h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

This is my first posting. I've had this paradox that has bugged me for years. I think it's a true paradox. Here it is:

The infinity paradox:

There is an infinite amount of integers, 1, 2, 3, 4, and so on, until infinity. But what if you count all the negative numbers? You can count backwards from zero, and there would be an infinite number of them too. Both are infinite. But surely, if you have an infinite amount of numbers on one side of the scale, and you have an infinite amount on the other, you have twice as many numbers, don't you? If you counted all the numbers, both positive and negative, you'd still have an infinite amount, yet there's twice as many.

So how can there be an infinity that is twice as large as the other?

I think this was kind of answered in one of the paradox examples.

One of them briefly mentioned 2 different types of infinity. It was about going across a room and saying that there are an infinite amount of distances within that room.

This is the first type of infinity, which is called "infinitely divisible" in the entry. Basically you set some sort of paramater and say you can split it again and again and it just keeps getting smaller.

The second one is "infinitely extendable" and I suppose the most obvious one is that there are an infinite amount of numbers. As you pointed out you can keep adding (or subtracting) one and you never get to the end.

The simple answer to your "paradox" though is that you can't count infinity, so there is a problem with the logic in the argument. Therefore it's not a paradox, since a paradox stands up to logic but there are two opposing forces.

Something that just occured to me though is the question of how many types of infinity there are. I mentioned 2 earlier, but the example you gave doesn't seem to fit. Come to think of it, its a kind of extendable infinity but you've just given it a condition of 'everything after zero'.

My brain hurts now!

"To infinity and beyond!"
-Buzz Lightyear

I have a hazy recollection of a TV programme (so don't quote me), but I think it's to do with what type of infinity you are talking with. You can count up to infinity going in one direction on the number line, and then to negative infinity, in the other direction. They're in the same plane, so you've actually got 2 x infinity (or 2I). Any multiple of infinity, can be collapsed down to infinity again, so you can't have 2I, because it will always equal I.

However I seem to recall (again, don't quote me, I'm probably wrong) believe that you can have a qualitatively different sort of infinity if you multiply them, infinity raised to the power of 2 for example. Also, infinity raised to the power of infinity.

Something like that.

Don't quote me!

It's good to have an answer!
It's a tricky one, and I do actually recall studies about multiplying infinities, but I can't actually recall any info from it whatsoever. D'oh!

The arguement about a room having different infinities makes sense, and I think that's probably the answer to it. So it's not really a paradox, but it's sort of getting there...

I also remember something to do with aleph, does that have any relevance?

Ooooh! Here you go, after a bit of Googling:
http://www.geocities.com/thesciencefiles/infinity/page.html

It talks about Aleph Zero (being the 'infinity' that we're all mostly familiar with) and Aleph One (being the number of points on a line) where Aleph One is bigger than Aleph Zero. Now, if you draw a bunch of lines, (say an infinite (or Aleph Zero) number of them), then... er... I have to read the article again...

the problem is that you are limiting infinity at 0. Your saying that 0 is a limit on positive infinity and negative infinity. But there all part of the same scale of infinity. 0 is just a number which comes in the midle of this scale not the begining. The scale starts at infinity and ends at infinity. Thus it is never ending (or orobouros, if you like red dwarf.) Thus it is not a paradox because there is no such thing as 2x infinity, but it hasnt been doubled just split, from infinity to infinity.

I dont exctly remember what my professor saying. But I took proof at my university and I remember my teacher said that there are different levels of infinity.

There is a level, for example, between 1 and 2. There are infinite numbers between 1 and 2 (1.05, 1.005, 1.0005 and so on for example). Yet there are infinite numbers between 1 and 3 also.

But there are more numbers between 1 and 3, then in 1 and 2, since the level of infinity is x greater in 1 to 3 then 1 to 2.

(i used x since this level increase is not actually one, but I forgot what it is)

But one infinity can never be bigger than another because there is no end to either one. There is this thing I read about once called Hibbert's Hotel or something of that nature, it goes like this.

