# Infinity

in·fi·nite

1. Having no boundaries or limits.
2. Immeasurably great or large; boundless: infinite patience; a discovery of infinite importance.

Isn’t infinity all relative? For example, there are infinite numbers between 1 and 2, but we can still count pass 2. There are infinite numbers between point A and point B, yet we still can arrive at point B. Why? Well for an ant to walk 250 miles (A to B) it would take nearly forever, but for a person in a car it doesn’t take too long, a person in a jet it takes even shorter time. What we see as infinitely long (the universe) very well could be a foot step if we were monstrously big, right? So, what about monstrously small? Could “intelligent” life very well be living under our skin, smaller than an electron, do you think it would mirror existence of our life. What if all our actions were a mirror on another “dimension” being that infinity may be a loop and repeats itself and that of which is infinitely small and is in all of us is us, thus affecting the stability of every atom, as well as the whole universe. What do you think? I’m not sure if this makes sense, maybe you can help with this.

See this thread for a little background material.

Basically, the everyday use of the word “infinity” is completely different from the formal use. A set is infinite if it has a proper subset (i.e., one which does not contain every member of the set) which has the same number of members. See the thread I referenced above for more exposition.

A truly infinitely long* line would be infinite on any scale–size is irrelevant. While it’s possibly that arbitrarily small objects could exist, there probably can’t be intelligent life on a sufficiently small scale, at least with the physics we’re used to.

btw, “Having no boundaries or limits” is a poor definition for infinite. While the surface of a sphere has neither boundaries nor limits, no one would claim that it’s infinite in any sense other than the number of points it contains.

*I’ll explain what this means later, if no one else does first.

Man, put that pipe down.

Nope.

We cheat. When I count “1, 2, 3…”, I’m skipping over the billuns and billuns of real numbers between 1 and 2, between 2 and 3, and so forth.

No, because the definition of infinity means that it stays infinity no matter how big/fast/tall you are. What you’re describing is just Really Big But Not Infinitely So.

Well, Stephen Hawkings’ A Brief History of Time includes speculation that time itself is a repeating loop, so there’s some possibility that you may be right. But we don’t know enough about the universe (yet) to say whether or not you are.

Life smaller than an electron? Well, then, it’s momentum is pretty much all over the map. There goes any chance at an ordered system, which is what we think is a basic requirement of life. You have a problem with that?

ok, thanks for clearing me up everyone.
On a side note, i heard that some scientists believe they’ve “seen” the end of the universe, has anyone ever heard of this and how they base that?

In math class the teacher said, it starts at (3,0) and then shoots off into infinity, that left me wondering, how can something have a beginning and be infinite? but i suppose it makes a little more sense now. After that my mind started to wander.

I’m gonna blame this one on them.
http://www.dictionary.com/search?q=infinite

Infinity in math can be murky. bhb, are you aware of the difference between aleph-null and aleph-one sized infinities?

I was speculating about infinity, and I came up with something that’s either out of a pipe, or else could actually be profound. I have no math education beyond high school, so I couldn’t either shoot it down or show it to be completely consistant, so I offer it here:

I was thinking about the classic conic sections- ellipses, parabolas and hyperbolas. An ellipse has two “foci”, points that define the ellipse. As the foci get farther away from each other, the ellipse becomes more and more elongated, and the curve at either end of the ellipse gets closer to the curve of a parabola. Finally, you get a parabola, which could be considered an ellipse where the foci are infinitely far apart (although at that point, you have to define the curve of the parabola using a single focus and a straight line).

Now, if your curve gets even wider it’s no longer a parabola, it’s a hyperbola, which again is defined from two foci except that the second focus is now in back of the first, and the definition yields two mirror-image curves, one centered on each focus. Could it be that this second curve is actually the end of an ellipse? That a hyperbola could be considered an ellipse who’s foci are a greater than infinite distance apart. That in effect the classic number line of integers “loops around”, so that it’s really a circle? This would mean that infinity equals negative infinity.

Crazy I know, but it seems an interesting analogy. And there is a point to this speculation, which is that often times infinity pops up in equations in ways that can’t be dealt with. If switching the value sign on infinity would make an equation solvable, or you could “redefine” infinity in the same way that an infinitely eccentric ellipse is redefined as a parabola, then maybe this would be a useful mathematical tool.

Sorry, Lumpy, but your ideas are closer to being out of a pipe than being profound. Still, they’re interesting, and I had to think for a moment as to how to respond.

Basically, you seem to be treating infinity just like any other number, which is bad, cause infinity’s not a number. There’s no such thing as being more than an infinite distance from another thing, cause you can just pick something further out that’s infinitely far away.

