Double-infinity

This is a strange thing for even me to envision. I have had a nagging thought about what exactly infinity is. I just subconciously thought about it in eighth grade and forgot, then remembered today and decided to explore.

In geometry, I remember when I wondered if the Ray was infinite in length. It seemed so, but the problem was that it had a beginning. I wondered then what the line was. It was, well, two rays back to back. Infinity * 2, and that got me thinking if perhaps there is a separate infinity.

Picture a coordinate plane, with an inequality of Y > X. Half of the plane is shaded in, and it becomes infinite in solutions. But it has another half of, as well, infinite non-solutions. How can this be on the same plane? I always thought that there was an extra infinity, making there exist two infinities on the same plane. So, I figure that there has to be a double infinity.

This was kinda bruised when I graphed a Y > X^2, because the inner parabola was infinite in solutions, and the outer one as well, but they weren't half-planes, because it wasn't a line.

Perhaps this isn't a new idea, just one I haven't heard elsewhere. Any thoughts?

[a different tim]

There's more than one infinity. This is because, as I understand it, rather than being a number as such, infinity can be seen as an instruction:"Keep on going".

Consider - there is an infinite number of, er, numbers, right? But there is also an infinite number of odd numbers. Intuitively, this is smaller than the first infinity. Also, there is an infinite number of decimals between 1 and 2, and this must be larger than the first infinity......sometimes this doesn't matter (odd numbers and numbers in general are the same infinity even though one is intuitively "bigger", because you can map the two series onto each other in a 1 to 1 correspondence without lhaving to leave anything out) and sometimes it does (the number of irrational numbers is a different, larger, infinity to the number of rational numbers).

The guy you want, who mathematically codified this and figured out how to use these different infinities in (some) calculations, is Georg Cantor. Then again, he was clinically insane.

To elaborate:

The set of integers is infinitely large. Subsets of this set, e.g. all the odd integers, are also infinitely large, they are the same size as the original set. This kind of infinity was called aleph-null by Cantor.

As ADT pointed out, there are an infinite number of real numbers between every integer, so the set of real numbers is infinitely bigger than the set of integers. Cantor called this aleph-one.

This process can continue as well. By taking the power set (the set that includes all the subsets) of any infinite set, you get an infinite set that is infinetly bigger! For example, the power set of the set of all real numbers would be aleph-two, its power set would be aleph-three, and so on.

[Choobus]

Cantors set theory addresses this. Basically, there are different classes of infinity.

didn't see that ADT beat me top the exact same reference. The "instructions" model is not correct though. It's to do with sets. You can have sets with infinitely large subsets, or you can have the set of all sets, which is a larger infinity because it contains all the others. There are mathematical proceedures for classifying various sets and so you can actually say that this infinity is bigger than that infinity. The degrees of infiniteness are the aleph numbers. It's very interesting mathematics, but can lead to massive brain shutdowns and a need for vodka.

Choobus wrote

It's very interesting mathematics, but can lead to massive brain shutdowns and a need for vodka.

All too true! :lol: I just need a better grasp on math first before delving into another section of it. Just thought it was a bit interesting, because it kinda seems like infinity is, well, finite. Like how one half-plane has infinite solutions, but that doesn't mean that it has all the solutions. Very much headache work.

[Choobus]

yeah, set theory, group theory and the order paramater all fuck with your head (at least, I think so).

Go here for a quick description of infinity.

Go here for a quick description of infinity.

[psyadam]

From "Zero" by Charles Schafe:

"In Cantor's mind there were an infinite number of infinities--the transfinite numbers--each nested in the other. Aleph0 is smaller than Aleph1, which is smaller than Aleph2, which is smaller than aleph3, and so forth. At the top of the chain sits the ultimate infinity that engulfs all other infinities: God, the infinity the defies all comprehension.

Unfortunately for Cantor, not everyone had the same vision of God. Leopold Kronecker was an eminent professor at the University of Berlin, and one of Cantor's teachers. Kronecker believed that God would never allow such ugliness as the irrationals, much less an ever-increasing set of Ressian-doll infinities. The integers represented the purity of God, while the irrationals and other bizarre sets of numbers were abominations...

Disgusted with Cantor, Kronecker launched vitriolic attacks against Cantor's work and made it extremely difficult for him to publish papers. When Cantor applied for a position at the University of Berlin in 1883, he was rejected; he had to settle for a professorship at the much less prestigious University of Halle instead. The same year, he wrote a defense against Kronecker's attacks. Then, in 1884, the depressed Cantor has his first mental breakdown."

