I was reading an article, multiverse, in which the author mentions infinity squared. I’ve encountered this before, and it would appear that some math geeks believe that infinity and infinity squared can indeed exist seperately.

How so?

Infinity squared equals infinity. Just like zero times zero equals zero. I wish math questions were usually as user friendly as that.

Well I agree, **Not In Anger**. But I’ve read where some think differently. I guess I *could* find a cite.

BTW; One times one equals one. One is closer to infinity than to zero.

Boring so far.

I believe infinity squared is a separate entity from infinity because of the fact that it can be either the result of multiplying infinity * infinity or -infinity*-infinity. Since infinity and negative infinity both exist separately, then infinity ^ 2 must be different. Just a guess/hazy memory of what a professor once said.

As cardinal numbers, they are the same, but as ordinals, they are different.

As you may know, there are different orders of infinity. For simplicity, I’ll just take omega (or aleph-naught, as some might call it), the set of natural numbers: {0, 1, 2, 3,…}

omega[sup]2[/sup] is different from omega–think of it as omega x omega, or omega copies of omega. It looks like this:

{0, 1, 2, 3,…

omega, omega + 1, omega + 2, omega + 3,…

omega+omega, omega+omega+1, omega+omega+2, omega+omega+3,…

omega+omega+omega, omega+omega+omega+1,…

.

.

}

In other words, picture a “copy” of the natural numbers. Next, picture another copy of the natural numbers after *that*. And another after *that*, and another after *that*,… So that, ultimately, you have as many copies of the natural numbers, one after the other, as there were natural numbers *themselves* in the first place.

That is the ordinal known as omega[sup]2[/sup].

There exists a bijection between the Cantor set, which has outer measure zero, and the Cartesian product of countably infinite copies of the real line, each of which has outer measure infinity. This bijection, sometimes called the Peano space-filling curve, demonstrates that the two sets are in some sense the same size. Both have the cardinality of the continuum, c.

Compared to this craziness, infinity^2 = infinity is considerably easier to wrap your mind around.

Here’s one idea.

So jazz it up a little.

**amore ac studio**, I thought a space filling curve was a continuous map from the reals onto the plane (or some higher dimension).

As for the OP, in addition to what I said above, I suppose it’s also possible that the writer confused infinity^2 with 2^infinity.

For any cardinal number (including infinite) x, 2[sup]x[/sup] is always strictly larger than x itself. For example, omega (the natural numbers as I mentioned above). If you take 2[sup]omega[/sup] (which can be described as the collection of *all possible subsets* of the natural numbers), that’s a strictly larger infinity than omega itself. 2[sup]omega[/sup] is also the cardinality of the reals (so the “size” of the set of reals is strictly larger than the “size” of the set of natural numbers).

Unfortunately, we have *no idea* how much larger 2[sup]omega[/sup] (size of the reals) is than omega (size of the naturals). 2[sup]omega[/sup] could be the next higher cardinal than omega, known as omega-1 (this is known as the Continuum Hypothesis), or it could be *much* larger. In fact, it’s consistent with standard (ZFC) set theory for 2[sup]omega[/sup] to be *any* cardinal larger than omega, so long as that cardinal has cofinality omega.

That’s the usual definition, which is why I said the bijection from the Cantor set to R[sup]infinity[/sup] was only *sometimes* called the space-filling curve. You’ll note that a bijection from the Cantor set to R[sup]n[/sup] can be constructed via the composition of the Devil’s staircase (which is continuous and monotone increasing) with a bijection from [0,1] to R, followed by the usual space-filling curve for R[sup]n[/sup]. The construction of a bijection from the Cantor set to R[sup]infinity[/sup] is roughly analogous.

From the Introduction of the page that **mangeorge** cited:

I’m not about to read Chapter 3 at this hour, and I don’t know whether this is just crackpottery or not, but from this paragraph it is clear that his Infinite Mathematics is distinctly not the infinite math that Cantor defined.

There are several ways to extend the arithmetic operations to apply to infinite numbers. **Cabbage** has already discussed Cantor’s cardinal and ordinal numbers (where infinite numbers are called transfinite). Cantor’s cardinal and ordinal numbers are discrete. John Conway’s surreal numbers are continuous. You can do just about any operation on his infinite numbers that you can do on finite numbers – take square roots, subtract them, divide by them. Any surreal number can be squared, and if the number is greater than 1, the square will be greater than the number.

The author of the page **mangeorge** cited mentions in the first few paragraphs that it’s obvious that 1/infinity = 0. I’m going to look it over tomorrow, but I’m already a bit cautious.

