Infinity: a concept or number?

[revo74]

Is infinity a mathematical concept or a number? Please elaborate. Is there any debate over this topic or is there a consensus amongst academia?

 

[micromass]

See https://www.physicsforums.com/showthread.php?t=507003

 

[HallsofIvy]

We can't give a complete answer to that until you define what you mean by "number".

If you are referring to the ordinary "real number system" (and its subsets) then, no, "infinity" is not a number.

There are, however, ways of extending the real number system, one of which adds 'infinity" and "negative infinity" as numbers. Caution- the usual laws of arithmetic do not generally hold in such a system- you still can't say "infinity - infinity= 0".

 

[uperkurk]

Infinity is never classes as a number. Take for example

Infinity - Infinity = ?

You can't add, subtract, divide or multiply by infinity.

 

[Whovian]

Infinity is never classes as a number. Take for example

Infinity - Infinity = ?

You can't add, subtract, divide or multiply by infinity.

Did you read HallsofIvy's response? It's classified as a number to which the normal rules of arithmetic don't apply sometimes.

 

[micromass]

Infinity is never classes as a number. Take for example

Infinity - Infinity = ?

You can't add, subtract, divide or multiply by infinity.

You can do \infty + \infty if you want to. So you can surely add them. But you can't subtract them (that is: you can't add infinity and negative infinity).

 

[revo74]

We can't give a complete answer to that until you define what you mean by "number".

If you are referring to the ordinary "real number system" (and its subsets) then, no, "infinity" is not a number.

There are, however, ways of extending the real number system, one of which adds 'infinity" and "negative infinity" as numbers. Caution- the usual laws of arithmetic do not generally hold in such a system- you still can't say "infinity - infinity= 0".

Didn't Hilbert and Cantor believe infinity was a number?

If you extended the real number system to include infinity then what isn't it indeed classified as a number?

 

[micromass]

Didn't Hilbert and Cantor believe infinity was a number?

Got any citation for that?

If you extended the real number system to include infinity then what isn't it indeed classified as a number?

Because you extend the real numbers, which means it wasn't in the real numbers to begin with. Infinity is, by definition, not a real number. It is an extended real number though. It's not a real number for good reason, because not all the usual operations work with infinity.

 

[SteveL27]

Is infinity a mathematical concept or a number? Please elaborate. Is there any debate over this topic or is there a consensus amongst academia?

I'd like to know your answer a simpler question. Is 6 a concept or a number? Isn't a number just a concept?

What I mean is, there is no 6 in the physical world. There are six apples, six days, six sandwiches. But the number 6, in isolation, is just an abstract concept in our minds.

So how is 6 any different than infinity? Aren't both just mathematical concepts?

 

[uperkurk]

Didn't Hilbert and Cantor believe infinity was a number?

If you extended the real number system to include infinity then what isn't it indeed classified as a number?

Extending the number system to include infinity would ultimately give infinity a value. Infinity does not have any direct value so it is not a number.

In case you're not sure, open up MatLab or any kind of programming compiler and you'll see that when you try to perform any calcuation using the word infinity you'll get an error or -0

 

[Mandlebra]

If you are interested in doing some algebra with ω (the "smallest" infinity), you should read up on Conway's surreal numbers (often also referred to as games due to how they arose during the study of go and similar games). However, the surreals are not for the mathematically loose as their definition and doing proofs can be quite tedious!

The reason I mention the surreals is that in this system the expression ω and ω-1 are different beasts!

 

[jreelawg]

I say it's not a number because it's not quantifiable.

 

[SteveL27]

I say it's not a number because it's not quantifiable.

How about pi = 3.14159265... continuing forever. Is it quantifiable? Is it a number?

How about the imaginary unit i? Is it quantifiable? Is it a number?

How about 10^101010101010?

Just for comparison, the estimated number of atoms in the universe is 10⁸⁰, way less than just 10¹⁰¹⁰.

Is it quantifiable? How many is it? Could anyone count that high? Is it a number?

How about the integers mod 5? Are they quantifiable? Are they numbers?

Just trying to understand what you mean by "quantifiable."

 

[Number Nine]

Extending the number system to include infinity would ultimately give infinity a value. Infinity does not have any direct value so it is not a number.

What is the value of aleph null? Methinks a good course in abstract algebra would clear up a lot of these restricted notions of what constitutes a number.

 

[uperkurk]

How about pi = 3.14159265... continuing forever. Is it quantifiable? Is it a number?

PI's value is not infinity but the concept is.

Think about this, which of these numbers would you regard as being larger?

9 followed by ∞ 9's or 1 followed by ∞ 1's?

The 9's have a higher value but they are equal to each because they have ∞ numbers trailing...

 

[micromass]

9 followed by ∞ 9's or 1 followed by ∞ 1's?

None of these are real numbers.

 

[uperkurk]

What is the value of aleph null? Methinks a good course in abstract algebra would clear up a lot of these restricted notions of what constitutes a number.

Regardless of how people try to justify infinity, the bottom line is if something goes on forever it can never have a number, a value yes but not a number

 

[Number Nine]

Regardless of how people try to justify infinity, the bottom line is if something goes on forever it can never have a number, a value yes but not a number

You seem to be treating "number" as synonymous with "real number"; which is to say, "something you can count with". Number systems are just algebraic structures; the elements of those structures are called numbers, whatever they may be. It may disturb you to know that every real number is actually a collection of intervals on the rational number line (a classic construction of the real numbers: Dedekind cuts). We call each of these intervals a real number because it's convenient, and they behave the way we expect real number to behave.

