Are There Different Values for Infinity?

[Hessam]

Some of my classmates were mentioning today that there are "different" values for infinity, like there is more than one infinity... can someone please help me? explain exactly what that means and what these values are please

 

[DeadWolfe]

Infinity doesn't exist.

There are infinite series which diverge, and have different values at finite points, but they are not equal to infinity and any point.

For instance, the sum or all whole numbers, and the sum of all even numbers.

The second one will grow faster, but it is not a "larger infinity".

Neither is infinity.

 

[Bob3141592]

Originally posted by Hessam
Some of my classmates were mentioning today that there are "different" values for infinity, like there is more than one infinity... can someone please help me? explain exactly what that means and what these values are please

I suspect they're talking about the cardinal numbers developed by Cantor. Basically, one value of infinity is the "countable infinity" that is, the number of positive integers. As the previous poster mentioned, that's also the same as the number of even positive integers, even though it might seem there should be twice as many positive integers as even positive integers. Surprisingly, it's also the number of rational numbers, positive and negative. These are all countable infinities.

But when you get to all the numbers, including the irrationals, then you have a set where you cannot simply count all it's elements. It's also a line, as in the number line. These are different degrees of infinity. That is to say, there is no way to put all of the numbers on a line into a one to one corrospondence with the integers. Again, perhaps surprisingly, the number of points on a short line segment is the same as the number of points on a long line segment, or even an infinitely long ray, and also all the points on a plane, or in a 3-D space, or even in an n-D space, as long as n is finite.

The cardinal numbers are denoted by the firstHebrew letter, aleph. The countable infinity is also called aleph sub zero, while the infinity of the continuum, the number of points on the line, is called aleph sub one.

What might aleph sub two be? It would be the set of all possible curves through space, or the set of all the permutations by which the points on a line could be folded or jumbled up. That would also be all of the possible histories of all the possible universes that might be (or even just of our one that is, since they'd all have the same infinite size). (Disclaimer for physicists: I'm discounting quantizing effects and assumming the universe is infinite and unbounded).

There are in fact an infinite number of such infinities, though I've never heard of an analogy for what aleph sub three might be. Cantor might have had a notion of it, but surely he was insane by then. I try not to think about it; it hurts my head too much.

It can be shown that adding infinities doesn't change their order, and neither does multiplying them. To go from one infinity to another you have to raise two (or larger) to the prior infinity, and that will take you to the next. It's all tied into set theory, which shows that the number of subsets of a set of n elements is 2^n. The number or subsets of an infinite set is a higher order of infinity than the original set.

At the turn of the last century (1900 thereabouts) there was a huge debate about whether there could be infinities in between other infinities. That is, is there a distinct infinity between the countable infinity and the infinity of the continuum? As it turns out, the answer is yes or no. Or maybe that should read "yes and no."

Rather than go into that, I'd recommend the excellent book Mathematics: The Loss of Certainty by Morris Kline. It's well worth the effort of a careful reading.

 

[matt grime]

It is easeier and less confusin to say Aleph_0 is not infinity.

Infinity is an abstract noun, coming from the adjective infinite. It causes lots confusion, sadly.

Aleph_0 is the cardinality of the set of natural numbers. It is often then termed the 'size' of the natural numbers which causes people serious issues, because they don't define 'size'.

So, two sets have the same cardinality (size) if there is a 1-1 correspondence between their elements.

N and Z have the same cardinality:

for z in Z, z >0 send z to 2z, for z<0 send z to -2z-1 and send 0 to 0. This gives a 1-1 correspondence between Z and N.

N and Q have the same cardinality.

if S_i is a family of finite sets and i in N is the index, then the union over all i of S_i has the same cardinality as N.

For a short hand we declare that anything that has the same cardinality as N, has cardinality aleph_0. But this is not a substitute for the infinity symbol you use in sums and integrals.

The question now ought to be: are there infinite sets which do not have cardinality aleph_0.

The answer is yes: P(N), the power set of N cannot be put in correspondence with N (and be careful cos there are some cranks out there to whom that is as a red rag to a bull).

