Raving Atheists Forum - View Single Post - Double-infinity

psyadam · 03-16-2006, 01:32 AM

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.

03-16-2006, 01:32 AM	#9
psyadam Senior Member Join Date: Feb 2005 Posts: 835	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.