Infinite infinities

[Bignose]

That does not mean these functions generate different infinities. It just means they approach the point at infinity at different rates.

 

I didn't say that they were different. Just that infinity minus infinity doesn't necessarily equal zero, and I gave an example of that.

[Fred56]

But hang on...

 

In other words infinity actually has an infinite number of values, but is also actually none of them (in actuality).

 

Isn't this a self-contradiction? “Infinity can be any of an infinite number of values (the set of values is infinite) but cannot be any of these (the set of values is empty).” Is the set infinite or empty (surely it can't be both)?

[D H]

I didn't say that they were different. Just that infinity minus infinity doesn't necessarily equal zero, and I gave an example of that.

Infinity minus infinity is not zero, or anything else. It is undefined. It has no meaning.

 

DH:

I may be wrong but I was under the impression that there are only 2 "sizes" of infinities:

aleph 0: The size of the integers, primes, rationals etc...

aleph 1: The size of the real numbers, R^n, C^n etc...

 

The continuum hypothesis is unproven, but generally thought to be true. It states that there is no cardinality between aleph 0 and aleph 1.

I once asked an acquaintance of mine, who has many mathematics degrees, about sets larger than aleph 1. If I understood him, he said that he could not imagine that there were any.

 

[math]\aleph_0[/math] is, by definition, the cardinality of the integers. [math]\aleph_1[/math] is, by definition, the first infinity larger than aleph-null. The cardinality of the reals may or may not be [math]\aleph_1[/math]. The continuum hypothesis says that this is the case.

 

Have you heard of Godel's undecidability theorems? Some statements in mathematics can neither be proved or disproved. This was deemed a curiosity for some time until Paul Cohen showed that the continuum hypothesis itself is undecidable. He won the Field's Medal (mathematicians equivalent of the Nobel Prize) for this work.

 

There are indeed an uncountably infinite number of infinities. The set of all subsets of the integers maps to the reals. The same arguments (Cantor diagonalization) that show that this set is "bigger" than the integers also show that the set of all subsets of the reals is "bigger" than the reals. This process of forming the set of all subsets of an infinite set continues infinitely -- and uncountably so.

[Fred56]

Infinity minus infinity is not zero, or anything else. It is undefined. It has no meaning.

 

Isn't this a problem when canceling or reducing infinite terms (either side of an equation or ratio)? Unless mathematical or algebraic infinity behaves "normally"?

[Bignose]

Infinity minus infinity is not zero, or anything else. It is undefined. It has no meaning.

 

I know, I was showing fattyjwoods, in response to his question "well if you minus infinty by infinity isnt it zero." That it isn't zero. I'm not trying to answer every case, I'm not trying to explore all possibilities, just trying to give some simple examples that show that infinity isn't just a "regular" number.

[Mr Skeptic]

Isn't this a problem when canceling or reducing infinite terms (either side of an equation or ratio)? Unless mathematical or algebraic infinity behaves "normally"?

 

You're supposed to cancel out the terms that will become infinite before taking the limit where they do become infinite.

[Fred56]

You're supposed to cancel out the terms that will become infinite before taking the limit where they do become infinite.

Is that another way of saying that something asymptotic can be infinite (have an infinite bound)? Because the limit isn't reached, the terms have an eigenvalue and so cancel, so there is an “asymptotic” infinity?

But infinity also can have no “real” value, only an undefined one? The set isn't empty, there's just no way to select a value (no order for starters), but it still contains all possible values, or something?

 

Some more from my cerebral measure-space:

 

Infinity can be asymptotic or it can be any of an infinite number of values, none of which can be defined. Which is like saying that the set of values is the infinite set, but the set of (defined) values is the empty set. Mathematicians get around this conundrum by insisting that infinity is a process (which never ends) and so we can get arbitrarily close to infinity (in fact infinitely close) without ever assuming a definite value, but can claim that our infinite term does have a value: infinity.