Proof: suppose f:N -->P(N) is any injection, define S to be the set of t in N such that t is not in f(t).
Exercise, consider if t is in S or not and derive contradictions iIt is easeier and less confusin to say Aleph_0 is not infinity. So we need another symbol for all the sets of the same cardinality as P(N). It so happens that the real numbers have the same cardinality as P(N), and they have the cardinality of the continuum, labelled C, or 2^aleph_0, in analogy with the fact that there are 2^n elements in the power set of a finite set with n elements.IMPORTANT: c is not exactly Aleph_1 despite what has been said above. The statement c=aleph_1 is the continuum hypothesis whose truth is dependent on the set theory being used. Something Bob alludes to.So what is aleph_1, aleph_2 etc?

We denote by |S| the cardinality of S. We say |S|<=|T| if there is an injection from S to T. With this ordering Aleph_1 is the smallest cardinal greater than aleph_0. Then aleph_2 is the one after that etc. There is something very deep going on there: how do we know that we can totally order cardinals like this? That we can is to do with the axiom of choice.

I'm sorry to say that Bob's post also contains some other errors mathematically, but that would require me to explain regular cardinals, which is beyond the scope of this post. As for the number of curves through space etc, well, that is more than overly dependent on the continuum hypothesis. No commment on the physics of history.

 

[cookiemonster]

Just out of curiosity, what math class(es) would cover these kinds of topics?

cookiemonster

 

[matt grime]

Look for courses with titles involving the words set theory or discrete mathematics

 

[cookiemonster]

Oh good, now I'll know what to stay away from. ;)

That stuff is scary! Think I'll stick to physics...

cookiemonster

 

[BasketDaN]

Positive and negative infinity both exist, but nothing else. You cannot have one more than infinity, or half infinity.

 

[matt grime]

Originally posted by BasketDaN
Positive and negative infinity both exist, but nothing else. You cannot have one more than infinity, or half infinity.

That is either wrong, or ill-informed. Sorry to be blunt.

Infinite is a well understood adjective; not finite. Infinity is the abstract noun we associate to it, usually with bad consequences such as above. To say that both negative and positive infinity exist would require you to state what you mean by infinity and exist, and is in any case not related to the concept of cardinality. If you put these vague notions onto a firm footing then N the natural numbers and its power set are a 'different' size in the sense that there is no bijection between them. They are infinite sets, to say *the* cardinality of either *is* infinity is not correct. Aleph-0 is not infinity, it is not the infinity in the sum from one to infinity, which just denotes that we do not terminate the sum at a finite point.

 

[BasketDaN]

It's really not that complicated at all. Infinity is a very commonly used term in calculus to describe a specific quality. The two differnet forms of this are positive and negative infinity. For instance, the Limit as n-->0+ of 1/(n) is inifnity. And the limit as n-->0 of ln(n) is negative infinity.

 

[DeadWolfe]

Originally posted by BasketDaN
It's really not that complicated at all. Infinity is a very commonly used term in calculus to describe a specific quality. The two differnet forms of this are positive and negative infinity. For instance, the Limit as n-->0+ of 1/(n) is inifnity. And the limit as n-->0 of ln(n) is negative infinity.

As I said, it's not infinity, it's a divergent value.

Not the same thing.

 

[Hurkyl]

You're both right (or both wrong, depending on how you want to look at it!)


There is a topological space called the "extended real numbers" which consists of the real numbers plus two points we call -&infin; and +&infin;. In this topological space,

$$\lim_{x \rightarrow 0^+} 1/x = +\infty$$

is a convergent limit.


If you are merely working over the real numbers, however, then this is a divergent limit.


There is another standard thing called the real projective line, which consists of the real numbers with only one additional point which we call &infin;. In this space,

$$\lim_{x \rightarrow 0} 1/x = \infty$$

is a convergent limit, whereas this limit does not exist over the reals, or the extended reals.


You don't have to stop at the "end" of the real line either; for instance, there is a horrendous 1-dimensional topological space called the long line which is so long that you can't even define a distance function on it!

