What is the Definition of Infinity?

[zeithief]

can i ask how do we define infinity? Is it a number that cannot be represented mathematically(like something divided by 0) or is it a number too large that our human mind can't think of? what do statements like 'The universe is infinite' mean? Is it too large to be represented or there are equations that we can use to calculate the size of the universe but the solution is something like a/0 or sqrt of a negative number?

 

[whozum]

The square root of a negative number is defined using the complex parameter i. Infinity isn't a number, rather a term described to use endless growth, or something without bounds. The two arent related.

 

[wisredz]

That's not a number that you can write down. It is something undefined, something that you cannot push any further. To understand this, graph the function 1/x. then look where the values of trhe function go as x approaches 0 from right. The graph goes up and up and up... It doesn't stop. Now, how many zeros you have add to get 1? Infinite. This is not a very good explanation and I know that but just try to understand that it is something undefined.

 

[quetzalcoatl9]

the idea of something being unbounded is important.

think of the surface of a sphere. the area is finite, yet unbounded...a 2-D creature living on such a surface would say that his universe is "infinite".

Galileo made an interesting observation:

1 2 3 4 5 6 ...
1 4 9 16 25 36...(the squares of the row above)

which set is bigger? it would seem like the set of squares would have fewer elements, since there are "gaps" and therefore might seem like a subset of the set of natural numbers. BUT, the two sets can clearly be put into a 1-1 correspondence. Hmm..

The concept of infinity gets more interesting. Do a google search on "cardinality" and infinite sets.

 

[εllipse]

think of the surface of a sphere. the area is finite, yet unbounded...a 2-D creature living on such a surface would say that his universe is "infinite".

This is incorrect. Cosmologists are currently in the works of trying to figure out if our universe is infinite or not. If our universe turns out to be spherically shaped, we would call that a closed and finite universe. Unless these 2-D creatures use the terms "infinite" or "finite" differently from us, they would conclude the same thing about their universe.

In fact, here's a quote from Albert Einstein that shows what the difference in the terms "unbound" and "infinite" is, and it even uses the same context of a spherically-shaped, two-dimensional world:

...Let us consider now a second two-dimensional existence, but this time on a spherical surface instead of a plane. The flat beings with their measuring-rods and other objects fit exactly on this surface and they are unable to leave it. Their whole universe of observation extends exclusively over the surface of the sphere. Are these beings able to regard the geometry of their universe as being plane geometry and their rods withal as the realisation of "distance"? They cannot do this. For if they attempt to realize a straight line, they will obtain a curve, which we "three-dimensional beings" designate as a great circle, i.e. a self contained line of definite finite length, which can be measured up by means of a measuring rod. Similarly, this universe has a finite area that can be compared with the area of a square constructed with rods. The great charm resulting from this consideration lies in the recognition of the fact that the universe of these beings is finite and yet has no limits.

In a spherical universe, there are simple tests can be carried out by the two-dimensional beings that would allow them to discover the finiteness of their universe. They would still, however, call it unbounded.

As to the question of what infinity is, I am not an authority on the subject, but my personal oppinion is that infinity should be thought of as a number, not a boundless variable. The idea that infinity is something "without bounds" implies that the first instant after you say "infinity" it means "100", a second later it means "1000", a second later "10000", and so on; as if infinity grows. Infinity does not grow, and in my oppinion can be defined as the value 1/0 without resulting to limits.

 

[matt grime]

The point about unbounded was to explain what a mathematician means when the use the word infinity, nothing to do with physics, an infinite series or sum to infinity is one that does not stop after a finite number of operations; that pretty much is all the term infinity denotes in mathematics. it causes far more problems than it should because people want to think physically.

 

[Icebreaker]

"Greater than any assignable number."

 

[mathwonk]

the approach by georg cantor was to quantify infinite counting as follows:

1) define a "bijection" between any two sets as a function f:S-->T which is both injective and surjective, i.e. such that for every point t in T there is excatly one point s in S with f(s) = t.

Intuitively there is a way to match up the points of S with the points of T, such that every point of T is matched with exactly one point of S and vice versa.

2) say that two sets have the same number of points if there exists a bijection between them,

3) say that a set is infinite if there is a bijection between it and some proper subset of itself.


example: the set S of integers is infinite because, if T is the proper subset of even integers, the map f:S-->T taking f(n) = 2n, is a bijection.

it also follows that there are different sizes of infinite sets, because cantor proved there can be no bijection between the set of integers and the set of real numbers.

 

[Hurkyl]

The first thing to realize is this:

"infinity" is a noun.
"infinite" is an adjective.

I think one of the problems the layman has is that whenever people talk about things that are "infinite", the layman mentally substitutes "infinity" and gets all confused.

This is unfortunate, because "infinite" is much more common than "infinity".


The only common use of infinity in mathematics is topological. One often likes to construct new topological spaces from old topological spaces by adding additional points. For example, suppose you're working with the open interval, (0, 1). One day, you grow weary of the fact your topological space has "open ends", so you decide to add two additional points to your space, giving you the closed interval [0, 1].

We can do a similar thing with the real line. It has two "open ends". So, as we sometimes like to do, we can add two "endpoints" to the real line. (In fact, in topological terms, this process is virtually the same as adding the endpoints to the aforementioned open interval) Because these points are "farther away" from zero than any other point we originally had, mathematicians opted to call them "points at infinity". In particular, we name them -∞ and +∞. Topologically speaking, we have this perfect analogy:

0 and 1 are to (0, 1) as -∞ and +∞ are to the real line


Incidentally, sometimes we only want to add a single point, but it's at both ends of the real line. We can intuit that we looped the real line into a "circle", because we "go off" towards infinity in one direction, and "come back" from the other side. This intuition is reasonable, because this situation is perfectly analogous to starting with a circle missing a point, then adding that point back. In this case, the "one-point compactification" of our topological space, we simply call the point ∞.


(Incidentally, there are reasonable ways to add much more than two points to the real line, but it becomes more complicated to describe)

This becomes more involved in higher dimensions. There are several reasonable ways to add "points at infinity" to the Euclidean plane. For example, we can add a "circle at infinity" to turn the plane into something that acts like a disk. We can add a "line at infinity" to get something extremely geometrically useful, called the projective plane. We can also add a single point at infinity, which makes it just like the surface of a sphere.

(And we can do other, much uglier, things)



Enough about the noun, let's get to the adjective. Infinite simply means "not finite".

In one context, (we use "infinite" in many circumstances) we would say "the universe is finite" would mean that there is an upper bound to the "distance" between two points.

So, saying "the universe is infinite" would mean that there is no upper bound to the "distance" between two points.

 

[mathwonk]

well you force me to mention a point of view i dislike, russells tautological definition of numbers.

i.e. three is sort of an adjective ro adverb also, but russell or frege or someone made it a noun by defining "three" as the class of all sets which are in bijection with the set:

{1,2,3}.


similarly: if you want to define "infinity" as a number you could say it is the clas of all sets which are bijectively equivalent to the integers.

then the proof that the reasl have more elemnts than the integers introduces the inconvenmient fact that there are mroe than one number deserving this name.

so it is usual instead of "infinity" to call this number assigned the integers by the name "aleph null", or "first infinity", and make up new ones for some other infinite numbers.

of course hurkyl knows all this better than i, but apparently has a different perspective.