 

A cosmic catch-all that can be “any value you like”, as long as you don't wish to define that value (other than as infinity). But infinity is also a sort of universal constant, with a value that is also universal, or can assume any value (except that it is assumed to be a very large value), and its inverse a very small value (an infinitisemal). Some believe that zero and infinity can be expressed as a ratio, but this seems almost a logical impossibility, since zero can not be anything (by definition). Division by an infinitely small value (one that is nearly zero) still yields infinity, but the infinite set of (arbitrarily large) values that infinity is assumed to have (without being able to select any actual value from the “value set”) are not discrete in any sense.

[sciencenoob]

Infinity and Zero are my two favourite numbers . You can do some amazing things with them and without them number theory would not be possible as we know it. In fact a lot of today's technology (computers included) could not be created (or work) without the concept of Zero.

 

Technically infinity is not a number. Only zero is a number. Infinity is a mathematical abstractation

[Fred56]

I would say that "an infinite set of arbitrarily large but non-discrete values" qualifies as a mathematical abstraction. Is that what it (infinity) is, though? Or is that just one of the "kinds" of infinity?

[uncool]

Gah...all the misunderstandings in here.

Guys, unless you are trying to work on the extended real line, infinity isn't a number. Usually, most people work on the normal real line, which does not include an infinity - only a limit to it. Even with limits to it, there is a lot of stuff you can't do.

Really, infinity is a placeholder for saying "It becomes too large for you to measure. It keeps going up, and up, and up..."

=Uncool-

[Fred56]

So it is a process then (as well as being "other" things)?

Mathematicians get around this conundrum by insisting that infinity is a process (which never ends) and so we can get arbitrarily close to infinity (in fact infinitely close) without ever assuming a definite value, but can claim that our infinite term does have a value: infinity.

 

And:

 

There is at least a philosophical barrier between real values and those that are “in” the set of infinite values. Real numbers are not in this set because they are said to “approach a limit”, which keeps them in the real number domain while allowing them arbitrary “largeness”. Infinite values are in a “domain” which is always beyond this. There are an infinity of real values that the real numbers can be, but a real value cannot itself be infinite, it only “approaches” infinity.

There are at least two sets here, then. The set of “real” values (which can be placed on some line which can extend arbitrarily), and the set of “non-real” or infinite values. The set of real values is not a subset of the second, and infinite numbers (not arbitrarily large reals) are excluded from the first.

 

Come on guys, surely I'm not completely correct here about all this? Doesn't anyone have any objections to my rant so far?

 

Can someone at least tell me if the following makes any sense?

 

Because Infinity can be all values, but no (single) value, is it like a guest in a hotel where the rooms are all full, but each contains not one single guest, so there is always a vacancy?

[D H]

Doesn't anyone have any objections to my rant so far?

Yes. A real number does not approach infinity. A real number has a fixed location on the number line.

Can someone at least tell me if the following makes any sense?

Because Infinity can be all values,

That makes no sense.

is it like a guest in a hotel where the rooms are all full, but each contains not one single guest, so there is always a vacancy?

You appear to have heard of the Hilbert hotel paradox, but your interpretation is not quite right.

[Fred56]

Because Infinity can be all values' date='

[/quote']

That makes no sense.

 

So how do you say something like "the infinite set (of values)"? How do you describe the "value" of infinity. Or isn't this possible (or logical)? Would it be better to say "infinity can be any indefinite value"? Or that "it is not denumerable"? I'm trying to avoid using terminology from set or number theory, and see if sticking to "ordinary" language runs into lots of problems, or what.

Could you expound a little? If a real number is fixed, how does it "assume" an infinite value? Although I studied calculus to 3rd year, the concept of approaching a limit wasn't covered in any great depth (perhaps because, once you understand how to use it, you don't need to keep thinking about what it really is, sort of).

[whaThehell]

[math] 0 = (1 - 1) = 1 - 1 + (\infty - \infty) = 1 + (-1 + \infty - \infty) = 1 + (\infty - \infty) = 1[/math] This would prove that 0 = 1. Do you now understand why [math]\infty - \infty[/math] is undefined rather than zero?

i spotted an error in this...1 + (-1 + infinity - infinity) = 1 + 1 + infinity + infinity since u open up the brackets...i think...correct me if im wrong...

