Infinity and our Existence

[devonin]

Also, whether gnr was just responding in the same way to prove a point or not, the 'Capitalizing Every Word' thing is incredibly annoying and poor communication, so you ought to try and not do it.

As for dvann's post: As gnr61 said, measurable numbers are not a fraction of infinity, because by its nature, infinity is not measurable.

There's an interesting question though...is 'infinity' technically a prime number? There are branches of mathematics that deal with differing sizes of infinities, such that one infinity could actually be larger than another, so it could be meaningful to say that the only factors of infinity are infinity and 1.

[Kilroy_x]

How could infinity have factors? It doesn't seem like infinity can even have a fixed value, at least not a discernible one, meaning its only provable factor is itself by identity.

[gnr61]

Originally Posted by Kilroy_x

How could infinity have factors? It doesn't seem like infinity can even have a fixed value, at least not a discernible one, meaning its only provable factor is itself by identity.

Non-prime until proven otherwise.

[devonin]

I'm referring here more to the Cantorian concept of transfinite numbers, and that there are sizes of infinities. Given sizes of infinity, I was half-joking that they qualified under the usual definition of primes, even though it would be impossible to prove.

[seltivo]

Ignore this post. everything in it is (according to someone who sounds smarter that me) wrong.(see post 117)
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If you multiply 1x2x3x4x...xn and then add one (I think it works if you subract one too, but I'm not shure), you will get a prime number. If, instead of stopping at n, you keep going to infinity, then add one you would get a prime number. This would make infinity prime.

However, even if infinity is prime, you gould just multiply it by two (not really changing much) and make it non prime.

This is where the whole "different sizes of infinity" thing comes in. In theory, multiplying infinity by two would give a larger infinity. But since infinity never ends, it would make sense that the second is no larger than the first.

So, it seems to me that some sizes of infinity are prime and some aren't.
How you would manage to tell them apart would be kinda confusing tough.

Another cool thing I thought I'd mention. There are (theoretically) infinitly more numbers between one and infinity than there are between one and zero. Yet, there is an infinite number of numbers between one and zero.

(If anything I said was wrong or didn't make sense, please tell me)

[shadowraikiri]

maybe infinity is tied to the universe. idk wait forget this what i just said is for a new thread!!!
Seltivo i have no idea what you just said, but i like it.xD and i agree.

[ddrissweet1]

who cares about infinity because numbers were created by humans anyways, so it really doesn't matter.

[seltivo]

I love your enthousiasm ddrissweet1. I'm glad you'r first post was so worth while.

On another note, I dont think infinity can be classified as a prime. I think it's more like zero since it can be devided by anything and still retain the same value.

[rebelrunner26]

i don't think you can really classify infinity as prime or composite because it's really more of a theory than a number

[seltivo]

ya, but it's still fun

[rebelrunner26]

true...but i think in order to discuss infinity and actually get somewhere with the discussion, it would probably be better to set some kind of guidelines prior to the start of the debate. otherwise people will start arguing off-topic details pertaining to infinity and submitting their own theories regarding its use while completely missing the point of the discussion....btw, i think we need a thread especially for infinity, unless we've already got one that i don't know about...

[devonin]

So start one, just if you are too specific in what you'll allow as subject matter pertaining to infinity, I suspect the thread won't live very long, but if you aren't specific enough, you seem like you'll be upset if it goes in other directions.

We just really can't have a reasonable discussion about infinity until we acknowledge that infinities are differently sized from one another, that one seems pretty integral, and yet seemingly not understood by too many people.

[Kilgamayan]

Originally Posted by seltivo

If you multiply 1x2x3x4x...xn and then add one (I think it works if you subract one too, but I'm not shure), you will get a prime number. If, instead of stopping at n, you keep going to infinity, then add one you would get a prime number. This would make infinity prime.

If you stop at any point to "add one" you cease to be dealing with infinity.

[devonin]

Originally Posted by Kilgamayan

If you stop at any point to "add one" you cease to be dealing with infinity.

Yes, and no. You can state 'X' as being an infinite series, and then also state X+1, and you get an infinity that is slightly larger than the previous infinity. We really need to get away from the 'infinity as a prime number' thing though, I meant it mostly as a joke.

[Kilgamayan]

The thing is, infinity only comes in two sizes, countable and uncountable, and neither of those sizes are quantifiable via any finite subset of the real numbers. Given X to be an infinite series, X+1 will be the "same size".

ap could probably explain it better than I, however.

But yes, I suggest moving away from the "infinity is prime" gag of an idea (and shame on anyone who took it seriously), mostly because infinity isn't an integer. >_>

[devonin]

The thing is, infinity only comes in two sizes, countable and uncountable, and neither of those sizes are quantifiable via any finite subset of the real numbers.

I disagree. Infinities come in,well, an infinite number of sizes. Here's a nice easy example:

Infinity A: The set of all Real Numbers between 1 and 2
Infinity B: The set of all Real Numbers between 1 and 3

To me, it seems clearly the case that Infinity B is simply larger, since the set contained in B contains within it the set contained in A And other numbers

They are both uncountable infinities, but they are not equal, and they are not the same size at all.

[aperson]

Ok time for the bad math fixathon

Originally Posted by seltivo

If you multiply 1x2x3x4x...xn and then add one (I think it works if you subract one too, but I'm not shure), you will get a prime number. If, instead of stopping at n, you keep going to infinity, then add one you would get a prime number. This would make infinity prime.