For comparison, you can define a distance function on the extended reals by:

$$d(x, y) = \left| \frac{x}{|x|+1} - \frac{y}{|y|+1} \right|$$

Where we define +&infin; / (|+&infin;| + 1) := 1, and similarly for -&infin;.


There are other, algebraic structures with nonfinite values. For instance, there is the surreal number &omega;, and things like &omega; - 1 and &omega; / 2 are distinct surreal numbers. The hyperreals also have nonfinite values in a similar way.


And, as mentioned, these have nothing to do with infinite cardinalities, such as the size of N or R. (Well, actually, the surreal numbers are somewhat related to them)

 

[DeadWolfe]

I kind of assumed we were working with the Real numbers.

Sorry.

 

[DrMatrix]

There are lots of ways to define infinity. In standard calculus, infinity is not a number -- it is a formal symbol. $$\lim_{x \rightarrow 0^+} 1/x = +\infty$$ is not a relationship between two numbers -- It is a statement that the limit does not exist.

We also have Cantor's cardinal numbers and his ordinal numbers that others have touched on. I'd like to make a minor correction to what Bob3141592 said. Aleph-1 is the cardinality of the reals only if we assume Cantor's continuum hypothesis. Without assuming CH, all we know is that the cardinality of the reals is greater than or equal to aleph-1.

Non-standard analysis defines infinities and infinitesimals.

There are other systems that define infinity. John Conway devised a system he calls surreal numbers where we have infinities, we can take half of aleph-null. We can take square roots of aleph-1. Basically treat infinities and infinitesimals as ordinary numbers similar to non-standard analysis.

In all the systems I've mentioned except standard calculus, infinity or rather infinities are numbers.

 

[phoenixthoth]

and then there is the absolute infinity which is bigger than all other infinities. look up quine's (sp?) new foundations and/or see http://diamond.boisestate.edu/~holmes/holmes/setbiblio.html for a list of references. yours truly is also constructing what may be a simpler and less dependent on esoteric stuff mathematical treatment of the absolutely infinite. see the recent posts under "the search for absolute infinity" thread for a link to the current version of that article. contains a brief review of the three valued logic preliminaries necessary to quickly dispose (ie circumvent) russell's paradox which, in binary logic standard set theory, prohibits a maximally infinite set...

 

[Nizzy]

I have a question, because there is 'no end' to the possibilities of real number values, we can say that the x-axis (of a graph) has infinite points, right?

Now say we add the y-axis. As the y-axis is also a line, it has infinite point values as well right? So it would be infinity; and infinity times infinity is infinity squared. But because infinity squared is still considered an infinite value, would the value of both an x-axis' and a y-axis' infinity be greater than just that of an x-axis'?

If that's the case, would adding a z-axis provide a greater value of infinity than the previous two equations?

Ah, thank you for helping!

 

[HallsofIvy]

I have a question, because there is 'no end' to the possibilities of real number values, we can say that the x-axis (of a graph) has infinite points, right?

Now say we add the y-axis. As the y-axis is also a line, it has infinite point values as well right? So it would be infinity; and infinity times infinity is infinity squared. But because infinity squared is still considered an infinite value, would the value of both an x-axis' and a y-axis' infinity be greater than just that of an x-axis'?

If that's the case, would adding a z-axis provide a greater value of infinity than the previous two equations?

Ah, thank you for helping!

Did you read the previous responses in this thread? You can't just say "infinity"- that word applies to a number of different concepts. If you are talking about "cardinality" then, yes, the x-axis contains an infinite number of points- more precisely "c" points. In fact, any interval on the x-axis, say from x= 0 to x= 1 also has "c" points. But the x and y axes together have "c" points (I would not call that "infinity squared". I would call it "infinity plus infinity" if I were casual enough to write things like that. The number of points in the plane might be considered "infinity times infinity" but that is still "c" points. The x, y, and z axes together have cardinality "c" and the set of all points in 3 dimensional space has cardinality "c".