There is a hotel that has infinite rooms. Starting at room #1 and going up forever. It is currently full, and there is no vacancies. One man runs in and says I need a room, please help me out. No problem says the guy at the desk. He gets on an intercom and says "Everyone please step out of your room, and move to the next room." At the same time, everyone does, leaving room #1 empty for the man. So he gets in.

So clearly you can see that any amount of people can stay at this hotel, even if it's "full" now the question another person proposed to me is, if an infinite amount of people came to this hotel and they had a nametag on for every possible decimal, this hotel couldn't hold them. Because the hotel has a room for every integer, but there are way more decimals than integers. There is actually and infinite amount of decimals for every one integer. 1.1, 1.2, 1.3, 1.02 ect... I disagree, even though it seems there are more people than rooms, there still around. No matter how many decimals you bring in, there will be an integer to hold them. There will never be an end to it, so all infinities are equal.

Hilbert's Hotel Problem is an illustration that there are no more integers {0,1,-1,2,-2,3,-3,...} than there are natural numbers {0,1,2,3,4,...} - both of those infinities are the same. As an equation:

#(Integers) = #(Natural numbers)

Hilbert's Hotel also illustrates that there are no more rational numbers {0,1,-1,2,-2,1/2,-1/2,3,-1,1/3,-1/3,2/3,-2/3,...} than natural numbers {0,1,2,3,4,...}, or in an equation:

#(Rationals) = #(Natural numbers)

The size of all these sets in the infinite number called Aleph-0.

The size of the set of Real numbers, including rationals and irrationals, is bigger than Aleph-0. One proof of this fact is Cantor's Diagonalization Argument, a variation of which can be found in the excellent Edited Entry A593552, "Bigger and Bigger Infinities".

Or just take a class in Set Theory, and you'll learn all about transfinite cardinals and transfinite ordinals, which can make your head swim in ways far too boggling to go into here...


GTB

GTB, that's true but the way Alex describes it he is right. If they are filing into the hotel one by one then as many as can file in will be able to be accommodated. It's also true that they can be accommodated if they are labelled by decimals which terminate (a subset of the rationals). I think this illustrates a deficiency in the Hilbert Hotel metaphor for cardinality.

Sure, there's no way to describe any collection of buses that will show up at the hotel with people numbering c (=the cardinality of the reals). HHP is a good illustration of the problem of proving that #Q = #N, but it doesn't help showing that #R > #N.

GTB

I was thinking about this, and i thought about the real value of intergers and asked myself what is actually between them. Say if instead of a whole number, you take a random object... you use an orange. Now you can have an infinate amount of oranges,and it is evident,that with real and substantial things, there are no inbetweens, or decimals.
Then i thought, what is a deciaml. It is simply a way of cutting down the space between one interger and the next. Thus, there can be nothing but infinate numbers in between each. So is it possible that the infinity between a whole number is man made.
If so, this means that the levels of infinity can be rather superficial, as there are an infinate amount of decimals between 0.2 and 0.3 or 0.03 and 0.04.
So infact you can actually see the levels of infinity on two simple levels (as well as deeper ones)one a purely superficial level, where infinity is a constant and to cut it up is ridiculously futile. This results in infinity acually being from -infinity to + infinity and is unalterable.
However, on a more intellectual level, the levels of infinty can be delved into. But it does seem that in order to do this, you must see the whole number as a pit stop, in a big chain of infinities. This chain would gives a near paradox, possiably resulting in all mathamatics or numbers being merely an illusion. This is because there is an infinate amount of anything to conquer between any whole number.
Thus, there must be a finite point where a decimal of 1 ceases to be 1 and becomes a 2. Concordantly, 1.999999999999 must become more or less a 2 somewhere down the line, otherwise no one would get anywhere. But if the whole number, is a finite number with nothing in between, this problem no longer occurs.
So i would like to determine where the decimal ends and the whole number begins.

Accidently unsubscribed... subscribing again...

Black Cheetah -