Can Infinity Truly Be Considered a Number?

[Asif]

Hi all !

I came across an article (link: http://www.math.toronto.edu/mathnet/answers/infinity.html#2)

The article states that in the context of a number system, infinity does not exist.

Now, being the dimwit that I am, I just wanted to ask/confirm this:

I THINK the article is saying that one cannot perform arithmetic operations like addition, subtraction etc. using infinity (please do correct me if I'm wrong).

If so, this would mean one simply cannot perform operations like:

infinity + infinity = infinity

OR

infinity - infinity = ?

OR

3 * infinity = infinity

Wouldn't all such operations be "illegal"? (this would of'course depend on whether I understood that article correctly).

Thanking you in advance,
Asif.

 

[tomkeus]

$$\infty$$ is not a number. I prefer to think of it as a limit. Say you have some function which limit in point a is $$\infty$$, then it means that for each M>0, there is a $$\delta$$ such that when $$|x-a|<\delta$$ then f(x)>M.

 

[Wong]

That's an interesting article.

I think that what the author means by "usual rules of addition" is that the "number system forms a/an (abelian) group under addition"? Infinity cannot be regarded as "a number" because +ve/-ve infinity by definition is an element which is greater/smaller than any number. Thus,

infinity + a = infinity (where a is any finite number)

But if we regard infinity as "a number" for which the "usual rules of addition/subtraction" hold, then we may subtract infinity from both sides to obtain

a = 0

a contradiction.

As to whether the operations you mention (i.e. (+ve)infinity + (+ve)infinity = (+ve)infinity) are legal, I think one may *define* them to be that way. Note that such relation is consistent with the ordering of elements (that +ve infinity is greater than any other elements), but at the expense that "the usual rules of addition/subtraction" do not hold for expressions involving "infinities".

 

[Zurtex]

As said, infinity is not a number.

Often it is just used as a mathematical short hand meaning slightly different things in different contexts. For example:

We could have some function of x, y = f(x). Some properties of this function maybe:

As $$x \rightarrow \infty$$
Then $$y \rightarrow \infty$$

This simple means as x gets bigger, y does not converge on a number and just keeps getting bigger.

There are mathematical systems for arithmetical operations on transfinite numbers, perhaps someone else will post about those.

 

[gazzo]

$$+\infty$$ and $$-\infty$$ are elements in the extended reals arn't they?

 

[HallsofIvy]

$$+\infty$$ and $$-\infty$$ are elements in the extended reals arn't they?

Yes, but in the originally quoted text they specifically referred to a number system with "rules of arithmetic" similar to standard arithmetic. As noted before, the "extended reals" don't follow that. The extended reals are geometric (strictly speaking, topological) extension, not algebraic.

 

[gazzo]

oh indeed.

 

[Asif]

Guys, y'all lost me by the 2nd post...lol

Lets try to make things simpler for idiots like myself.

Thus far, what I could gather is that my understanding of what the article was trying to say is not entirely (if at all) correct.

We can carry out operations like (+ve) infinity + (+ve) infinity, "but at the expense that "the usual rules of addition/subtraction" do not hold for expressions involving "infinities"."

If the "usual rules" don't apply, then what does apply? And if the usual rules do not apply, then how can one say infinity + infinity = infinity (isn't that making use of the usual rules?)

Pardon my ignorance,
bye,
Asif.

 

[JD]

Might it be the case that infinite numbers exist in theory only, rather than in actuality?
Do we need to count 'things' (whatever they might be)? For all I know, however, I could be completely wrong.

 

[Zurtex]

We can carry out operations like (+ve) infinity + (+ve) infinity, "but at the expense that "the usual rules of addition/subtraction" do not hold for expressions involving "infinities"."

If the "usual rules" don't apply, then what does apply? And if the usual rules do not apply, then how can one say infinity + infinity = infinity (isn't that making use of the usual rules?)

Pardon my ignorance,
bye,
Asif.

:uhh: Can you give me an example under the usual rules of addition where:

a is positive

and

a + a = a.

 

[matt grime]

Might it be the case that infinite numbers exist in theory only, rather than in actuality?
Do we need to count 'things' (whatever they might be)? For all I know, however, I could be completely wrong.

philosophically speaking, do any numbers exist in actuality? mathematically the existence of numbers in some physicaly sense is neither here nor there: if all snooks are green, and boojum's blue, then i can reason no boojum is a snook; that I've not told you they are, or aren't, things that exist is not important.