This rule is true for any number that's an element of Z+ (read: Integers). Inf is not an element of Z+, therefore infinity does not have any sense of primality.

However, even if infinity is prime, you gould just multiply it by two (not really changing much) and make it non prime.

Ok so infinity isn't prime and you were just rambling.

This is where the whole "different sizes of infinity" thing comes in. In theory, multiplying infinity by two would give a larger infinity. But since infinity never ends, it would make sense that the second is no larger than the first.

See this is what happens when people stumble across a few interesting math posts on sizes of infinity and now think they have some conceptual grasp of what's going on. When we're talking about different sizes of infinity, we're talking about infinite sets, as in, a collection of an infinite number of objects. The set of all even, positive integers is an infinite set. The set of all even and divisible by seven positive integers is an infinite set. The size of the set is denoted by what is called its cardinality. So the set of even positive integers less than ten has a cardinality of 4 {2, 4, 6, 8}.

Any set that is countably infinite bijects to the integers. Any set that is countably infinite has a cardinality of . (Read: Aleph-Not, which I'm going to call A_0 from here). Any set of size n+A_0, n*A_0, or A_0^n where n is any integer is still an A_0 sized set because it still bijects to the integers. It isn't until you take the Power set of an A_0 sized set that you get a larger infinity. The size of the power-set of an n sized set is 2^n, so 2^A_0 is a larger infinity than A_0.

As it turns out in mathematics, the size of the real number is the powerset of the size of the integers (Because every integer has an infinite string of integers after it). Therefore, the real numbers are considered uncountably infinite because they are larger than countably infinite sets, and cannot be bijected to the integers. (This can be demonstrated with a Diagonalization argument).

So, it seems to me that some sizes of infinity are prime and some aren't.
How you would manage to tell them apart would be kinda confusing tough.

So it seems, after a crash course in infinite sets, that this post is completely asinine and really makes no sense whatsoever.

Another cool thing I thought I'd mention. There are (theoretically) infinitly more numbers between one and infinity than there are between one and zero. Yet, there is an infinite number of numbers between one and zero.

Depends on your domain. Across the integers there are 0 numbers between 1 and 0 and infinitely numbers between 1 and infinity. In the reals, that's kind of a different story.

(If anything I said was wrong or didn't make sense, please tell me)

Everything was wrong.

The thing is, infinity only comes in two sizes, countable and uncountable, and neither of those sizes are quantifiable via any finite subset of the real numbers. Given X to be an infinite series, X+1 will be the "same size".

This isn't quite true; countable and uncountable describe all the different infinities you can run into, but there's more than one size of infinity in the uncountable sets. Taking the powerset of an infinite set always yields a larger infinite set, so taking the powerset of an uncountably infinite set yields a larger set. Therefore, there are an infinite number of infinities, and all but one of them are uncountable.

Also, a kind of interesting tangent is the Continuum Hypothesis which asks if there are any different sized infinities between countable and uncountable sets. However, the continuum hypothesis has been demonstrated to be axiomatically undecidable (it's unsolvable, and it can never be solved because of our formulation of mathematics), so this throws the possibility of even more sized infinities into a bit of a grey area.

Originally Posted by devonin

I disagree. Infinities come in,well, an infinite number of sizes. Here's a nice easy example:

Infinity A: The set of all Real Numbers between 1 and 2
Infinity B: The set of all Real Numbers between 1 and 3

To me, it seems clearly the case that Infinity B is simply larger, since the set contained in B contains within it the set contained in A And other numbers

They are both uncountable infinities, but they are not equal, and they are not the same size at all.

No these are both uncountably infinite sets which both have a cardinality of 2^A_0. You're right, there are an infinite size of infinities, but you're completely wrong on the reasoning. Go read the wiki article on diagonalization and bijection then come back.

[gnr61]

I didn't think math could make a good CT discussion, considering its set-in-stone, indisputable values, but this is turning out to be a pretty interesting read. Especially when people who know what the hell they're talking about follow up people who, well, don't.

[devonin]

Originally Posted by aperson

No these are both uncountably infinite sets

Originally Posted by myself

They are both uncountable infinities

Why the 'no' if you say what I say? My whole conclusion was not whether they were countable or not, my point was that by every logical standpoint, the one is larger than the other. Whether we can determine in a useful way, how much larger or not is irellevant to the point.

How can it be the case that numbers between 1 and 2 can be anything -but- smaller than the numbers between 1 and 3, since the numbers between 1 and 3 contain as part of their set, all the numbers between 1 and 2 -and- the numbers between 2 and 3?

[aperson]

Originally Posted by devonin

Why the 'no' if you say what I say? My whole conclusion was not whether they were countable or not, my point was that by every logical standpoint, the one is larger than the other. Whether we can determine in a useful way, how much larger or not is irellevant to the point.

And my point is that they are the same size and you are completely misunderstanding how infinity works. 2*A_0 is equicardinal to A_0. therefore any finitely sized domain of real number is equicardinal to any other finitely sized domain of real numbers. If they are equicardinal then they are "an equal sized infinity." I'm not saying at all what you're saying. I'm saying that the powerset of an infinite set is larger than the original set. You're saying that an integer multiple of an infinite set is larger than the original set. And my point, which has the backing from a lot of mathematicians over the 20th century, is that your different interval-length subsets of the reals are the same size.