Re: An infinite number question


Raymond Griffith wrote:
> On 1/22/06 1:28 AM, in article
> 1137911317.459299.301020@xxxxxxxxxxxxxxxxxxxxxxxxxxxx, "rev.goetz"
> <jimgoetz316@xxxxxxxxx> wrote:
>
> >
> > Raymond Griffith wrote:
> >> On 1/22/06 12:29 AM, in article
> >> 1137907765.563011.113340@xxxxxxxxxxxxxxxxxxxxxxxxxxxx, "rev.goetz"
> >> <jimgoetz316@xxxxxxxxx> wrote:
> >>
> >>>
> >>> flabbergasted wrote:
> >>>> rev.goetz wrote:
> >>>>> I am working on a theoretical issue that deals with infinity, and I
> >>>>> need to know the following: does an infinite number divided by an
> >>>>> infinite number equal 0 or 1?
> >>>>>
> >>>>> I tend to think that the answer is 0, but I could also see that it may
> >>>>> equal 1. Can anybody help me with this?
> >>>>
> >>>> Your question is meaningless without additional information. What kind
> >>>> of infinity?
> >>>>
> >>>> The concept of infinity includes quantities that are different sizes.
> >>>> For instance, there are an infinite number of integers and an infinite
> >>>> number of real numbers, but the two sets aren't the same size. The
> >>>> infinity of real numbers is bigger than the inifinity of integers.
> >>>>
> >>>> If you have a more practical problem, such as taking the limit of
> >>>> exp(x)/x as x goes to infinity, then you can use l'Hospital's rule. Of
> >>>> course, you would know that if you had ever had a calculus class. If
> >>>> you haven't studied calculus, then the chance of your producing
> >>>> anything more than mathematical gibberish is pretty small.
> >>>
> >>> Okay, I am dealing with probability where a proportion has an infinite
> >>> number of possibilities. And in this case, there are also an infinite
> >>> number of trials.
> >>>
> >>> So if a proportion has an infinite number of possibilities while there
> >>> are an infinite number of trials, then is the probability equal to 0?
> >>> Or is the probability equal to 1 with a standard deviation equal to 1?
> >>
> >> Let me take an example. Suppose you were taking a random value between 0 and
> >> 1, where every real number could be chosen.
> >>
> >> In that case, the probability of any particular numerical value being chosen
> >> would be 0. But the probability of the value chosen being, say, between 1/6
> >> and 1/2 would be 1/3.
> >>
> >> You would need to define your sample space a bit more carefully. Describe
> >> how you are choosing your values, and what values are able to be chosen. The
> >> kind of distribution for your values is also important. A normal
> >> distribution will look differently than a uniform distribution. If you are
> >> doing anything special in how your numbers are chosen, we need to know.
> >>
> >> If you do that, we can answer your question.
> >>
> >> Raymond E. Griffith
> >
> > I am looking at the probabilities of a hypothetical World Ensemble with
> > an infinite number of universes, which includes two models: 1) an
> > infinite number of past universes, 2) an infinite of parallel
> > universes. For example, Collins and Hawking (1973) noted that the
> > origin of the universe faced an infinite number of physical values, and
> > that a World Ensemble with an infinite number of universes would
> > exhaust all possible universes.
> >
> > And I break this down to my problem where a proportion has an infinite
> > number of possibilities that faces an infinite number of trials.
> >
> > Part of the problem involves and infinite number of possible values for
> > the four fundamental constants and an infinite number of possible
> > values for the Riemannian tensor constants. And I doubt that I could
> > list all of the possible factors involved.
> >
> > Is this enough information?
> >
> > Reference
> > C.B. Collins and S.W. Hawking, "Why is the Universe Isotropic?,"
> > The Astrophysical Journal 180 (1973), 317-44.
> >
>
> Hmmm. A little clearer, but we need more focus.
>
> Each possible universe could have a finite subset of an infinite set of
> physical values?
>
> Or
>
> Each possible universe could have the same finite set of physical
> characteristics (like having a speed of light, a gravitational constant, a
> mass of an electron, etc), but each set of characteristic could have an
> infinite set of values. Please note that an infinite set of values does not
> mean that values could not be confined within a range. One could confine the
> set and still have an infinite number of real-valued choices.
>
> I'm leaning closer to this one since you talk about the four fundamental
> constants, but I am unclear as to the infinite number of trials part. If by
> that you mean that there are an infinite number of values they might have
> taken on, then good. If you mean that different universes might have
> different variables they rely on as fundamental, and not just the values
> those variables took on, then I need more information.
>
> If your particular constants are real-valued, then the probability of
> hitting any one particular value would be zero. If your values are confined
> within a finite range, and you are able to choose values within a range,
> then we could assign a nonzero probability. For example, if your values were
> confined to, say, (0, 10] and you wished to know the probability of getting
> a value in [1,3], I could tell you it would be 20%. The real valued set of
> of possibilities is still infinite, and each possibility has a mass of zero.
> But the range of acceptable possibilities has mass.
>
> I understand that this is rather technical. This is why I am trying to pin
> down precisely what you have in mind.
>
> You have noted the four fundamental constants. I have noted with some
> amusement the design theorists trying to use the cosmological constants as
> proof of design because they had to be exactly such and such. The same
> argument was used for the genome, of course, and failed miserably (although
> it is still used by people who don't know the flexibility of the genome!). I
> have heard the discussion that if this number was different by such and such
> then life could not have occurred. I tend to add such mental disclaimers as
> "as we know it" to these kinds of claims because for all of our
> calculations, we really don't know how the different universe would behave.
>
> Ultimately, the probability of hitting a particular point in an interval is
> zero if the two following conditions hold:
>
> 1. Every real-value (or even every rational-value) within the interval could
> be chosen.
> 2. The choice is truly random.
>
> Practically speaking, though, in a simulated environment the probability of
> hitting a particular value is not zero. That is because our random-number
> generators cannot generate irrational values and our rational numbers are
> limited to a specific decimal length. That violates rule 1. And our best
> random number generators are cyclic (though with long, long cycles!), which
> means that some values *cannot* be chosen (violation of rule 2). Not to
> mention that the generation from a starting "seed" also violates the notion
> of true randomness.
>
> I have a TI-83 calculator. It is capable of generating 10^10 random numbers
> between 0 and 1. That is good enough for most work, but the production of a
> random number like 0.5682328953 tends to make us forget that this is still a
> rational number.
>
> The reason I bring this up is because it illustrates another point. We
> really do not know what the four fundamental constants are. We have very,
> very good measurements of them -- but only to a certain point. There is
> associated with our measurements a range of error. In a comparison of two
> universes, it would be possible (I suppose) to have two different sets of
> constant values which, while truly different, are also beyond our ability to
> distinguish between.
>
> I know I have rambled a bit. I hope this helps you. If you can clear up a
> bit more what you are working on, then perhaps I can be of more assistance.
>
> Regards,
>
> Raymond E. Griffith

My ultimate goal is to evaluate the claim made by Collins and Hawking
(1973) that an infinite number of universes would exhaust all possible
universes. I suspect that they have no mathematical basis for their
statement. And it seems like many scientists and philosophers accept
their statement without criticism.

I am in the early stages of looking into this, and this thread is
helping me to focus on the issue. I will reevaluate the material, and
develop a clearer approach to the issue.

In the mean time, here is my question:

Do you think that Collins and Hawking (1973) have a mathematical basis
for their claim that an infinite number of universes would exhaust all
possible universes?

.

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