that one has no knowledge of in the first place.

Please note that this is still a "Physics Forums" and unlike other other public forums, there are professional physicists, mathematicians, etc. on here. So if anyone tries to strengthen his/her arguments by invoking such subject matters, but using it based on lack of understanding or outright ignorance, then one should expect to be corrected. The Global PF Guidelines STILL apply in here, and that includes the spreading of misinformation regarding these subject matters.

Zz.
 
  • #27
ZapperZ said:
I don't have too. Even when this is a philosophy subforum, the issue of PHYSICS was invoked. When one claims that "... infinite results in physics are considered as meaningless and useless as paradoxical results..", then it is a valid response to cite EXACT examples of where such a claim is false. I have produced just a few of these examples in which such singularities are not only part of the theory and description of these real-world phenomena, but also have been detected. It is a physics issue, and there's no dumbing it down. It makes zero sense to me to make claims about something that one has no knowledge of in the first place.

Please note that this is still a "Physics Forums" and unlike other other public forums, there are professional physicists, mathematicians, etc. on here. So if anyone tries to strengthen his/her arguments by invoking such subject matters, but using it based on lack of understanding or outright ignorance, then one should expect to be corrected. The Global PF Guidelines STILL apply in here, and that includes the spreading of misinformation regarding these subject matters.
My point is that so-called "science" (physics and the like) can never know anything, never explain anything, but can only describe. These descriptions can be of use for human beings, in order to do certain things. Let me explain the difference between knowing and describing. Knowing would mean: being able to describe completely accurately. But "science" can only approximate complete accuracy in its descriptions. Thus quantum theory is a more accurate (namely more workable) description than atomic theory. But atomic theory had already been dismissed and quantum theory been anticipated by a philosopher, Friedrich Nietzsche, on a philosophical basis. Likewise, I - a contemporary philosopher - can already dismiss quantum theory, for the simple reason that a precise quantum cannot exist (as there is nothing to "cise", to cut). The idea of a "quantum" is atomic theory once again, or, to speak with Nietzsche, the soul superstition. Man cannot think but about "things", cannot be conscious but of "things". But atoms do not consist of "things".

Heidegger marked the end of philosophy: the point where philosophy became science. This science is not the science I have put between quotation marks above. It is best formulated by Neumann:

"I use "science" in its original meaning, scientia, knowledge of the way things really are. It is the only genuine science and it is readily available always and everywhere. [...]
Science is the simple realization that whatever is experienced -- a self, a world, the law of contradiction, a god or anything else -- is nothing apart from its being experienced. Science's reality is nothing but empty experiences, impressions as Hume called them. [...] Existence as a whole is now understood in such a way that it only exists in the first place insofar as it is produced by the man who perceives or produces it (durch den vorstellend-herstellenden Menschen [Heidegger, "The Age of the World-View"])... [...] this very thing, "that the world itself becomes a view" is the essence of science. In this regard, Nietzsche claims that, for science there are no facts, only interpretations or methods -- methods of experience, points of view. There is nothing inherently rigorous or mathematic in scientific method which, rightly understood as it rarely is, means nothing more than nihilist experience, any way (or method) of experiencing -- whether it be that of a tiger, an infant or an Einstein. Science is the realization that reality is nothing but mere experience, methods of perceiving or thinking. This "definition" of science, like all theories and thoughts, is no more than another empty experience or method."
[Harry Neumann, "Political Philosophy or Nihilist Science?"]

From your passionate nature, ZapperZ, which I have now witnessed for the second time - your political passion for "science" is evident from your attempt to appear superior to a "layman" by giving a display of your knowledge of hieroglyphics -, I suspect that for you it is political philosophy, not nihilist science ("philosophy" used in its original meaning, philosophia, love of knowledge - of the way things really are...).
 
  • #28
Sauwelios said:
Please tell me how one may conceive of the infinite.
Studying is usually a good way to learn a subject.
 
