a philosophy problem about infinity

[tmwong]

does it make sense if i define that the shape of a circle is infinity, which i write its equation as:

x^2+y^2 = ∞

i know that infinity means very very big until we cannot define how big is it. in this case, is the equation above still valid? can i still say that the shape of this infinity in the equation is a circle? or its size is not important or does not make sense already because infinity doesn't have shape since it's too big until we can't define what's the shape of that "thing"?

[matt grime]

Since infinity is not part of the plane the question, as a question of geometry, is meaningless. You need to extend the plane to include infinity, even then the most natural extension, that of the one point compactification of the Complexy plane, yields that the locus of points an "infinite distance from the origin" is the point at infinity.

[tmwong]

e.g: if i say the size of the universe is infinity but its shape is a sphere. does it make sense?

[matt grime]

It doesn't make mathematical sense unless you state it mathematically. Are you speaking topologically?

[tmwong]

if yes? is that mean the universe is expanding from a singularity until infinity size in a sphere shape in my example?

[tmwong]

however, if my statement just now was true, since its size is infinity, how could we know it's expanding in a sphere shape? and not in a cubid or other geometry shape? i found there is a paradox here.

[DeadWolfe]

Wow! You're right!

Also, I found another paradox. If there is an infinite amount of even numbers, and an infinite amount of numbers, are they the same? But one should be twice as big! Cool, huh?

Think I should send it to a math journal or something???

[kreil]

It's already been done, by George Cantor. Search for his name and you will find some pretty cool stuff about infinity/infinite sets.

[jcsd]

Wow! You're right!

Also, I found another paradox. If there is an infinite amount of even numbers, and an infinite amount of numbers, are they the same? But one should be twice as big! Cool, huh?

Think I should send it to a math journal or something???

What you have noiced is that infinte sets can have proper subsets that are infinite (but this is nothin new as indeed the defintion of an infinite set says somehting even stronger i.e. a set is infinite if there is a bijection (http://en.wikipedia.org/wiki/Bijection) between it and a proper subset of itself) . There is a bijection between the set of even numbers and the set of integers and there is also a bijection between the even numbers and the set of rationals, but there's no bijection between the set of even numbers and the ste of real numbers

[philosophking]

Right. Just to add to this: when a bijection between two sets is possible, then the cardinality of those two sets are equal. Thus, the cardinality of the set of even numbers equals the cardinality of the set of positive numbers/integers/rationals. Sometimes for short we call this "type" of infinity [aleph]_0. [aleph] is a hebrew letter.

What do I mean about different types of infinity? Well, there does not exist a bijection between the natural numbers and the real numbers. Google for "cantor's diagonal argument" for specifics. So in a sense, the cardinality of R (the reals) is larger than [aleph]_0. But both are infinity. Thus, the existence of different types of infinity.

[ex-xian]

It's already been done, by George Cantor. Search for his name and you will find some pretty cool stuff about infinity/infinite sets.
I thought he was being sarcastic.