Do you object to the existence of cardinal numbers? Ordinal numbers?

 

[micromass]

The thing is that there is no such thing as a "number" in mathematics. We do have

- Rational numbers
- Real numbers
- Surreal numbers
- Transfinite numbers
- p-adic numbers

and many more. Some of these systems allow some form of infinity. But mathematicians never speak of just numbers.

 

[Studiot]

One reason we have infintity or infinities is that it is sometimes possible and even useful to evaluate expressions of the type


\frac{\infty }{\infty }

Wallis product is a good example.

 

[phoenixthoth]

I'm surprised that unlimited hypernatural numbers haven't been mentioned much...

 

[coolul007]

Don't we use infinity to define a set of numbers, as in a series, to reach a limit or it is limitless. Isn't the underlying premise of Calculus infinite members of a set?

 

[HallsofIvy]

Yes, but that has nothing to do with the question of whether infinity "is a number or a concept".

 

[SteveL27]

Don't we use infinity to define a set of numbers, as in a series, to reach a limit or it is limitless. Isn't the underlying premise of Calculus infinite members of a set?

Well, yes and no. Logically if you build calculus from the ground up, you need a theory of the real numbers, which requires a theory of infinite sets. This is the modern view.

But historically, Newton and Leibniz developed calculus without any set theory and without a rigorous theory of limits. So you don't need infinite sets to do calculus, only to logically justify it.

 

[coolul007]

Yes, but that has nothing to do with the question of whether infinity "is a number or a concept".

As I so UN-eloquently put it, my intent was to show the use of infinity as a concept. i also agree with the previous posts that all numbers are a figment of our imagination, and we have given rules to how they are to behave.

 

[revo74]

The thing is that there is no such thing as a "number" in mathematics. We do have

- Rational numbers
- Real numbers
- Surreal numbers
- Transfinite numbers
- p-adic numbers

and many more. Some of these systems allow some form of infinity. But mathematicians never speak of just numbers.

So infinity is a number then?

 

[revo74]

You seem to be treating "number" as synonymous with "real number"; which is to say, "something you can count with". Number systems are just algebraic structures; the elements of those structures are called numbers, whatever they may be. It may disturb you to know that every real number is actually a collection of intervals on the rational number line (a classic construction of the real numbers: Dedekind cuts). We call each of these intervals a real number because it's convenient, and they behave the way we expect real number to behave.

Do you object to the existence of cardinal numbers? Ordinal numbers?

So you consider infinity to be a number?

 

[revo74]

I was told that Cantor possibly Hilbert as well, considered infinity to be a number. Is this true?

 

[coolul007]

However, with the exception of complex numbers, all "numbers" have a fixed location on a number line infinity does not. So, a concept of limitlessness.

 

[revo74]

Is there any official mathematical dictionaries that define infinity? It seems to me that there are various views on this topic. Is there any authority that decides such things?

 

[Whovian]

However, with the exception of complex numbers, all "numbers" have a fixed location on a number line infinity does not. So, a concept of limitlessness.

Sort of how it's defined in Calculus. If a limit to a real number of a function grows without bound, we call it "infinity." Same goes for a limit to infinity, but a limit to infinity is as a number grows without bound.

 

[micromass]

I was told that Cantor possibly Hilbert as well, considered infinity to be a number. Is this true?

No, this is not true. Unless you can come up with a citation.

So infinity is a number then?

Did you not read my entire post?? There is no such thing as a number.

 

[micromass]

Is there any official mathematical dictionaries that define infinity? It seems to me that there are various views on this topic. Is there any authority that decides such things?

There is also no such thing as "infinity". There are various interpretations of infinity throughout mathematics. There are ordinals, cardinals, extended reals, projective spaces, etc. These are all incarnations of infinity.

 

[Hitarth]

Infinity is not a real number then why sometime it is domain or range of a real function?

 

[Whovian]

Infinity is not a real number then why sometime it is domain or range of a real function?

It isn't truly in any domain or range. A range of something like (0,∞) means that the range of the function has no upper limit, not that the upper bound is ∞.

 

[micromass]

Indeed, things like [x,+\infty) only have real numbers has elements. So infinity is not a member of the set. Furthermore, the notation "+\infty" is nothing more than a notation. It is a notation for

[x,+\infty) = \{y\in \mathbb{R}~\vert~x\leq y\}

If you don't like the infinity, then other notations are

[x,\rightarrow )~\text{or}~\uparrow x

 

[Number Nine]

So you consider infinity to be a number?

...as part of what set? With what operations?

 

[Diffy]

I don't understand why people on this forum respond the way they do to questions like these.

People asking these types of questions have little to no mathematical training. Trying to explain something like the extended reals to them just confuses the matter doesn't it?

Can we just assume these people are talking about the Reals? Simply because they obviously haven't studied or heard of anything else? Can't we just assume the question is "Is infinity in the Real number system?"

This way when someone asks this question, we can simply say: NO.

I really hate these threads, that are posted so often. And even when linked to the FAQ, the OPs don't read it.

Can we lock this? There is no math in it.

 

[micromass]

Can we lock this? There is no math in it.

Agreed.