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Simple contradiction/flaw in many worlds theory/theory explanation.

  1. Nov 28, 2003 #1
    As it's simply explained, the many worlds theory states that everything has an infinite amount of possible outcomes and that a new reality is created in which one of each outcome occurs. This process of new reality creation keeps occuring infinately, creating more and more infinite amounts of realities.

    If my basic understanding and explanation of the teory is flawed, then disregard my post, as it is irrelevant.

    This theory states that for one event, you get an infinite number of outcomes, and in the next event, infinte outcomes again result. If you were to modify the theory to say that for every event there are 5 possible results, you would have one original event, then 5 events, then each of those 5 events would form 5 new realities, making a total of 25. So bassically, the formula for figuring the amount of realities in existance out would be to simply square the amount of realities that existed prior to the latest event which caused more realities to spawn.

    Using this, after the very first event, you'd have infinity realities, the second event would breed infinite realities squared, then infinite realities to the 4th power, infinite to the 8th power etc.

    But, there can be no number larger than infinite, so then one would have to say that either
    A.) This theory makes the number of infinite increase with each new event.
    B.) The actual number of new realities spawned from each event is infact less than infinity, but still very large.

    If one chooses A as their response, then they would also have to conclude that since infinity increases each time an event occurs, then the previous number of realities created by the last event was not infinity, but the square root of infinity.

    Have I just found a flaw in this widely accepted theory, or do I misunderstand it?
    If I misunderstand it, please explain it to me in more detail.
    If I found a flaw, I expect a cookie from everyone who believed the theory.
  2. jcsd
  3. Nov 28, 2003 #2
    Well, you probably misunderstand it, since almost everyone does, but aside from that, your basic argument regarding infinity is also flawed.

    Consider the space of all possible curves in Euclidean space. To define a curve by specifying all of the points on the curve, there are infinitely many choices for where you can put one point, infinitely many choices for where you can put another point, etc. (This is even more extravagant than your example, because we are allowing an infinity of possibilities for each choice, instead of a finite number.) Your same argument would apply to "prove" that it's impossible for curves to exist!

    The flaw is in your understanding of infinity. For instance, the Euclidean plane R2, which is the "square" of the real numbers R, is the same cardinality (same "size of infinity") as R itself. Crudely, infinity times infinity equals infinity.
  4. Nov 28, 2003 #3
    Assuming a Many-Worlds universe with a finite number of quanta, no matter how many finite bifurcating measurements you make, you still end up with a finite (although intractable) number of possible paths.

    Even if the quantum universe were infinite (in the sense of the counting numbers, integers or real numbers), there are still higher-order infinities which can represent larger cardinalities (counts exceeding the aforementioned number line zero cardinality), although I think Many-Worlds would conserve cardinality zero under the operation of exponentiation you mention.
  5. Nov 28, 2003 #4
    Re: Re: Simple contradiction/flaw in many worlds theory/theory explanation.

    x^2 can't equal x, regardless of what x stands for other than zero...
  6. Nov 28, 2003 #5
    Re: Re: Re: Simple contradiction/flaw in many worlds theory/theory explanation.

    If you want to treat infinity as a number, then x2 can indeed equal x, if x is infinity. (You also left out x=1 as something that equals itself when squared, by the way.)

    Of course, treating infinity as a number is problematic in the first place. But consider a set S with, say, 5 elements in it. The set SxS of pairs of elements has 52 = 25 elements in it. We can say that it has 25 elements, because we can place the elements of SxS in one-to-one correspondence with the elements of the set {1,2,...,25}.

    Now, consider the set of real numbers R. The set of pairs of real numbers, RxR = R2, can be placed in one-to-one correspondence with the set R itself. Thus, the "square" of an infinite set is the same "size" as the infinite set itself, unlike the square of a finite set.
  7. Nov 28, 2003 #6
    Re: Re: Re: Re: Simple contradiction/flaw in many worlds theory/theory explanation.

    UUUUUHHHHHHHHH I feel so stupid

    What's a one-to-one correspondence? The example you gave with the SxS example seemed to just be the numbers in the set S2...
  8. Nov 28, 2003 #7
    Re: Re: Re: Re: Re: Simple contradiction/flaw in many worlds theory/theory explanation.

    A bijective (one-to-one and onto) map. A map f:S->T is one-to-one if it doesn't map two elements in S onto the same element of T. It's onto if every element in T is mapped to by some element of S. A bijection has both those properties: it means that every element in S is mapped to one and only one element of T.

    There are no numbers in the set SxS. That set is the set of pairs of numbers. If S is the set {0,1,2,3,4} and T is the set {0,2,...,24}, then I can define a bijection f:SxS->T by:

    f((s1,s2)) = 5 s1 + s2

    where (s1,s2) is an element of SxS, i.e. s1 and s2 are both elements of S, e.g., (s1,s2)=(1,4) or something.

