Just a crazy question: Is there an Infinity Paradox?

[JustFiveMoreMinutes]

Maybe it's a known and I just haven't run across it or it's just one of those things that we don't think about (like to cross a room, first you have to get 1/2 there... but before that 1/4 the way there... etc.. so how do you cross a room?)

Anyway, the Paradox I'm thinking of came up when reading about parallel universes (Scientfic American).

If I have a double in an x-number of parallel universes that x-number should never reach infinity since logically there would be x-number of universes in which the conditions never came about that my double would come about in.

So I would exist in say 1/n of an infinite number of parallel universes....

but then 1/n of infinity still equals infinity.

So I can't exist in an infinite number of universes.. but then again, i can't simply exist in a fraction of infinite universes...


Where is the flaw?

[Orrex]

1. There might be, if we didn't have the mathematical concept of limits

Or "sums of infinite series."

[The_Casual_Observer]

2. Infinity leads to stuff like the Euler problem 1-1+1-1+1-1+1-1+1....=1/2

[wtbymark]

3. first, i would content that parallel universes are illogical...

..it would exclude the possibility of infinity in this one. Plus, add in the metaphysical argument of god vs. infinity and if you contend that parallel universes exist, you therefor also contend that god doesn't exist. It's quite a flawed argument in many ways. There is an infinity paradox, but it has to do with the existance, or non-existance, of a diety.

Science only seeks to quantify what philosphy has already proven.

[JustFiveMoreMinutes]

5. Most of theoretical quantum physics is counter-intuitive

... so 'illogical' can't be the only argument against a thought experiment.

[qazplm]

10. I see no reason

why parallel universes would exclude the possibility of infinity in this one, or why, even if it did, that would necessarily be something that cannot happen.

[TechBear_Seattle]

4. My engineer grandfather used to say about infinity....

"Infinity is any number too large for us to deal with at the moment."

x is some infinite value. I wish to determine 1/n of x, or x/n, where n is some finite value. The value x/n will either be finite or infinite.

If x/n is finite, then x must also be finite because we can determine the value of x through the equation x=n*(x/n); it is not "too large for us to deal with at the moment." Because it is not possible for a number to be both finite and infinite, and because we were given that x is infinite, this must be an incorrect conclusion.

If x/n is infinite, the question of relative size is meaningless. By definition, infinity is infinity and there is no way to make any kind of comparison between them. Greater than, less than, equal to, not equal to... it cannot be done mathematically because the operations are undefined for infinity.

So your proposition about 1/n of an infinite number of parallel universes is nonsense. There is no paradox, as your line of reasoning collapses at that point.

[JustFiveMoreMinutes]

6. Or perhaps it's only in the mathematical expression of x/n

And should instead by x - n where n is 0 to near infinity but never reaches infinity.

[TechBear_Seattle]

7. There is no such thing as "near infinity"

It is a binary state: either a number is finite or it is infinite. And again, operations such as addition and subtraction are not defined for infinite numbers.

[JustFiveMoreMinutes]

8. infinite thanks.

[cyborg_jim]

24. Numbers can't be infinite

Numbers are definite.

There is no number "infinity".

[phantom power]

9. A couple comments on inifnity that may help you...

one is that infinity comes in different varieties. The infinity you are talking about is aleph-0, the countable infinity. Like other infinities, it's a funny thing.

For example, the sequence {1, 2, 3, 4...} has the same cardinality as {2, 4, 6, 8 ...}. Both sets are countably infinite: their cardinality is aleph-0. However, it's easy to see that the first set properly contains the second.

The second comment is that there are also aleph-1 and aleph-2. The cardinality of parallel universes, if they exist, is probably aleph-1. That is the cardinality of the real numbers. It is a larger infinity than aleph-0.

[JustFiveMoreMinutes]

11. Thanks!

Trying to use mathematics to just put forth a concept probably threw me off the hard mathematics.

So trying again, the possible number of L(ives) in this and all parallel universes would be something like

L = x - n

where L is the number of Lives probable lived/living/will live in parallel universes
x = total number of parallel universes
n = total number of parallel universes where one doesn't exist.

and n is limited by ( 0 LTE n LT x )

((x cant equal n since I'm here writing this note! <smile> ))

[Salviati]

19. The problem is that you're dealing with infinities, and they work different than regular numbers.

In your notation, All of L, x, and n can be infinite, and of the same size.