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  • #29
Sauwelios said:
My point is that so-called "science" (physics and the like) can never know anything, never explain anything, but can only describe. These descriptions can be of use for human beings, in order to do certain things. Let me explain the difference between knowing and describing. Knowing would mean: being able to describe completely accurately. But "science" can only approximate complete accuracy in its descriptions. Thus quantum theory is a more accurate (namely more workable) description than atomic theory. But atomic theory had already been dismissed and quantum theory been anticipated by a philosopher, Friedrich Nietzsche, on a philosophical basis. Likewise, I - a contemporary philosopher - can already dismiss quantum theory, for the simple reason that a precise quantum cannot exist (as there is nothing to "cise", to cut). The idea of a "quantum" is atomic theory once again, or, to speak with Nietzsche, the soul superstition. Man cannot think but about "things", cannot be conscious but of "things". But atoms do not consist of "things".

Heidegger marked the end of philosophy: the point where philosophy became science. This science is not the science I have put between quotation marks above. It is best formulated by Neumann:

"I use "science" in its original meaning, scientia, knowledge of the way things really are. It is the only genuine science and it is readily available always and everywhere. [...]
Science is the simple realization that whatever is experienced -- a self, a world, the law of contradiction, a god or anything else -- is nothing apart from its being experienced. Science's reality is nothing but empty experiences, impressions as Hume called them. [...] Existence as a whole is now understood in such a way that it only exists in the first place insofar as it is produced by the man who perceives or produces it (durch den vorstellend-herstellenden Menschen [Heidegger, "The Age of the World-View"])... [...] this very thing, "that the world itself becomes a view" is the essence of science. In this regard, Nietzsche claims that, for science there are no facts, only interpretations or methods -- methods of experience, points of view. There is nothing inherently rigorous or mathematic in scientific method which, rightly understood as it rarely is, means nothing more than nihilist experience, any way (or method) of experiencing -- whether it be that of a tiger, an infant or an Einstein. Science is the realization that reality is nothing but mere experience, methods of perceiving or thinking. This "definition" of science, like all theories and thoughts, is no more than another empty experience or method."
[Harry Neumann, "Political Philosophy or Nihilist Science?"]

From your passionate nature, ZapperZ, which I have now witnessed for the second time - your political passion for "science" is evident from your attempt to appear superior to a "layman" by giving a display of your knowledge of hieroglyphics -, I suspect that for you it is political philosophy, not nihilist science ("philosophy" used in its original meaning, philosophia, love of knowledge - of the way things really are...).

You have somehow either hijacked this thread, or turned it into what it is not- the function of science.

You haven't been able to dispute the FACT that before someone can use something in his/her favor, one has to actually understand what it is. Someone invoked something in physics to support a point, but it was used in the wrong manner. I never made ANY argument that science knows everything, and so, I have no clue on where this diatribe is coming from.

If you argue about something that you have only a superficial knowledge of, would you take your opinion seriously? Or more importantly, do you think you have arrived at a conclusion that is based on the intimate knowledge of the subject matter rather than based on ignorance?

The whole concept of "infinity" has a strong mathematical base. Whole chapters on complex analysis are devoted to it. It is an extreme oversight if anyone dealing with the issue of "infinity" to be ignorant of the mathematics involved. And then to argue that it isn't based on "real world" while ignoring the application of such concepts in many parts of physics is just plain wrong.

No matter how much you try to derail and confuse the issue, those are the issues in which you haven't tackled. Stick to the topic and produce an argument on why all those examples I gave are not valid. It is not my fault that you are unable to comprehend basic mathematics. I am not the one who was asking about things I know nothing about.

Zz.
 
  • #30
This thread has wandered way too far from the original topic.
 

1. What is infinity?

Infinity is a concept that refers to something that has no limit or end. It is often described as being endless or boundless.

2. Is infinity a number?

No, infinity is not a number. It is a concept that represents something that is uncountable or limitless. In mathematics, infinity is often used as a symbol to represent numbers that are larger than any finite number.

3. Can infinity be measured?

No, infinity cannot be measured because it has no specific value or limit. It is a concept that represents something that is endless and cannot be quantified.

4. Is infinity a real or abstract concept?

Infinity is considered to be an abstract concept because it cannot be physically observed or measured. However, it is used in many fields of study, such as mathematics, physics, and philosophy, to help explain and understand various phenomena.

5. Can infinity be divided or multiplied?

No, infinity cannot be divided or multiplied because it is not a number. It is a concept that represents something that is limitless and cannot be contained or altered by mathematical operations.

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