Oo = -oo? positive infinity be equivalent to negative infinity?

[Loren Booda]

In what regard might positive infinity be equivalent to negative infinity?

 

[chroot]

In no sense at all that I'm aware of, but I'm not a mathematician.

- Warren

 

[Hurkyl]

In what regard might positive infinity be equivalent to negative infinity?

Define your terms. Then one may speak about what might be true.

If, for example, by "positive infinity" and "negative infinity" you mean the endpoints of the extended real line, and by "equivalent" you mean equal, then in no way might positive infinity be equivalent to negative infinity.

 

[Curious3141]

Makes no real sense as you posted it, but if you wanted to make some physical sense of the nonsensical, here goes : Negative absolute temperatures are "hotter" than all positive absolute temperatures (in that a body with a neg. absolute temperature will spontaneously transfer heat to a body with a pos. absolute temp. Low magnitude neg absolute temps are the hottest (just "below" zero kelvin). Higher magnitude neg absolute temps are colder (but still hotter than pos. infinity temperatures). When you go "right to the ends" of the scale, neg. infinity is the "same" temperature as pos. infinity, in that a body with a temp of -infinity will be in thermal equilibrium with one of +infinity.

 

[Integral]

Consider what happens to the image formed by a simple lens as the object moves through the focal point.

Is not this some form of direct connection between $+ \infty$ and $- \infty$ ?

 

[Loren Booda]

If 2 times infinity is infinity, and 3 times infinity is infinity, thus a constant time infinity is infinity (proved by Cantor as aleph-0?), then it seems to follow that -1 times infinity [negative infinity] is infinity, or -2 times infinity [2 times negative infinity] is infinity, etc.

 

[Loren Booda]

Does the limit of C/x as x approaches zero from a positive direction equal the limit of C/x as x approaches zero from a negative direction?

 

[Hurkyl]

If 2 times infinity is infinity, and 3 times infinity is infinity, thus a constant time infinity is infinity (proved by Cantor as aleph-0?), then it seems to follow that -1 times infinity [negative infinity] is infinity, or -2 times infinity [2 times negative infinity] is infinity, etc.

Define your terms: only then can we start having discussions like this.

For example, if by "infinity" you mean any infinite cardinal number, and by "times" you mean the multiplication of cardinal numbers, then "-1 times infinity" has no meaning whatsoever.

Does the limit of C/x as x approaches zero from a positive direction equal the limit of C/x as x approaches zero from a negative direction?

Again, define what you're talking about. Only then can such a question be meaningful. For example:

If we're working in the reals, then neither of these limits exist, so they can't be equal. (unless C = 0, in which case both limits exist and are equal)

If we're working in the projective reals, then both limits are equal.

If we're working in the extended reals, then both limits exist and unequal. (unless C = 0, in which case both limits are equal)

Until you state what you mean, we cannot have any sensible discussions along these lines.

 

[Loren Booda]

Hurkyl,

Your explanation is sensible enough for me. Does 2 [a positive number] times infinity [an infinite cardinal number] have meaning? How does one distinguish between projective and extended reals, in laymans terms?

 

[Werg22]

I'm not sure in what terms do you ask this question, but infinity does not exist as a number. You can define infinity in terms of the upper bound of the positive real. Multiplying infinity has no meaning unless you it is for the sake of evaluating limits... how can infinity equal minus infinity? Equality is property of finite numbers...

 

[matt grime]

Hurkyl,

Your explanation is sensible enough for me. Does 2 [a positive number] times infinity [an infinite cardinal number] have meaning?

2A is the cardinality of A disjoint union A, this is equal to A, in cardinal arithmetic for an infinite cardinal A.

How does one distinguish between projective and extended reals, in laymans terms?

by their definitions.

 

[matt grime]

how can infinity equal minus infinity? Equality is property of finite numbers...

No it isn't.

 

[fopc]

Is it possible the word infinite has two meanings in mathematics? Apparently, Georg Cantor thought so. Here's a quote from him.

"In spite of the essential difference between the conceptions of the potential and the actual infinite, the former signifying a variable finite magnitude increasing beyond all finite limits, while the latter is a fixed, constant quantity lying beyond all finite magnitudes, it happens only
too often that the one is mistaken for the other."