As to what these alleged infinite numbers are, then we may offer the following interpretation (it is not the only one):

we mean cardinal numbers, and by addition, we mean that the result of adding |X|+|Y| is the cardinality of XuY, where X and Y are some disjoint sets in the equivalence classes of cardinal numbers. Then Aleph-0+aleph-0=aleph-0

 

[JD]

philosophically speaking, do any numbers exist in actuality? mathematically the existence of numbers in some physicaly sense is neither here nor there: if all snooks are green, and boojum's blue, then i can reason no boojum is a snook; that I've not told you they are, or aren't, things that exist is not important.

Given that no cap need be put (if a finite number of things is not being counted) on a number series (as you can always add one to the end of the series) and if numbers do not actually exist (whatever existence might exactly mean in this context), is there a valid argument against infinity?

 

[Asif]

Hi Zurtex !

You said: "Can you give me an example under the usual rules of addition where:

a is positive

and

a + a = a."

You've given my rusty brain cells a jump start...hehe...so I guess this means (+ve) infinity + (+ve) infinity = (+ve) infinity isn't really making use of the "normal"/"usual" rules of addition.

<sigh> Math !

Bye,
Asif.

 

[matt grime]

JD, you'd have to say what you mean by infinity. As has been pointed out, the term infinity means a multitude of things, all linked by the idea of being 'not finite'. It would alse depend on your personal view point of what it means for something to 'exist' mathematically.

 

[JD]

JD, you'd have to say what you mean by infinity. As has been pointed out, the term infinity means a multitude of things, all linked by the idea of being 'not finite'. It would alse depend on your personal view point of what it means for something to 'exist' mathematically.

That's the difficulty I suppose. Infinity is impossible to visualise, to understand in a rational sense. At the same time, though, it is impossible to visualise a number series (which is not counting a definite number of objects) that cannot be added to indefinitely. So this is impossible in both senses.
Existence wise, the number would not necessarily have to be applied to anything. I suggested that infinity might be merely theoretical, but that suggests a limitation.

 

[matt grime]

No, that's not what i meant about indicating what you mean by infinity. Infinity's meaning is clearly understood dependent on context, though it is always better use some compound adjective form involving the word infinite when that is possible.

For instance we know fully what we mean when we say "sum from 1 to infinity", and that is better written as "sum of n in N"

We know what we mean when we say 1/x tends to infinity as x tends to zero (from above), we mean that for all L>0, there exists d>0 such that 0<x<d implies 1/x >L

The "point at infinity" is the point we add to the complex plane, say, as a topological object such that it is compact in the norm topology (sorry, that's probably too high-faluting for this discussion), and can be visualized in many ways.

 

[JD]

No, that's not what i meant about indicating what you mean by infinity. Infinity's meaning is clearly understood dependent on context, though it is always better use some compound adjective form involving the word infinite when that is possible.

The context that I am interested in is the application of infinite energy to bring slower-than-light-speed objects up to light speed or slow FTL objects down to light speed.

 

[homology]

actually it really depends on what you're doing. You folks would probably be interested in Abraham Robinson's Nonstandard Analysis (usually abbreviated NSA) in which there are infinitely large numbers and infinitely small numbers. In NSA there are no (or very few) limits. You just "use" infinities and infinitesimals.

http://members.tripod.com/PhilipApps/nonstandard.html

is a link with some stuff and many more links for the interested parties.

Kevin

 

[homology]

Oh I should add that Robinson also proved that a theorem true in NSA is also true in ordinary analysis and vice versa.

Kevin

 

[matt grime]

The context that I am interested in is the application of infinite energy to bring slower-than-light-speed objects up to light speed or slow FTL objects down to light speed.

in that case i will be moving on from the discussion.

 

[rayjohn01]

If it helps I avoid that question by considering 'infinity' to mean 'never ending' this avoids any 'number' , but does not prevent a discussion of change rates. That is counting a straight line of trees is not quite the same as counting a two dimensional field of tress when BOTH lines(x,y) are 'never ending'.
I know I'm a little radical -- I claim that there are NO real numbers that includes integers.
Given a 'number line' say going from near 0 to near 10 , but without any other labels
I believe it is impossible to find any number ----- a stab resulting in
1.00000000000001 does not constitute finding 1.0 , nor does finding
1.001 001 001 001 001 ... mean it repeats ( hence a rational number).
Any process of finding without preconceived labels appears to result in 'infinite' or never ending procedures -- so where are they ?
Ray.

