Infinity Taken to the Infinity Power an Infinite Number of Times.

[kaminogan]

it makes sense, although its real world possibility can be questioned,

say you have a graph, said graph has a line,

as you know, lines are infinitely long,

now add another line,

you now have two infinitely long lines on the same graph,

so what does that make the graph ? infinity to the second power,

you can go even further and add a third dimension to the graph (3D graph) making its total size infinity to the third power,

[CookingWithGas]

What is worse
Than zombie blank verse?

[Czarcasm]

Than zombie blank verse?
I think I would have to say
it’s zombie haiku

[Isilder]

The OP’s example is trivial, so it has trivial explanation

What is Inf - Inf ? Undefined as yet.

But what if we make definitions.
a= Inf.
b=a+1 … Evaluate… Inf.

So what is b-a ? defined as 1

My point: Maths on Infinity occurs in an algebraic setting. The example of infinite occurring on a graph really isn’t the same.

[Chronos]

This may seem surprising, but there are a number of different algebraic rules one can adopt concerning infinity, and adopting different such rules can result in multiple different, but still sensible and usable, models of algebra. One can, for instance, rule that infinity is not a number, and that it therefore has no place in algebra: That algebra without infinity is fine and valid (though it has awkward features such as that 1/0 is undefined). Or, one can introduce an algebra where anything nonzero divided by zero is defined as infinity, and put in a few more rules for dealing with infinity, and that can work out, too. Or one can introduce separate positive and negative infinities, such that any positive number divided by zero is positive infinity, and any negative number divided by zero is negative infinity, and again, with a few more rules, that can work out, too. You may (or may not) also wish to define, in such a model, a distinct positive and negative zero. In any model which includes infinity, you end up with the problem of how to deal with infinity - infinity, but really, that’s no worse than the problem you have with 0/0 (and in fact is just a different manifestation of the same problem).

[Itself]

@BigT

The thing everyone always leaves out in these talks is that you necessarily take a section out of the infinite set before you can compare them. You then notice that pattern, and notice that it will continue infinite times. The one to one mapping has to occur with a finite set that is a set proportion from an infinte one.

I don’t know what you mean by that.

@BigT

Lets say you take all positive integers less than 100, and compare it with all positive odd integers less than 100. You notice that there is a 1:2 mapping. (1, 1&2), (3, 3&4) (5, 5&6), etc. Therefore, in that set, there are twice as many positive integers as odd positive integers.

Now, it doesn’t matter where you make this arbitrary cutoff. As long as the cuttoff is at the same point, you will have a 1:2 mapping. So that means there are twice as many positive integers as odd positive integers, even thought the sets are infinite.

When mathematicians talk about the “size” of a set, we mean a very specific thing: Two sets A, B have the same size if and only if there exists a one-to-one map f: A -> B. It doesn’t have anything to do with the way A and B sit in some other set C, or how A and B are related to each other, etc. In fact, we don’t care what the elements of A and B specifically are: the sets {1, 2, 3} and {20, -19, 18} and {the 3-sphere, the real function x -> x^2, the empty set} have the same size, the one-to-one map being obvious. There are other ways to define a notion of ‘size’: asymptotic density, which is sort of like what you were takling about above; Schnirelmann density, which is sort of a worst-case scenario of the above; switching to ordered sets and looking at ordinals rather than cardinals; and so on. But again, we’re talking here about the size, or cardinality, of sets, and that’s inarguably the definition I mentioned above. If you want to come up with your own notion of size, fine, but you have to define it more precisely.

In any case, the odd integers and even integers have the same cardinality, since the map f(n) = n + 1 is a bijection from the former to the latter.

[Itself]

@josephi

Given infinity is represented here by j… and think in terms of graphically representing j (Cartesian for the heck of it)… How do you visualize j^j^j (ad nauseum)?

Cardinals, even infinite ones, are not natural, rational, real, etc. numbers and therefore don’t have operations like exponentiation defined a priori, any more than it makes sense to talk about the real square root of -1 or the largest power of 2 dividing \pi. Actually, there is an arithmetic for caridnals, and there is an exponentiation operator there: |X|^|Y| = |Map(Y, X)| (with the latter often written as X^Y). To justify the notation, note that the power set 2^X can be considered as the set of maps f:{0, 1} -> X, with f being identified with f(0). It works formally like you might expect.

There are different things that are called infinity: infinite cardinals, infinite ordinals, the extended real number, the point at infinity on the Riemann sphere, etc. They have different properties and different rules for manipulating them.

@josephi

My thought is an infinite number of spheres within spheres. Math can’t define the cosmos, as it is as incomprehensible as any being who could create it. Think outside the rigors… visualize how the network this construct would appear if you could see it with your eyes… daunting, but it might be quantum in nature, given any number of infinite particles exist simultaneously in an infinite number of probabilities… perhaps?

Seems legit.

[Itself]

@Itself

In any case, the odd integers and even integers have the same cardinality, since the map f(n) = n + 1 is a bijection from the former to the latter.

…and the set of all integers and the set of odd integers have the same cardinality, the map f(n) = 2n - 1 being a bijection between them.

[Saint_Cad]

@Mangetout

The point is that the infinity of integers is fixed in size - you can’t invent a new integer between 1 and 2, but you can do that with real numbers - therefore, the infinity of real numbers is larger than the the infinity of integers

Then explain how rational numbers are likewise dense yet the same cardnality as the integers.

[race_to_the_bottom]

Nobody really understands infinity. The way I remind myself of that is by assuming an infinite universe, or multiverse, probably. I have a piece of shale which is cleaved along the fracture plane so that the two pieces fit together perfectly. I tell myself that if the Universe is infinite, that there are an infinite number of pieces like this in the Universe which fit together exactly like these two. Furthermore, there are an infinite number of people who look exactly like me, and like you, and everyone else who has ever lived and will ever live and who is holding identical pieces in every possible way and every tick of the Planck clock, 5.39 x10 -E44 seconds.

And there is STILL other infinities you can add to these. In fact, an INFINITE number.

Infinity is a real mindfuck.

[Little_Pig]

if (Grok(infinity))
then bliss();

/* Hey Kids! Gimme an else. */

[Senegoid]

Throughout my primary and secondary school mathematical education, it was routinely beaten into our heads that “Infinity is not a number!!!”

Just what infinity is, was never really addressed very well. But we were certainly not allowed to think that 1/0 might ever by any stretch of the imagination, be defined or definable or contain any useful information.

The implication that I think was there, was that infinity is not a thing at all, at least not at thing that can be discussed at all. It was like a forbidden taboo subject.

So when some of us went on to higher education, and specifically higher math, it took a bit of reconditioning of our battered brains to come to think of infinity as something that could meaningfully be talked about, or even thought about.

I wonder if that was the kind of mindset that such luminaries as Georg Cantor had to overcome before he could do his thinking on the subject.