Universe [Probably] 250+ Times Bigger Than What Is Observable

[Requia ☣]

this is a tangent, and I apologize for the threadjack...

but I'm vaguely recalling a mathematician ... Georg Cantor? whose theories involved varying sizes of infinity.


The idea being that there's an infinite number of fractions between each integer. And there's an infinite number of integers too. So the total number of fractions is actually [total number of integers] to the infinite power.

so you end up with this paradox where there are larger and smaller infinities

Those have a 1 to 1 correspondence as well actually:

1 1/2
2 1/3
3 2/3
4 1/4
5 2/4
6 3/4
7 1/5
etc

It makes no sense to me but it apparently convinced one of my math professors.

[Igor]

Yes, there is the same amount of rational numbers (fractions) as there are whole numbers. Like Requia said, you can set up a one-to-one correspondence between each fraction and each whole number.


What you can't do is set up a similar correspondence between the natural numbers and the irrational numbers (like pi and root[2]). So, in a sense, there are more of those than there are natural numbers. This only happens because the number of digits in an irrational number is infinite itself. Cantor had a nice little proof of this, which I hope wiki can explain, because I'm running out of time right now:
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

[Requia ☣]

Hmm, if certain theories about the nature of space (specifically that there are minimum units of lengths and time) are correct, then there's no reason you couldn't do a 1 to 1 with matter and space.

So there would actually be an equal amount of matter and space, even though there's more space than matter.

Go paradoxes!

I wonder which screwy axiom led to this one.

[Triple Zero]

What Igor said.  It doesn't matter if there is Infinity, or Infinity+1.  Both amounts are infinitie.  You will never have one bigger than the other, because both never end.

Math tricks aside.

No.

That's why they call the countable infinity Aleph-Nul. There are types of infinity that are truly bigger than that. Not just in an infinity+1 way, but in a very fundamental level. The most common example is the infinity of countable numbers (Aleph Nul) versus the infinity of Real numbers, which are uncountable, and called Aleph One.

Whether Aleph One type infinities actually exist in "reality" outside Mathematics, is another question, on which Bucky Fuller has to say a thing or two.

What's beyond Aleph One, like Aleph Two and onwards, either they haven't been able yet to (mathematically) figure out what that would look like, or I don't understand the theory well enough to be able to grasp whether they have or not.

[Triple Zero]

this is a tangent, and I apologize for the threadjack...

but I'm vaguely recalling a mathematician ... Georg Cantor? whose theories involved varying sizes of infinity.


The idea being that there's an infinite number of fractions between each integer. And there's an infinite number of integers too. So the total number of fractions is actually [total number of integers] to the infinite power.

so you end up with this paradox where there are larger and smaller infinities

Actually, the fractions, aka Rational numbers also belong to Aleph Null, or countable infinity. It works like this, you can write every fraction as p/q, where p and q are both integers, okay? So now you interpret these p and q as coordinates on a plane, a grid. Then you start in the origin (which is 0/0, but we're going to skip the impossible fractions) and then twirl around the origin in a tight spiral. Now you've got a countable progression, that will ultimately hit every point in the plane (skipping all the p/0 ones cause they're not proper fractions), so you're going to hit every possible fraction at some point. Makes them countable, makes them just as big as the set of Natural numbers.

It's only when you also add the Irrational numbers (such as sqrt(2), Pi, e and such) that you really fill up all the tiny gaps between integers and fractions, and then you get the Real numbers, and you get a set that is impossible to count anymore.

Looking at it in another way, the decimal expansion of a fraction always repeats at some point. 1/3 = 0.3333333... and 1/7 = 0.14285714285714... that's the rational numbers. As soon as you also allow decimal expansions that do not repeat, ever, you get a huge inifnity of numbers that are not fractions, and you get the set of Real numbers. Pi has a decimal expansion that (provably) never repeats. But also constructed numbers such as 0.12345678910111213141516171819202121... never repeat and are therefore irrational.

Personally, I hate the set of Real numbers because it's got to do with the Axiom of Choice (which I don't quite understand) and the Banach-Tarski Paradox (which somebody once explained to me satisfactory at a party except I was drunk and now I forgot) which allows for stupid paradoxes such as cutting a sphere into five pieces and reassembling those pieces into two spheres of the same size. Which is patently ridiculous, so I'm siding with Bucky Fuller here on the non-existance in reality of these larger sets of Infinities.