Perhaps the wording is a bit dated, but the message still comes through.

The symbol representing the "potential" in the above is usually given by oo. The "actual" are the transfinite cardinal numbers.

In your posts, you appear to use the two concepts indiscriminately.

For example in your post #9, "... infinity [an infinite cardinal number] ...", you are referring to transfinite cardinals numbers.
Yet, in your OP you are referring to infinity in the context of the symbol oo, which would be Cantor's potential infinite.

Perhaps you don't hold Cantor's work in any regard, and it's your intention to recognize only one notion of inifinite. Certainly that's your prerogative.
But I will say that not everyone endorses that opinion.

There is an arithmetic defined on the transfinte cardinal numbers. Let c be a cardinal. The "product" -1*c is meaningless irrespective of the actual definition of the product of two cardinal numbers. Why?
Ask yourself, is -1 a transfinite cardinal number? Is -1 even a finite cardinal number?

With respect to oo, I would treat +oo, -oo as intact symbols and not as the result of some form of multiplication. Whether there are contexts where these symbols "matchup" in some sense (as suggested in other posts), I'm not qualified to comment on.

 

[Gib Z]

Yes there is a difference between Potential and Actual Infinities. An actual infinity is something like, the number of elements in the reals. But potential infinity is more like, this. Say we have a computer game where someone has no limit to the amount of items they can buy. They can buy as many as they want, an "infinite" amount, but really it just means it can go on unbounded but really not infinity.

 

[Loren Booda]

What are the definitions of projective reals and extended reals?

 

[matt grime]

Projective - 1 point compactification: homeomorphic to a circle. aka RP^1 = {lines through (0,0) in R^2} = S^1/(x=-x)=S^1

Extended - 2 point compactification: homeomorphic to [0,1]

 

[Werg22]

No it isn't.

What do you mean? Of course if you were to talk about sets and other stuff, there's a tons of meanings for what equality is, but Loren's question was algebraic I believe.

Edit: One could say infinity equals infinity all he wants, but it wouldn't mean anything mathematically, let alone infinity = - infinity.

 

[NateTG]

As Hurkyl pointed out, $\infty$ is not a well-defined symbol - what it means depends heavily on context. In some situations, like: http://en.wikipedia.org/wiki/Riemann_sphere

The notion $\infty = -\infty$ might make sense, and be true.

 

[Loren Booda]

Werg22,

I attempted to keep my opening post general ("equivalent," rather than "equal," in some "regard" between the two "infinities"), although the thread title does suggest an algebraic relation. The thread spun out seems to have recounted various definitions associated with these terms.

 

[matt grime]

What do you mean? Of course if you were to talk about sets and other stuff, there's a tons of meanings for what equality is,

noooo. there's only one. Your opbviosuly nonsensical assertion was that equality is something that pertains to numbers alone. I don't have to elucidate just why that is complete rubbish do I?

but Loren's question was algebraic I believe.

and what does tha have to do with the price of fish?

Edit: One could say infinity equals infinity all he wants, but it wouldn't mean anything mathematically, let alone infinity = - infinity.

Really? So, Aleph_0+1=/=Aleph_0, or does it? Is there a meaning there?

 

[Werg22]

noooo. there's only one. Your opbviosuly nonsensical assertion was that equality is something that pertains to numbers alone. I don't have to elucidate just why that is complete rubbish do I?

I'm afraid you lost me. There's only 1 definition of equality, yet equality doesn't pertain to numbers alone? I agree that equality doesn't pertain to numbers alone, but I don't quite understand what you mean by "only one".

 

[matt grime]

Two numbers are equal if they are identical, two sets are the equal if they are identical. Two things are equal if they are equal - there is not a different notion of equality for sets as there is for numbers. Your initial assertion was that equality was the preserve of (fininte) numbers. Precisely the same notion of equality can be applied to, say, infinite cardinals (a cardinal is an equivalence class of sets, approximately, once we have got some set theoretic problems out the way), two infinite cardinals are equal if they are the same equivalence class. Two representatives from an equivalence class are merely isomorphic (unless thay happen to be the same representative when they are equal).