Frankly, it looks like the author of the page is either:

[ul]

[li]Not using the concept of infinity the way we have used it in all previous threads here, and the way I’ve always seen it used in the serious texts, or[/li][li]Completely incompetent at the math he professes to teach to others and is making it up as he goes along.[/li][/ul]It’s a pity that more of the site isn’t up yet, but when people begin to say that they’ve developed a whole new branch of math, I begin to wonder about how many wheels they have on the ground.

Quote of interest:

Eh, very few things are as significant to math as the Calculus. *Damned* few, in fact. Makes one wonder why I’ve not heard of it, or of these people, before.

Anyway, let’s take a little example from the site:

As he says (and has been quoted in this thread), normal mathematical operators work on infinity.

He also states that 1/inf = 0.

Therefore, let’s do a little algebra.

inf * 1/inf = 0 * inf

1 = 0

Ah, a little inconsistency! Unless he wants to retract one or more of the above assumptions he’s made, he’s fallen into a fairly serious trap.

I’m not a mathematician but it seems to me that anything other than zero *times* infinity = infinity.

p.s. I only entered this thread because the board lied and told me it had zero posts. I wanted to be the first with a witty remark after a ham(p)ster eaten OP.

Derleth, **zero * infinity** is not necessarily zero. It is usually undefined (or indeterminate).

That is precisely how integral calculus works. Integral calculus shows instances where you can define it. Or more to the point how you can calculate it.

Take for example determining the area under a curve. Calculus takes the area and breaks it up into an infinite number of parts, each with zero thickness (and therefore 0 area).

Totaling up all the areas you get :

0 + 0 + 0 + 0 + 0 + … (an infinite number of times or [0 * infinity] ) = the area under the curve. Depending on the curve, that area can be zero, +/- infinity, or any number in between.

a visual representation:

http://hyperphysics.phy-astr.gsu.edu/hbase/integ.html

Those infinite mathemtatic people look like cranks (or jokers, or overenthusiastic high-schoolers) to me. Their method is just defining a symbol oo equal to 1/0, and, if I read the site correctly, assuming that it can then obey all the normal arithmetic rules, which paints them into a corner immediately:

1/oo=0=(1/oo)/2=(1/2oo) rearranging gets 2oo=oo, which they explicitely say it doesn’t (alternatively, they say you can cancel oo, so 2=1)

There *are* number systems that have an infinity obeying normal rules of arithmetic - see hyperreal numbers and surreal numbers - but they have to have infinitessimals (numbers greater than zero, but less than any positive real number) as well. The site gropes at the edges of this by saying they use “a positive and a negative zero.”

It’s even possible to use infinitessimals in calculus instead of limits, though it was ages before anyone did it rigorously, despite everyone thinking that way

I’m not a mathematician either (I’m dabbling in a bit of complex analysis at university, but that’s just for fun). But from what I understand, the symbol “infinity” is never actually defined in such a way that makes, for example, “2*inf = inf” any more meaningful than “k) + :mad: -& = / )(((()(” (at least not in the more ‘traditional’ branches of mathematics).

However, “infinity” can be *part* of another ‘symbol’. In this case, an appropriate ‘symbol’ would be the limit. So we could say something like “lim(x->inf) 2x = inf”, which expresses the same idea as “2*inf = inf”, except that it’s nice, meaningful, and well-defined (so long as we make certain obvious assumptions, like x being a real number). As an added bonus, it’s also true!

Try not to think of “lim(x->inf) 2x = inf” as an equation, despite the equals sign. In particular, don’t try anything silly like rewriting it as “(lim(x->inf) 2x)/inf = 0”, or we’ll be back where we started. Another way of writing “lim(x->inf) 2x = inf” is “As x->inf, 2x->inf”, which doesn’t have an equals sign. This way is probably nicer, but it won’t impress the girls/guys/whatever. The formal definition (and proof) involves more epsilons and deltas than is appropriate for my first post to this message board :). But basically it says that “as a variable x increases without bound, 2x also increases without bound.” Note the absence of any mention of equality.

A great thing about using this limit notation is that it doesn’t foul up the rules of arithmetic any more than necessary. If infinity was a symbol we could plug into equations willy-nilly, we couln’t even be sure that obvious statements like “x = x” are true anymore!! Granted, mathematics is already quite dirty with all the problems that zero causes, but why make things worse than they already are?

So yeah, basically you’re right. Another way of saying what you want to say is that as a variable x increases without bound, x multiplied by any non-zero number also increases without bound. Well, I guess that’s not quite right, because if that non-zero number is negative, it *decreases* without bound. So, in a nutshell,

lim(x->inf) nx = inf, where n>0

lim(x->inf) nx = -inf, where n<0

These can be proven, but just not by me right now.

(now I’d better brace myself for my post to be ripped apart by people who actually know what they’re talking about)

What’s the infinite root of infinity? Or infinity to the infinite power?