    Thus, I can say that the set SxS and the set T have the same cardinality ("are the same size"), because for every element in SxS, there is one and only one element of T.

    However, there is no bijection between SxS and S, i.e. the elements of those two sets cannot be placed in one-to-one correspondence with each other. (There is no onto map from S to SxS, which means that S is smaller than SxS.)

    On the other hand, there is a bijection between R and RxR, which means that the real line and the plane "have the same number of points". Here is one such bijection: interleave the digits of the two real numbers that make up an element of RxR, and you will get a real number (an element of R). Conversely, given any real number in R, I can turn it into an element of RxR by pulling out alternating digits and constructing two real numbers from them. To every point in R there corresponds one and only one point of RxR using this interleaving, so there are the same "number" of elements in R as there are in RxR.

    Example of interleaving:

    Given the element (0.1532,0.5438) of RxR, I can convert it into the element 0.15543328 of R; given the element 0.2854325710 of R, I can convert it into the element (0.25351,0.84270) of RxR.
  9. Nov 29, 2003 #8
    Re: Re: Re: Re: Re: Re: Simple contradiction/flaw in many worlds theory/theory explanation.

    You've lost me totally.
  10. Nov 29, 2003 #9
    Re: Re: Re: Re: Re: Re: Re: Simple contradiction/flaw in many worlds theory/theory explanation.

    There are as many points in the real line as there are points in the plane, because to every point in the real line you can associate a unique point in the plane, and vice versa.
  11. Nov 29, 2003 #10
    First, the example in the first post is of the form y=5^x, not y[now]=y[prev]^2.
    Second, here are the infinity rules(where 8=infinity and n=any real number):
    8*n=8, except 8*0=0
    8+n=8 and 8-n=8, except 8+8=8, 8-8=0
    8/n=8, except 8/8=1, 8/-8=-1, 8/0=?
    8^n=8, except 8^0=1, 8^(n<0)=0
    n^8=8, except (n<0)^8=-8, 0^8=0
    [n]rt8=8 except [8]rt8=0, [0]rt8=?, [n<0]rt8=0
    Okay, this is taking longer than I thought, and I'm not sure about some of it, since it doesn't come up often. If you take an intro calculus class, they cover all this while analysing functions with the concept of limit.
    Last edited: Nov 29, 2003
  12. Nov 29, 2003 #11
    It doesn't make sense to me that ifinity=infinity+6
  13. Nov 29, 2003 #12
    Well, infinity is not a particular number, it is a peculiar number.
  14. Nov 29, 2003 #13
    Well, if you try to treat infinity as if it was a number, you're going to run into oddities like that. If you have an infinite set of elements and add six more elements to it, you still have an infinite set of elements.
  15. Nov 29, 2003 #14
  16. Nov 29, 2003 #15
    If I understand, his paradox is only an example of Godel's incompleteness theorem.
  17. Nov 29, 2003 #16
    Your guess is as good as mine. From what I understood, it sounded like he was saying R doesn't equal R due to the fact that it does equal R...
  18. Nov 29, 2003 #17
    Well, maybe it is just an example of how one can't be too careful with infinity.
  19. Nov 29, 2003 #18
    Infinity is more a concept than a number. Cantor had a really neat way of showing orders of infinity (like oo and oo+6) equivalent by using a table which implies that such infinite sets' members map one-to-one.

    Finite sets and infinite sets behave differently, much like an algebraic solution differs from a differential limit.
  20. Dec 1, 2003 #19
    what russell's paradox shows is that in two valued logic, there can be no maximal infinity which is to say an infinity greater than all other infinities.

    note that RxR is of the same type of infinity as R.

    N, the set of natural numbers, is one level down roughly speaking.

    when you have an infinite set x, it's powerset (set of all its subsets) is always greater (look up cantor's diagonal argument). hence, N<P(N)<P(P(N))<... and so there is no largest infinity because its powerset would always be bigger. well, at least this suggests that a maximal infinity will never be achieved "from below" starting with N and by applying even an infinite amount of power set operations (and even any kind of infinite amount). as far as i can tell, the existence of a maximal infinity must be axiomatic though that's something russell's paradox rules out. blocked from below and blocked from above. in two valued logic, that is.

    technically, in the proof that |RxR|=|R|, one must treat the situation in which real numbers have two different decimal expansions a bit more carefully. for example, you have to say that your map from R to RxR does the same thing that it does to 1 as it does to 0.9999.... or at least deal with that somehow. it's got the right spirit as far as i can tell.

    so if there are x "events" (an unfortunate word for the abstraction of particles) in "instant 1" then there would be 2x events in "instant 2" which perhaps means one unit of plank time later. in the next, there are (2^2)x events. in general, (2^t)x. if x is finite, then so is the whole sequence x, 2x, 4x, 8x, 16x,... but if x is infinite, it reduces to being x, x, x, x, ....