Cantor's mathematics were a great leap for mathematicians and logic. His mental breakdown was caused by Kronecker and his religious insanity, not for any other reason.




I am sortof a mathematician, but then I'm sure there are mathematicians alot better than I am that can explain infinity alot better, but I'll tell you what I know about it (which isn't much).

Mathworld has a rather short fact list about infinity:

http://mathworld.wolfram.com/Infinity.html

Some more facts it didn't really say:

- infinity is not a "number". It's not even an imaginary one. 0 is a number, but infinity is not. While technically dividing by zero is undefined, as the calculus limit is infinity, it is often useful to treat the answer of a division by zero as infinity and also the division by infinity as being 0.

- let x be a real number
- then infinity * x = infinity
infinity / x = infinity
infinity - x = infinity
infinity + x = infinity, all all the resultant infinities are equal

of course, there is also a negative infinity (-infinity). It is arguable to whether -infinity = +infinity. Often mathematicians define things like this to help them in their proofs. In other words, at some point it can often be more of a matter of convenience than a science.

if you are doing set theory, then you can talk about an infinity being greater than another infinity, but I don't know much about set theory and it's rather a hairy topic from what I understand and it doesn't use normal arithmetic as shown above but instead uses unions and intersections and complements I believe.

The following forms are "indeterminate" which means that it could essentially be any real or imaginary number or infinity:

infinity - infinity
infinity / infinity
0^0
infinity^x, where x is any number or infinity
x^infinity, where x is any number or infinity

I believe infinity + infinity = infinity and infinity * infinity = infinity but I could be wrong. It's interesting that infinity is a vortex that once you get it doing arithmetic using infinity either gets you infinity at best or something indeterminate at worse. 0 acts the same way in some respects, but infinity is even worse.

[psyadam]

From "Zero" by Charles Schafe:

"In Cantor's mind there were an infinite number of infinities--the transfinite numbers--each nested in the other. Aleph0 is smaller than Aleph1, which is smaller than Aleph2, which is smaller than aleph3, and so forth. At the top of the chain sits the ultimate infinity that engulfs all other infinities: God, the infinity the defies all comprehension.

Unfortunately for Cantor, not everyone had the same vision of God. Leopold Kronecker was an eminent professor at the University of Berlin, and one of Cantor's teachers. Kronecker believed that God would never allow such ugliness as the irrationals, much less an ever-increasing set of Ressian-doll infinities. The integers represented the purity of God, while the irrationals and other bizarre sets of numbers were abominations...

Disgusted with Cantor, Kronecker launched vitriolic attacks against Cantor's work and made it extremely difficult for him to publish papers. When Cantor applied for a position at the University of Berlin in 1883, he was rejected; he had to settle for a professorship at the much less prestigious University of Halle instead. The same year, he wrote a defense against Kronecker's attacks. Then, in 1884, the depressed Cantor has his first mental breakdown."

Cantor's mathematics were a great leap for mathematicians and logic. His mental breakdown was caused by Kronecker and his religious insanity, not for any other reason.




I am sortof a mathematician, but then I'm sure there are mathematicians alot better than I am that can explain infinity alot better, but I'll tell you what I know about it (which isn't much).

Mathworld has a rather short fact list about infinity:

http://mathworld.wolfram.com/Infinity.html

Some more facts it didn't really say:

- infinity is not a "number". It's not even an imaginary one. 0 is a number, but infinity is not. While technically dividing by zero is undefined, as the calculus limit is infinity, it is often useful to treat the answer of a division by zero as infinity and also the division by infinity as being 0.

- let x be a real number
- then infinity * x = infinity
infinity / x = infinity
infinity - x = infinity
infinity + x = infinity, all all the resultant infinities are equal

of course, there is also a negative infinity (-infinity). It is arguable to whether -infinity = +infinity. Often mathematicians define things like this to help them in their proofs. In other words, at some point it can often be more of a matter of convenience than a science.

if you are doing set theory, then you can talk about an infinity being greater than another infinity, but I don't know much about set theory and it's rather a hairy topic from what I understand and it doesn't use normal arithmetic as shown above but instead uses unions and intersections and complements I believe.

The following forms are "indeterminate" which means that it could essentially be any real or imaginary number or infinity:

infinity - infinity
infinity / infinity
0^0
infinity^x, where x is any number or infinity
x^infinity, where x is any number or infinity

I believe infinity + infinity = infinity and infinity * infinity = infinity but I could be wrong. It's interesting that infinity is a vortex that once you get it doing arithmetic using infinity either gets you infinity at best or something indeterminate at worse. 0 acts the same way in some respects, but infinity is even worse.