[Bignose]

i spotted an error in this...1 + (-1 + infinity - infinity) = 1 + 1 + infinity + infinity since u open up the brackets...i think...correct me if im wrong...

 

Ok, you are wrong. Never mind the infinity adding and subtracting that is indeterminate... but why would all the terms on your right hand side become additions just from removing the parentheses?

 

5 + (6 - 7) does not in any way equal 5 + 6 + 7, which is what you posted.

[Fred56]

Infinity can be something very large, so large it is uncountable, or something very small, as small as you like, or infinitisemally small. Newton's concept of infinitisemal quantities was the key to understanding the dynamics of motion. Pythagoras' relation between the sides of a right triangle holds for very small triangles, so when approximating the slope of some curve, any triangle can be made arbitrarily small, even "infinitely" small. Once this is understood, it can be seen that as the triangle gets smaller, the hypotenuse's slope “approaches” the slope of the curve. From the triangle's point of view, the curve “approaches” a straight line. The curvature “disappears”. The process of shrinking the triangle's dimensions does not have to complete. Because we know it can be made arbitrarily small, we also know that this regression doesn't actually need to reach any “destination”, we only need to understand that it can be made as small as we like, until it approaches the “size” of some point of interest. And this arbitrary smallness will mean that the hypotenuse's slope will, at some remove, be equal to the slope of the curve at that point. The triangle has a fixed, but near infinitely small, size.

[uncool]

Fred the hypotenuse is a length. The slope is a y-value divided by an x-value. These are different units, so it's impossible for the two to be the same. Are you saying that the slope will be approached?

There is no such thing as an infinitely small triangle. A triangle always has finite size, by the definition of a triangle. There really isn't an infinitesimal quantity - just a limit to 0. Nothing more, nothing less. All infinities are is really saying "If you get really close" or "if you get really, really large...", not a limit to an actual number.

and of all the topics which have been on, so many others are the ones I see contain wisdom and knowledge. This very topic renders me so emotional that I could run out of the database memory. I simply can not comment a single thing without embarking upon a sojourn of consensus ridicularum.

 

i even see people's replies which I always am interested in seeing what they have to say but this is just too much. enjoy the posts till it becomes degenerate and the inductions and deductions mentioned from the OP's post will become redundant.

 

 

 

Sorry, just have to, cos i started reading it all instead of browsing but from bottom up i think.

 

I mean, the point of infinity... one specific point? lol , yes lol. I mean if you are speaking from a metaphysical point , then this becomes interesting. If you mention it as a valid mathematical definition, then there is just no sense in it.

 

 

(not the part about funtion behaviour, but how you worded 'the' point AT infinity).

Actually, there is such a thing as a point "at infinity" in some senses of the word. Projective geometry is the same as affine geometry, except that all parallel lines meet at a point, known as the point at infinity. Now, you are right in that there are more than one - there is a point for every direction - but there still is one. There even is a line at infinity for the plane, and a plane at infinity for projective space. This geometry is very useful because it is so much more symmetric than affine that its conclusions are much nicer.

 

In other words infinity actually has an infinite number of values, but is also actually none of them (in actuality).

 

How is it none of them?

 

Isn't this a problem when canceling or reducing infinite terms (either side of an equation or ratio)? Unless mathematical or algebraic infinity behaves "normally"?

 

You usually don't cancel or reduce infinite terms directly. Usually, as said before, it's done by reducing terms before the sequence (if it is a sequence) approaches infinity.

For example: the limit as x goes to infinity of x + 1 divided by x is 1, even though both numerator and denominator go to infinity.

 

So how do you say something like "the infinite set (of values)"? How do you describe the "value" of infinity. Or isn't this possible (or logical)? Would it be better to say "infinity can be any indefinite value"? Or that "it is not denumerable"? I'm trying to avoid using terminology from set or number theory, and see if sticking to "ordinary" language runs into lots of problems, or what.

Could you expound a little? If a real number is fixed, how does it "assume" an infinite value? Although I studied calculus to 3rd year, the concept of approaching a limit wasn't covered in any great depth (perhaps because, once you understand how to use it, you don't need to keep thinking about what it really is, sort of).

