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In any discussion of the fundamental
structure of the Universe the word "infinity" invariably pops up. It's usually
meant to evoke a sense of wonder, but we may well ask what precisely do we mean
by that word? Actually, we can mean any one of several things by it, but here I
want to consider only the mathematical meaning. I can do that because in the
late 1800's Georg Cantor worked out the technical descriptions of an endless
number of infinities. The one I want to use is the smallest of them, the one
that Cantor called Aleph-Null.
Imagine that all of the counting
numbers, absolutely all of them, have been written down on a straight line. You
can see that the sequence has a beginning, at the number zero, but does it have
an end? To answer that question suppose that I show you a very big number and
claim that it's the last number in the sequence. You know that's nonsense
because you know that you need only add one to my number to generate the next
number in the sequence and you know that you can apply that procedure over and
over again and never reach a point at which you are compelled to stop. Thus, the
sequence has no end, which is just what the Latin word infinitas means.
Even though we can never come to the
end of the sequence of counting numbers, we nonetheless want to pretend that we
can do so and talk about the hypothetical last number in the sequence, the
number that can never be reached. We call that unreachable number Aleph-Null. We
know that we cannot draw that number in the way that we draw the counting
numbers, even the very largest that mathematicians use, so Cantor designated it
with the Hebrew letter Aleph to which a subscript of zero has been appended
(that subscript distinguishes it from Aleph-One, Aleph-Two, and the other alephs
that represent progressively bigger infinities). You might think that there's
nothing more that we can say about Aleph-Null, but what made Cantor's work
significant was that he showed how to extend the rules of arithmetic to include
operations carried out on Aleph-Null.
For our purpose I need only work out
one of those rules. Recall the sequence of numbers from above and imagine that
Cantor has come along, counted along that sequence to any one of its members,
and then, starting from that point, drawn the same sequence next to ours. Thus,
for example, if Cantor chooses to draw his zero next to our fifty-three, then he
will draw his one next to our fifty-four, his two next to our fifty-five, and so
on. He can go on drawing his numbers next to ours all the way along our
sequence, all the way toward the end that has no end, and as he does so he can
make an extraordinary claim: he can claim that a) we and he have drawn exactly
the same sequences, containing exactly the same elements, and b) that, even
though his sequence started somewhere past where ours did, his sequence will not
extend beyond ours.
But, we object, surely his sequence
must extend fifty-three elements beyond ours; after all, he began his sequence
where we had already drawn fifty-three elements and we and he have drawn all the
same elements, matched in pairs. Why would he not have fifty-three elements left
over when he comes to our last number? And, Cantor asks, what would that last
number be? The very asking of that question opens our mind's eye to the
understanding that his sequence will not extend beyond ours because there is no
last number relative to which there can be a beyond.
Having thus made the necessary preparation, we now subtract Cantor's sequence from ours. We accomplish that subtraction through the simple expedient of eliminating all matched pairs of numbers from the double sequence. Thus, we eliminate his zero and our fifty-three, his one and our fifty-four, his two and our fifty-five, and so on, eliminating all of the paired elements toward the end that has no end. When we have finished the elimination we have left only the unpaired elements, the first fifty-three elements of our sequence (0, 1, 2, 3, 4, ..., 50, 51, 52). We have thus subtracted, one from the other, two things that are the same and have come up with a nonzero remainder. From that fact we infer a simple rule for infinities:
Aleph-Null subtracted from Aleph-Null yields
any finite number as
That rule means that subtracting one
Aleph-Null from another, whether they come from counting different sequences or
from counting the same sequence at different times, is not guaranteed to yield
zero as the difference. Indeed, because we have no reason to believe that any
one finite number is more probable than any other as the difference, we can
state that the likelihood of the difference being zero is itself effectively
equal to zero. That proposition necessitates an important constraint upon the
laws of physics, one that will enable us to use the conservation laws to
describe space and time as something more than merely there and then.
The generic conservation law tells us that some property possessed by matter can be neither created nor destroyed, but can only be transferred from one body to another. That law can also be stated as a subtraction: if we are given an isolated system of bodies possessing some quantity of a conserved property, then subtracting the amount of that property that the system possesses at one time from the amount that it possesses at some other time must always necessarily yield zero. Viewing that rule in the light of the subtraction rule for Aleph-Null, I can now state
THE FINITE-VALUE THEOREM:
No conserved property can ever exist in an infinite quantity.
If that theorem were not
true to Reality, then the subtraction in the conservation law would not yield a
necessary zero and the property in question would not be necessarily conserved.
To get a feel for what that rule
means, let's consider something that is not conserved. Space is a good example.
We describe it as being made up of points and we describe the points as having
zero extent. How many things of zero extent must we take to span a nonzero
extent? An infinite number at the very least, of course. Every volume of space,
however small, contains an infinite number of points, which means that the
fundamental measure of space, extent, is not subject to a conservation law.
Space can expand or contract freely.
In contrast, consider angular
momentum. That is a conserved property that we describe as the product of a
mass, a velocity, and a distance. The finite-value theorem tells us that the
Universe must be so structured that there is no possibility of that product
being infinite. But we are free to set up our revolving body in any way that we
want within the Universe, so we must infer that none of the factors in that
product can possibly be infinite: that is, we infer that there cannot be an
infinite amount of mass in the Universe; there can be no motion at infinite
speed; and there can be no infinite distance.
Those three inferences may not seem impressive. I found them disappointing at first myself; they seemed to say so little about the physics of space and time. But, as I discovered, they are part of the combination that lets us unlock the deeper secrets of the Universe.
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