    the nice thing about (2^t)x is that it can handle the case that time is actually discrete but very fine (ie something like divisible up to 10^(-43) seconds) or truly continuous.

    let's run this backwards and let t approach either 0 or -oo. if t can't be less than 0, then we have that the number of events at time zero is x, the number of particles the universe started with be that finite or infinite.

    if t secretly goes all the way back to -oo, then we have a more interesting situation in terms of the number of particles the universe started with. if x is finite, roughly speaking (2^-oo)x would be 0 which would indicate the universe started with no events and no particles. that would seem to contradict not only common sense, whatever that is, but the conservation of mass/energy. now what if x is infinite? then (2^-oo)x is not so clear. i think a more careful analysis of what kind of infinity -oo is and what kind of infinity x is would be in order for that one.

    it seems to be the case that upon iteration that events may be double counted. if you draw a diagram of three dots, each of which represents an event (eg a particle), and draw a total of six arrows (each dot shoots out two arrows) from the three dots you seem to get six dots. but what if arrow #1 sends event 1 to the same place arrow #3 sends event 2? then you've double counted and what could have been six separate events turn out to be five. therefore, (2^t)x is just an upper bound for the number of events.

    a scary thought arises when one considers that. in bifurcation diagrams, sometimes all the noisy events suddenly collapse into three events. from a huge number of bifurcated paths all of a sudden, poof, you have three events. will the universe suddenly become three particles one day and then suddenly become something like it once was the next day? well, that's not quite right because what i meant by event is the initial point of a new subuniverse. and so perhaps one day the intractible number of parallels will suddenly collapse into a small number of different sub-universes only to suddenly expand back into something of what it once was.

    if i haven't crossed the line into absurdity yet, let me try a little harder. perhaps any episode of precognition, prophetic dreams, deja vu, etc, are really instances of arrow #1 sending event/universe 1 to the same place that arrow #3 sends event/universe 2. and the deja vu feeling isn't permanent because the number of world lines (yes i know i'm using that wrong) re-expand from a reduced number back to a higher number, the subuniverses go back to being separate.

    still not absurd enough for ya? ok, how 'bout this. the subuniverses can be brought together with our minds or, perhaps more accurately, our minds can navigate the myriad realm of subuniverses and this is exactly where dreams come from; we navigate other worlds every night.

    some people claim to be able to navigate while awake though the words they use might be "i practice magick."

    i'll keep trying to come up with something even more absurd than that but my mind just blew up.
    Last edited: Dec 1, 2003
  21. Dec 2, 2003 #20

    ok, from what you said, I gathered that 3=6=5 is a statement which russels paradox supports... or maybe just 6=5...
  22. Dec 2, 2003 #21
    russell's paradox implies that there are statements that are both true and false though that's more accurately stated as neither true nor false.

    as far as i can tell, 3=5 or 5=6 are JUST false. as far as i can tell, one can't conclude 3=5 from russell's paradox.

    the law of logic russell's paradox is involved with is the law of the excluded middle: gotta be true or false but not both.

    the law of logic you could violate to get 3=5 is the law of identity, that everything equals itself. and then you've got 3=3 and 3!=3 (!= means does not equal). well, if 3!=3, maybe it's 5. so 3=3 and 3=5. again, this comes from violating the rules though, which is a small demonstration of why they shouldn't be violated.
  23. Dec 2, 2003 #22
    Are you sure about that last one? After all there is more than one type of infinity! If I were to subtract the infinity associated to the number of integers (aleph_0) from the infinity associated to the number of reals (aleph_1), then I don't think that I'll end up with zero...
  24. Dec 2, 2003 #23
    yeah, |P(N)|-|N|=|P(N)|, not 0.

    let n be the largest number you can think of. i'll conjecture that your n is greater than 1 for i need that assumption in what follows.

    let f(x) be a function defined on the reals such that f(x)=n^x.

    now consider g which is the nth iterate of f.

    now consider the limit of (g(x) - x/n) as x goes to infinity.

    in this case, oo-oo=oo, not 0.

    that's a pretty brutal example, like killing bacteria with a nuclear bomb. something a bit more elegant, maybe?

    let g(x)=x^(1.000000000000001) - x or just x^2 - x if you like.

    then the limit of g(x) as x approaches infinity is infinity, not zero. oo-oo=oo.

    you can also constuct examples where oo-oo=r for any real number r as well as r=oo and r=-oo.

    in conclusion, oo-oo is not 0.
  25. Dec 2, 2003 #24
    Similarly, oo/oo is not defined.
    For example (in each case take the limit x -> oo):

    1) x^2 / x --> oo / oo = oo

    2) x / x^2 --> oo / oo = 0

    3) c x / x --> oo / oo = c for any constant c (assuming c != oo).

    Infinity is fun!
  26. Dec 2, 2003 #25
    I was assuming that all the infinities used were the same type.
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