 

There are a lot of values that infinity, when you try to use it as a quantity, can be. It can be the number of integers - which also happens to be the number of rational numbers, etc. It can be the number of real numbers. It can be just about anything. However, this infinity is different from the infinity used in calculus - this is a set-theory infinity.

A real number doesn't assume an infinite value. A real variable can be increased past any bound. If it does so, then it "approaches" infinity. However, this can only be done with a variable - not an actual real number.

 

The idea of a limit is the following: Let's consider the function f(x), and how it behaves as it increases. If L is a number such that for any epsilon there exists an M such that for any x > M, |f(x) - L| is less than epsilon, then L is considered to be the limit of f(x) as x goes to infinity. This differs from the usual limit because usually there is a range of values that is finite in length, as opposed to an entire ray.

=Uncool-

[Fred56]

Thanks for all that. It seems to me that an understanding of "the infinite" is fraught with all sorts of problems and difficulties (especially language). You first need to understand such concepts as "a real value", "a variable", and "regression" and so on. The concept of "taking a limit" was explained to me, I recall, in a similar way to the explanation I used above. That the hypotenuse's slope (sorry should have specified that), approaches the slope of the curve at some point. This point doesn't have to be "reached", the triangle just has to be small enough and it is easy to "extend" this idea of "small enough" to "small as possible", or "infinitisemally small", except you don't have to get that far, just accept that it is logically possible. There are semantic difficulties when using "ordinary" language that require a more formal grammar, which is all that number and set theory I need to get my head around. My earlier posts on all this have factual errors, I see.

 

In Hilbert's hotel, are the guests all natural numbers, or have I jumped to an incorrect conclusion? What if they are all infinite series with non-finite results (values)? Or infinities?

[uncool]

Thanks for all that. It seems to me that an understanding of "the infinite" is fraught with all sorts of problems and difficulties (especially language). You first need to understand such concepts as "a real value", "a variable", and "regression" and so on. The concept of "taking a limit" was explained to me, I recall, in a similar way to the explanation I used above. That the hypotenuse's slope (sorry should have specified that), approaches the slope of the curve at some point. This point doesn't have to be "reached", the triangle just has to be small enough and it is easy to "extend" this idea of "small enough" to "small as possible", or "infinitisemally small", except you don't have to get that far, just accept that it is logically possible. There are semantic difficulties when using "ordinary" language that require a more formal grammar, which is all that number and set theory I need to get my head around. My earlier posts on all this have factual errors, I see.

 

In Hilbert's hotel, are the guests all natural numbers, or have I jumped to an incorrect conclusion? What if they are all infinite series with non-finite results (values)? Or infinities?

 

The guests are basically placeholders, while the rooms are the natural numbers. The guests represent the idea of a set isomorphism.

=Uncool-

[Fred56]

I see. The guests are all isomorphic discrete sets, and the rooms are the ordinal mapping? Or something like that?

Please do keep up the comments. This getting more interesting (as I said, I've only done a bit of 3rd yr math, the furthest I got was transforms in the s-domain, not much set theory).

[foodchain]

Well between the numbers 1 and 2 you can have infinity but of course you can get from 1 to 2 which I think why such is a pointer to infinity being treated somewhat the way it is. Sort of like that statement about constantly dividing a space as a way to move towards it invokes infinity if a I remember correctly, but of course in reality there is a finite amount of space to move, such as a couple of feet on a sidewalk.

 

You can do math problems that show someone did an infinite amount of work and I am sure somewhere in chemistry infinity has popped up:D I think more or less its a product of math and not so necessarily reality. As far as infinite infinities go, I don’t really understand what you mean. I am sure such exists in math but I don’t know the correct technical reality of such, as in nature I don’t know if infinite forms can appear from chaos say for how a snowflake may come to look for example taking into account any possible geometry or structure it could posses, but I don’t really know if those would be the same concepts, chaos theory and infinity that is. Not to say its chaos theory behind the variance of snowflake appearance, just that I don’t understand the role of infinity in chaos theory.