 

[JD]

I believe it is impossible to find any number ----- a stab resulting in
1.00000000000001 does not constitute finding 1.0 , nor does finding
1.001 001 001 001 001 ... mean it repeats ( hence a rational number).
Any process of finding without preconceived labels appears to result in 'infinite' or never ending procedures -- so where are they ?
Ray.

Can anything be actually measured then? There could always be (in theory, if not in practice) finer scales (beyond the minimum scales utilised in design, manufacture etc) that could potentially be employed.

 

[matt grime]

If it helps I avoid that question by considering 'infinity' to mean 'never ending' this avoids any 'number' , but does not prevent a discussion of change rates. That is counting a straight line of trees is not quite the same as counting a two dimensional field of tress when BOTH lines(x,y) are 'never ending'.
I know I'm a little radical -- I claim that there are NO real numbers that includes integers.
Given a 'number line' say going from near 0 to near 10 , but without any other labels
I believe it is impossible to find any number ----- a stab resulting in
1.00000000000001 does not constitute finding 1.0 , nor does finding
1.001 001 001 001 001 ... mean it repeats ( hence a rational number).
Any process of finding without preconceived labels appears to result in 'infinite' or never ending procedures -- so where are they ?
Ray.

the real numbers are not "the number line" so this argument is not valid. the number line is merely a convenient and unrigorous way to explain these things in simple terms without the need to talk about localizations and completions of metric spaces.

 

[rayjohn01]

to JD

Yes you can to some accuracy but only in the relative sense-- i.e pick a stick call it '1' compare anything else in stick units , for a real object ( unknown) you will never find the precise value because it entails a never ending quest for accuracy which you cannot confirm. The only thing which could be accurate is one made of 'n' sticks -- but how could that be ? You picked the stick arbitrarilly there's no reason for anything else to be 'a priori'
related and how would you prove it anyway -- you end up with the same problem.
It's a bit like asking are two ceasium clocks 'the same' one would like to think so but in practice they drift with respect to the other ( not always smoothly).
This picture suggests that '1' is an arbitrary label with no inherent meaning which mathematicians do not like -- but to me it's they that have the problem the physicists just accept what nature says within the limits of measurement. Ray.

 

[ExecNight]

One can not talk about inifnity inside a universe where everything is finite..

So infinity is just a toy for the mathematicians to play with

 

[matt grime]

eh? no, 1 is the unique identity element in the multiplicative sense for a ring (in this case of the integers), we have no issues abot trying to explain its meaning and relevance to the real world, rayjohn, we're perfectly happy with it

 

[rayjohn01]

To Matt

Matt I am well aware of what mathematicians think of '1' but it does not apply in the real world -- it's quite impossible to confirm that any formed identity is the same as the original -- except by assumption as in maths.
Maths can be reduced to grouping of related "identical" quantities per addition multiplication etc. etc. the real world does not allow such simple occurences.
Take two identical electrons ( spatially separated and free from influence)
It is now impossible to say that they are provably identical -- if you now group them they tend to split identities occupying different states .
The mathematical identity '1' is a concept just as '0' and the rules and the results of maths follow -- but it has to my memory been proven that this structure can never be logically consistent -- I wonder why ?
Ray.

 

[master_coda]

It is now impossible to say that they are provably identical -- if you now group them they tend to split identities occupying different states .
The mathematical identity '1' is a concept just as '0' and the rules and the results of maths follow -- but it has to my memory been proven that this structure can never be logically consistent -- I wonder why ?
Ray.

It's impossible to say that anything about the "real world" is provably true. Any field of study that requires it's results to be consistent with the physical universe has this problem.

Remember, an 'electron' is just an abstract concept, just like '1' or '0'. You can prove things about the abstraction, but that doesn't prove anything about the world because you have to assume that the abstraction is a correct description of a physical object.

And, there are systems of arithmetic that involve '1' and '0' that are provably consistent. Normal arithmetic is not, but this is a consequence of logic and not the real world.