Let's put it this way, Let's imagine that we can count up all these parallel universes. We're going to get infinity, so how can we make sense of that?

Well, ultimately the way we count is by matching up things with other things, for example, when we count our chickens, we match them up with the whole numbers until we run out of chickens. Then the last numbered chicken tells us how many we have. We've got the same number of chickens as we do numbers.

We can do the same thing with sets of an infinite number of things, we simply make a rule that produces a pattern to match each thing with a number. For example, if we wanted to compare the quantity of even numbers to the quantity of odd numbers, I could use the rule:

Even <=> Even -1

Matching each even number with the odd number one below it. So we see that they come in pairs, and we're never going to run out of either of them. So there are an infinite number of even and odd numbers, and the same size of infinity for both.


But, infinity doesn't work like normal numbers, for example, let's compare the even numbers with ALL whole numbers. I could use the matching rule:

Whole Number <=> (Whole Number) * 2

Which matches each whole number with the even number that is twice as big. Once again they come in pairs, and we're never going to run out of either. So it appears that there are also the same number of even numbers as whole numbers, which is the same as odd numbers.

This size of infinity is called "countably infinite" because of the fact that it can be counted, i.e. arranged in some kind of ordered list, numbered by the whole numbers.

Other things that are countably infinite are all rational numbers, i.e. all numbers that can be written as fractions. All fractions between 0 and 1 are countably infinite. This number is called the cardinality of the set, and for normal finite sets, the cardinality gives the number of things in the set. The cardinality of anything that is countably infinite is aleph-0.


There is another size of infinity however, one that is larger than aleph-0. This would be the cardinality of all the real numbers, this includes all rational numbers, and all irrational numbers (like pi, the square root of 2, the golden ratio, anything that can be written as a non-repeating decimal...)

It's actually pretty easy to show this, we can prove it by contradiciton (It's called Cantor's diagonalization proof) as follows:

Let's assume that the real numbers are countable. That means that must be able to generate a list of ALL real numbers that might look like the one below:


1) 1.02158645498216.....
2) 0.326589456216562....
3) 1.2352994249316654...
4) 4.23455469875165...
5) and so on...

But given any such list, I can now produce a number not included in the list. For the first digit of this number I will take the first number in the list and choose a number that does not match the first digit of that number, e.g. anything but 1. For the second digit, I will choose one that doesn't match the 2nd digit of the 2nd number. For the 3rd digit, I choose one that doesn't match the 3rd digit of the 3rd number, and so on.

The number I produce in this manner cannot be in the list, because it has at least 1 digit different from every number in the list. But my list was supposed to contain ALL real numbers, so therefore my assumption that such a list can be made must be wrong.

So this size of infinity, which is basically the number of points in some sort of continuum, is called aleph-1 and is a larger cardinality than aleph-0.

[skepticscott]

14. Where have you found information

on the proof that there is an aleph-2, and what sets have that cardinality?

[phantom power]

15. Am I wrong about aleph-2 ? I thought there was an aleph-2.

The set of curves through all possible points, or something. It isn't anything I've ever seen a proof for.

[skepticscott]

16. I couldn't find anything on it

If there is such a thing, there's nothing about it on the net, and I haven't read anything about it in any books on infinities.

[phantom power]

20. I did a bit of digging. I think I should have said "beth-2," not "aleph-2."

Definitions are here:
http://www.nationmaster.com/encyclopedia/Uncountable-set

[liberaldiva]

12. Meanwhile a lurking freepers head exploded attempting to

comprehend the complexities in this thread...

[pokerfan]

13. To infinity, and beyond!

Was there a recent article on the Many-worlds interpretation?

[HiFructosePronSyrup]

17. The first response was correct.

An infinite series can have a finite sum.

There's an infinite number of fractions between 1 and 2.

That does not preclude their being an infinite number of fractions between 3 and 4.

[Sandrine for you]

22. you"r right, in fact, Aristotle debunk this paradoxe...

simply the difference between the potential infinity and the actual infinity.

[RagAss]

18. "You" don't exist in any universe.

[Orrex]

21. Oh yeah? Well neither do 'you!'

:evilgrin:

[Jim__]

23. Yes. For instance, Hilbert's Grand Hotel.

From wikipedia( the link is http:// then the rest of the address is en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel - the 's causes problem in the link):

Consider a hypothetical hotel with infinitely many rooms, all of which are occupied - that is to say every room contains a guest. Suppose a new guest arrives and wishes to be accommodated in the hotel. If the hotel had only finitely many rooms, then it can be clearly seen that the request could not be fulfilled, but because the hotel has infinitely many rooms then if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1. By extension it is possible to make room for a countably infinite number of new clients: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room N to room 2N, and all the odd-numbered rooms will be free for the new guests.