[lakmilis]

Golden rule: there are no infinities in nature.

wrong, nature itself is infinity, thus all subsets are finite.

 

almost but you left out the major set X, of which all subsets derive from ,)

 

well if you minus infinty by infinity isnt it zero. same goes for if you minus a billion from a billion

 

 

no, thats the whole point of infinity versus finity, a billion minus a billion is 0, infinity minus infinity, strictly speaking in mathematics, is not proven to be either 0 nor infinity but it is non in between.

 

it as someone said yields to infinity however , not in the sense its infinitely large, but rather to the notion that it remains illdefined

 

therefore, infinity minus infinity tends to stick to infinity rather to approaching 0

[Fred56]

Maybe it would be safer to say something like "infinity - infinity can be zero, if the infinities have the same cardinality (size)"?

 

Where is this guy?

 

Infinity is something that can never be reached, by definition. When Mathematicians say a value “approaches” infinity, this is understood to mean that some variable, either a real or natural number, can always increase. It doesn't “become” infinite but the concept that it is able to “approach” an infinite size is easy to understand. This concept of a value that can be arbitrarily large and approach the infinite, is the key to understanding a lot of other mathematical ideas.

 

Saying something like: “an infinite number”, or “an infinite value”, seems to run into problems: a number can not “be” infinite and infinity has no (determinable) value. We are able to conceive the infinite in terms of some process, some counting or ordering which can continue indefinitely, but we cannot conceive the infinite. This isn't a big problem, because mathematics lets us deal with it by assuming that, while it is "attainable" in some sense, it doesn't need to actually be “attained”.

 

Saying: “an infinite set is countable” sounds odd too, but it just means that there is some ordering, like a series or progression, which maps to the natural numbers. There is no such ordering available to the set of real numbers, and this set is said to be uncountable.

Saying: "infinities can not have any value, but can have size", sounds a bit paradoxical, except that "size" can't be defined any further than being larger, smaller, or equal, in terms of infinites. So "size" is a property of infinities which can only be defined as relative (to some other infinity).

[Mr Skeptic]

You could say that there are an infinite number of finite integers -- go ahead and count them if you don't believe me. But wouldn't that mean that finite=infinite?

[Fred56]

Next stop Hotel Infinity

 

A certain entity (whom shall remain nameless) decides to check in to the Golbert-Hidel for the night. As he goes to sign the register he notices it appears to be already full, and every name looks very much like his own. "No need to sign, sir" the desk clerk says, and the new guest is given a key, but as he goes up to his room, he forgets which one it is. He stops to ask a porter if he can find the room for him.

“Certainly, sir, can I have your name?”, the porter asks, at which point the guest realises he has completely forgotten it.

The porter smiles, asks the guest for his key, and turns and unlocks the nearest room.

“It's not a problem sir”, he says, “The rooms all have the same lock and they're all empty anyway”.

 

Version 1.1

A certain entity (whom shall remain nameless) decides to check in to the Golbert-Hidel for the night. The new arrival goes to the desk, to find two registers, one already apparently filled up with names, the other open at the first blank page. He turns the last page of the already filled book and sees that there are more pages also filled in, with signatures that all look very similar, but somehow vague. When he tries to look closely, the writing sort of wriggles and squirms out of his focus. Turning another page, he can see the same thing. He notices that every signature looks very much like his own. Puzzled, he asks the desk clerk where he is supposed to sign.

"No need, sir", the desk clerk says, “We know you are here already”.

“That's room number infinity for you, sir”, and the new guest is given a key with an odd-looking 8 on it, drawn sort of sideways.

“Plus one, of course, sir”, the clerk says, and the guest, with a wry look, hands over a tip.

“Have a pleasant stay, sir”, the desk clerk says.

As he goes up to his room, he realises he has completely forgotten which one it was the clerk told him. He stops to ask a porter if he can find the room for him.

“Certainly, sir, if I might have your name?”, the porter asks, at which point the guest realises he has absolutely no idea.The porter smiles, asks the guest for his key, and turns and unlocks the nearest room.

“It's not a problem sir”, he says, “The rooms all use the same key and they're all empty anyway”.