 

[CrankFan]

but it has to my memory been proven that this structure can never be logically consistent -- I wonder why ?
Ray.

I'm sort of fuzzy on your meaning here. Which structure are you talking about when you say, from memory: it's been proven it can't be logically consistent?

And, there are systems of arithmetic that involve '1' and '0' that are provably consistent. Normal arithmetic is not, but this is a consequence of logic and not the real world.

If I understand you correctly you're saying that normal arithmetic isn't consistent, which would be shocking if true. Maybe you meant to write something else here?

 

[master_coda]

If I understand you correctly you're saying that normal arithmetic isn't consistent, which would be shocking if true. Maybe you meant to write something else here?

Normal arithmetic isn't provably consistent. Of course it isn't known to be inconsistent, and we don't have any reason (right now) to believe that it isn't consistent. But we can't prove it.

Perhaps you misinterpreted what I was saying: I meant "not provably consistent" but it kinds of sounds like I'm saying "not consistent".

 

[CrankFan]

Normal arithmetic isn't provably consistent. Of course it isn't known to be inconsistent, and we don't have any reason (right now) to believe that it isn't consistent. But we can't prove it.

Right.

Perhaps you misinterpreted what I was saying: I meant "not provably consistent" but it kinds of sounds like I'm saying "not consistent".

I did misunderstand your meaning; but now it's clear to me.

 

[JonF]

Matt I am well aware of what mathematicians think of '1' but it does not apply in the real world -- it's quite impossible to confirm that any formed identity is the same as the original -- except by assumption as in maths. .

Just because we can not measure the difference between two object in a perfectly accurate way it doesn’t mean that two objects can not be similar. It simply means in practicality we can not make (or find) similar objects. All of this is kinda moot though, because if you needed use mathematic for some application to the level of accuracy you describe, you would include a rang of error in your calculation. Relating to your example with sticks: You may call some arbitrary amount of “stick mass” 1. To keep your data accurate in measurement error the next time you measure a stick don’t call that value 1, call it constant that is c which satisfy: .999<c<1.001. You may not be able to find exactly 1 but you sure can find a value it is in-between.

Maths can be reduced to grouping of related "identical" quantities per addition multiplication etc. etc. the real world does not allow such simple occurences.

I beg to differ. That is exactly what the real world allows. In the real world these simple occurrences just interact with each other and there are a infinite amount of them.

it's quite impossible to confirm that any formed identity is the same as the original … The mathematical identity '1' is a concept just as '0' and the rules and the results of maths follow

This is a silly argument, that can be said of anything describe in symbolism, which is just about everything. The thing is with math its rules are consistent with reality.

Also you have a small point with 1 (but see my previous response to one of your quotes) but you got no ground with zero. In reality comparing two “ones” may require an inequality but you can compare two zero’s with no such tricks.

 

[Philocrat]

The article is merely confirming the existence of infinity in a non-existent way. In fact, the clearer picture is that infinity is quantitatively finite. That is, infinite and finite quantities are mathematically interchangeable. I personally think that there is no need to formalize this relation as everything is describable in both ways.

 

[quartodeciman]

It is unfortunate that we still tend to speak of "infinity" as a number. Suppose we spoke of its contrary as "finity". Then we would get stuff like:

finity + finity = finity
finity - finity = finity
finity*finity = finity

. Phooey on that! With discrimination between +ve infinity and -ve infinity, the situation only gets worse:

Is (+ve finity) + (-ve finity) equal to -ve finity, 0 or +ve finity?

Instead of infinities, I would prefer something like "transfinite numbers" as a class, but very individual and well-discriminated. Maybe there is even a better name (one lecturer at MIT tried "macho numbers", versus "wimps").

Alas, it is hard to undo centuries of habit!

------

A comment on non-standard numbers: they are great for providing proofs of theorems, returning at the end to the original domain with only ordinary finite numbers. But trying to use non-standard numbers to produce results in an extended domain as the final result leaves us with a problem: there are pairs of non-standard numbers for which we can't tell their order:
In other words, we can't tell for some non-standard numbers x and y whether x < y, x=y or x > y. That is OK in the non-standard analysis usage, because those numbers don't occur in the final result. But if we want results in the extended system, then we are stuck.

But maybe someone has already found a constructible free hyperfilter or Lindstrom measure function and I don't know it. Happy thought!