(BTW before you wonder, yes the 5 pieces to cut that sphere are indeed very very fractal and impossibly shaped)

[Triple Zero]

Oh it seems that Requia and Igor said pretty much what I said.

Requia if the fraction correspondence doesn't make sense to you, check my spiral example.

Cantor's diagonal proof is one of those cool Reductio ad Absurdum proofs, my favourite kind Basically it says that, ASSUME the set of Real numbers is countable. That means you can make an infinite list of all the Real numbers. The diagonal proof then shows a way, given this list, to construct a Real number that is not on the list, meaning that the list cannot have been complete. WHAAA CONTRADICTION therefore one of the assumptions must be wrong, and the only assumption was that Real numbers are countable, therefore they are not.

[Golden Applesauce]

What's beyond Aleph One, like Aleph Two and onwards, either they haven't been able yet to (mathematically) figure out what that would look like, or I don't understand the theory well enough to be able to grasp whether they have or not.

AFAIK the Aleph numbers are kind of sketchy, mostly because the Continuum Hypothesis (which says that the Aleph numbers and Beth numbers correspond to each other nicely) has been proven to be independent of the main formulation of set theory - that is, you could have two worlds that both abide by the axioms of ZF(C), in one of which the continuum hypothesis is true, in one of which it is false, and neither world would be paradoxical.

But in a system where the Continuum Hypothesis is true, sets of various Aleph numbers are easy to come by.

Aleph 0:
Counting numbers
Integers, and any infinite subset thereof (prime numbers, even numbers, perfect squares, etc.)
Rational numbers
Turing machines
Set of all finite subsets of any of the above

Aleph 1:
Real numbers
Irrational numbers
Polynomials
Set of all subsets of any Aleph 0 set (e.g., the set of all subsets of the integers.)
Points in N-dimensional Euclidean space
Set of all sequences of integers or real numbers
Set of all continuous functions from the real numbers to the real numbers

Aleph 2:
Set of all subsets of the real numbers (or any other Aleph 1 set.)
Set of all functions from the real numbers to the real numbers

Aleph 3:
Set of all subsets of any Aleph 2 set

Aleph 4:
Set of all subsets of any Aleph 3 set, etc.

[The Good Reverend Roger]

If the universe is bigger than 4/3*(Pi)*R^3, where R is the speed of light times the age of the universe, I am calling shennanigans on God.

End of fucking story.

[Triple Zero]

If the universe is bigger than 4/3*(Pi)*R^3, where R is the speed of light times the age of the universe, I am calling shennanigans on God.

End of fucking story.

Good point. I'm with you, banging at his door, demanding the situation to be rectified, not only immediately but also retroactively, and he better get his shit RIGHT this time, or we kick him out of the group and have somebody else be the GM for a while.

[Nephew Twiddleton]

But it could be that the Big Bang that formed us is only a localized event. What if there were some other Big Bang that occurred simultaneously 154 quadrillion light years away (random number). We would never know about it, but it would still be part of the Universe.

[LMNO]

If the universe is bigger than 4/3*(Pi)*R^3, where R is the speed of light times the age of the universe, I am calling shennanigans on God.

End of fucking story.

Ok, I think I've got my thought I keep having*.  Thanks, TGRR.  I think I'll be using that for a long time.





















*The RWHN definition of "belief".

[The Good Reverend Roger]

But it could be that the Big Bang that formed us is only a localized event. What if there were some other Big Bang that occurred simultaneously 154 quadrillion light years away (random number). We would never know about it, but it would still be part of the Universe.

There wouldn't be any distance between the two big bangs.  Just saying.

[Jasper]

As I understand it, the Big Bang was a dilation of space itself.  It's not that more universe is being added onto what we've got, it's that what we've got is being stretched, distances keep getting greater, like two points on a balloon as it inflates (except that matter itself is not expanding, otherwise nobody would notice anything).

[The Good Reverend Roger]

As I understand it, the Big Bang was a dilation of space itself.  It's not that more universe is being added onto what we've got, it's that what we've got is being stretched, distances keep getting greater, like two points on a balloon as it inflates (except that matter itself is not expanding, otherwise nobody would notice anything).

I'm noticing.

The other galaxies are moving away from us because they hate us.

[Jasper]

It's probably for the best.  Containment is key with things like this.  It's not enough to keep us at the bottom of a gravity well, light years from anything worth ruining.  It is also necessary to run.