And there’s no geometric or analytic consideration that’ll allow you to consider a parabola as an ellipse with infinitely distant foci. Recall that an ellipse is the set of points [symbol]a[/symbol] such that d([symbol]a[/symbol], f[sub]1[/sub]) + d([symbol]a[/symbol], f[sub]2[/sub]) = k, a constant (d is euclidean distance, and f[sub]1[/sub] and f[sub]2[/sub] are the foci). Put in infinity for k, and there’s no [symbol]a[/symbol] that satisfies that equation.

You might want to look into the polar form of the equation for a conic section, as you get one equation that describes any conic section (with a suitable choice of parameters, of course).

Even if the universe is finite, infinity isn’t. Infinity never ends. It might just be a concept and not reality. But that’s what it is.

Depends on what you mean by “never ends”. Are you OK with the fact that the number of points on the surface of a sphere is infinite, even though it has a finite surface area?

ultrafilter, the two recognized sizes of infinity do show that one can be “farther away” from one infinity… in fact, that one can be infinitely farther away. No?

Also, there are “eccentricity” formulas for all conic sections in cartesian coordinates, if one wants to avoid the polar method. Six of one, half a dozen of another, though.

No, actually. The infinity used in measuring distances has nothing to do with the infinity used in measuring the size of a set. Neither has anything to do with the infinities that show up in limits, or in the length/area/volume of a shape. In fact, there is no single concept of “infinity” that’s meaningful in all contexts.

I don’t know. Certainly the way metrics are conventionally defined, no two points are infinitly far apart. But I think the notion of distance could be generalized to handle infinite distances. For example, the class of ordinals can be thought of as a very long line, and the distance between two ordinals can be measured on it by subtracting the smaller from the larger (that is, by taking as the distance the order type of the complement of the smaller ordinal in the larger).

Another aproach might be to generalize the notion of measure. As I’ve seen it, the definition of measure is built around the notion of countably infinite sums, but I wonder if this could be replaced with higher orders of infinity. I know that infinite sums (series) can be defined on uncountably many terms, so maybe a theory of integration could be built on this.

Regarding infinity being infinite at any scale: of course this is true, as posters have said. I think it is interesting, though, that if you use a non-constant scale, then you can “fit” an infinite amount of space in a finite amount of space. Consider, for example, the Riemann Sphere (pdf file), or more simply, the diffeomorphism between the real line and the interval (0,1).

Haven’t seen anything like that. I always thought that points were allowed to be infinitely far apart in a metric, but you may have some knowledge I don’t. Any links on sums of uncountably many terms?

I suppose it depends on how you define metric. I’m thinking of the definition in e.g. Rudin’s Principles of Analysis (which I think I actually remember you quoting in some thread somewhere, so I won’t bother to quote it back at you). In that definition, the metric d is defined as a function taking ordered pairs of a set into the nonnegative reals (and satisfying some other properties), so, by that definition, no two points will be infinitely far apart. The distance between any two points will always be some real number.

Of course, I’m sure that there are other definitions. For example, I believe physicists define “metric” in a way which allows distinct points to have zero distance between them, which isn’t allowed under Rudin’s definition.

The definition with which I am familiar is from Applied Analysis, by John Hunter and Bruno Nachtergaele, available online here. The definition is on page 136 in chapter 6.

This definition is valid for summing terms in Banach spaces, which are vector spaces on which a metric has been defined, with respect to which the Cauchy convergence of a sequence implies the convergence of that sequence (ie, complete normed vector spaces).

Oh, and for those interested in how infinity is treated in mathematics, I think a good accessible presentation is Rudy Rucker’s Infinity and the Mind. See also this recent thread, in which I made the same book recommendation (though inadvertantly under a sock, since I was using someone else’s computer).

In the metrics used in relativity, two distinct points can, indeed, have zero distance between them, referred to as a “null separation”. Note, incidentally, that this is not transitive: Distance from A to B = 0 and distance from B to C = 0 does not necessarily imply that distance from A to C = 0. On the other hand, unless there’s something in the “and satisfying some other properties” that precludes it, I don’t see how this is contary to Rudin’s definition: Zero is a perfectly valid nonnegative real.

Rudin’s definition requires that the distance between two points be positive if they are distinct.

For mathematician’s, metrics have to satisfy the “triangle inequality”, which states that, for any points A, B, and C, the distance from A to C is less than or equal to the sum of the distance from A to B and the distance from B to C. From what you say above, it sounds like the physicist metric need not satisfy this, at least not when two of the points involved have a null seperation between them. Can the triangle inequality also not hold in cases not involving null seperations?

As ultafilter has indicated, by Rudin’s definition, if d is a metric and x and y are points in a metric space, d(x,y) = 0 if and only if x=y.

I probably shouldn’t speak of mathematician vs. physicist definitions, though. I suppose that mathematicians studying Riemannian manifolds use the same definition physicists use. But I’m less familiar with that usage.