...

Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity, for instance an argument for the existence of God posed by the Christian philosopher William Lane Craig is roughly as follows;

Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. Since a temporal sequence receding infinitely into the past would constitute such an actual infinite, time must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.

It must be noted that Hilbert's hotel does not merely require a hotel of infinite magnitude to accommodate additional guests, but also involves the performing of supertasks. It could then be argued that it is unclear from Craig's argument whether this intuition of the fallacy of the hotel is really an indication of the physical impossibility of an actual infinite, or merely the practical impossibility of a supertask. A causal chain receding infinitely into the past need not involve supertasks.

Saint Thomas Aquinas' - arguably the most influential Christian theologian ever - made a well-known attempt to prove the existence of God through infinite regressions in his Summa Theologica.

Of course that's not the paradox you asked about. But it looks similar. You say you can't exist in an infinite number of parallel universes. But I don't see why not. After all (2 * infinity) = infinity. So, if there are an infinite number of universes, you could exist in half of them (an infinite number) and also not exist in the half of them - another infinite number.

[scubadude]

25. As a derivation of your paradox, how about this one?

The distance from point A to B=C.

For each distance C there is a distance D that equals C/2. For each distance D there is a distance D/2 and so on ad infinitum.

In other words if you keep going half way to a destination then half way again how do you ever get to the destination? Can't you keep dividing the distance by half ifinitely?

I have remembered this is called Zeno's Paradox.

Scuba

[cyborg_jim]

26. You take an infinite number of steps.

[Morpheal]

27. INFINITY AND PARALLEL UNIVERSES - NOT QUITE RIGHT

Not all that difficult to understand but it is easy to become mixed up about it.

There are two definitions for infinity.

One is absolute infinity. It is the idea that there could be something that actually has no end, no beginning, and extends forever in terms of
one or more properties of that, whatever it is.

The other is mathematical infinity. This is different from absolute infinity. It really means that you cannot give a definite value defining the
property of something. For instance you know a point exists, but you cannot give it meaningful coordinates, so you say it is at infinity. Actually
you know it exists but you have no idea where it really is. The same type of thinking can apply to other properties other than location.

Remember though that a lot of things can be said in mathematical language that are mathematically true. They are logical within the rules of
the language of mathematics. However logical and true within math does not make them true about the real universe. Mathematics, or
"mythematics" as I jokingly called it to annoy some physicists, can create logical and true imaginary worlds that have never existed, do not
exist and will likely never exist anywhere. So be careful with math. It can tell tall tales and create fictions same as any language.

As to parallel universes, it seems unlikely. 5 dimensional space time, as opposed to Einstein's 4D space-time is being worked on at places such
as U of Waterloo (see articles by the Cosmology Group there, etc.). A new group is forming there to take the paradigm shifts involved further.
5D provides inclusion of all the wierd and wonderful things that 4D had no place for. They were simply anomalies, outside the existing
generally accepted scientific explanations for all and everything.

In fact once you get there, to 5D, you can get branching quantum cause effect chains, but a whole parallel universe is improbable at best.
That branching of quantum histories due to 5D interactions leads to 6D, to include that "many worlds" parallelism. Sort of a mostly micro
cosmic parallel universes, but not like the science fiction stories have played with. The quantum level does not have a lot of obviously
visible effect our day to day lives even when a quantum causality chain splits and goes in divergent directions. At least not usually. Such
branchings are mostly limited to a very few, compared to the total number of quantum causalities.

6D takes us to 7D and that is it for irreducible dimensions. (Physicists often use the term dimension differently and are not really talking about
irreducible dimensions, but of something else.)

7D is probability. The fact that some cause effect chains are not determined to end up as a purely deterministic chain. Instead they are
purely random either / or results completely free of any cause effect determinism deciding the end result being one or the other possibility.
At least that is the simplest way of understanding it. That is essential for free will, or otherwise everything is determinism and there would
only be an illusion of non deterministic choice. In that universe intelligent activity would never really evolve.

That's it !

The magnificent 7.


Cheers.

Robert Morpheal

[BlooInBloo]

28. No. There are only people who don't know much math.