Revisiting the Finite-Value Theorem
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In AThe Shape of Space and Einstein= s Postulates@ I introduced the finite-value theorem, which I related to the conservation laws and used with conservation of angular momentum to necessitate the finitude of the extent of space. I wrote that essay before I discovered the error that Georg Cantor made in his work on infinity, so I incorporated the same error into my essay; I treated infinity as if it denotes an actual number rather than denoting the mathematical concept of endlessness. Now I want to revisit the finite-value theorem and correct that error.
When we describe a physical system we assign to certain of its features numbers that we obtain through various methods of comparison with certain standards. Thus we give bodies length, width, depth, mass, energy, linear momentum, electric charge, and other properties. We have a conservation law pertaining to one or another of those properties if we must say of some collection of bodies not interacting with any other phenomena that the amounts of the property possessed by those bodies adds up to the same number at all times, regardless of how those bodies interact with each other. If we have N bodies in the collection and we designate the amount of the property held by the i-th body at a time t1 by Pi(t1) and designate the amount of the property held by the j-th body at a time t2 by Pj(t2), then we express our conservation law by writing
in which the Greek letter sigma indicates summation over all N bodies.
To gain a clearer picture of what that equation means we can imagine the property under consideration as consisting of a set of Štherial tags, each tag bearing a unique index number and all tags having the same size. We then use P to represent the number of tags that a body carries. Equation 1 then expresses the idea that the bodies may exchange tags when they interact, one body perhaps giving more tags than it receives, but that no interaction can destroy tags or create new ones. The sole exception to that rule, and it does not properly stand apart from the rule, comes to us when we have both positive and negative versions of the tags: in that case a negative tag can cancel out a positive one and thus annihilate them both or some phenomenon can create a positive tag and a negative tag together.
Now I ask whether the tags can be elements of an infinite set. Recall that we define an infinite set, as Cantor did, as a set in which we can make a unique, bijective, one-to-one matching between the elements of the whole set and all of the elements of one of its proper subsets. In the classic example we match each and every positive integer with its double, thereby matching the set of the natural numbers with the set of the positive even numbers. If we want to subtract the set of the even numbers from the set of the counting numbers, then that matching tells us that we should expect to obtain the empty set as a result. But we also know that if we remove the subset of the even numbers from the full set of the counting numbers, we should have the subset of the odd numbers left as the difference. Because we get two mutually contradictory results from the same valid manipulation of the sets, we must dismiss from the realm of mathematics the premise that makes the contradiction possible. The only premise that we have available for dismissal is our tacit assumption that subtraction of infinite sets yields a definite conclusion. We must thus declare that the operation of subtraction is undefined for infinite sets.
Equation 1 stands as an absolute sine qua non of a conservation law: it must apply to all possible manifestations of the conserved quantity. That fact necessitates that all possible manifestations of the conserved quantity come under the rule of subtraction (among others). And that fact, combined with the fact that we cannot define the operation of subtraction for infinite sets, necessitates that the tags representing any conserved quantity cannot be elements of an infinite set. Thus we have the finite-value theorem.
Like any good theorem, this one has interesting consequences. Each of the tags that we assign to a body denotes a standard unit of the conserved quantity and represents some object or phenomenon by comparison with which we count the conserved quantity. If we measure the mass of a body in kilograms, then each tag corresponds to the inertia of a squat platinum-iridium cylinder kept in a vault somewhere in Paris, France. We can replace each kilogram tag with one thousand tags, each denoting one gram, and then we can split those tags even further, thereby multiplying the number of tags that we assign to a given body. But we cannot continue that process endlessly and thereby create an infinite set of mass-tags: we must reach a point beyond which we can no longer subdivide the tags. With mass we will discover that any attempt to split the lightest possible particle requires so much energy that the process we apply merely creates new particles and antiparticles, none of which carries less mass than the original particle does. We might call this Leukippos= theorem, after the Greek philosopher who first conceived the idea of the atomos (the uncuttable) and the idea that matter consists of atoms in ever-shifting agglomerations.
Let us note that the principle of least action also involves a subtraction that must yield a perfect zero, so action comes under the finite-value theorem as if it were a conserved quantity. Thus we infer the existence of an absolute minimum quantity of action, which we denote as Planck=s constant. When we pursue the consequences of that fact we will obtain the quantum theory.
Now I want to look at angular momentum. As part of developing the quantum theory we will discover that an absolute minimum of angular momentum exists and equals one half aitch-bar (that is, one half of Planck=s constant divided by two pi). But I want to look at macroscopic manifestations of angular momentum. On the microscale the angular momentum tags come to us as indivisible wholes, but on the macroscale each angular-momentum tag comes to us as a collage of three other tags, brought together in an act of multiplication: one tag represents mass, one represents a velocity, and the other represents a distance measured in a direction perpendicular to the direction pointed by the velocity. We have already seen that even if the mass-tag represents all of the matter in the Universe, we cannot subdivide it into an infinite set of smaller tags.
The velocity-tag represents the ratio of a distance crossed to a time elapsed. Assume that the tag represents one meter per second. We know that tag cannot be an element of an infinite set (if it were, then angular-momentum tags would also be elements of an infinite set), so we infer that the Universe has a natural speed limit, a finite number of meter-per-second tags, of which no body can have more. I know that I can add velocity-tags to a body or subtract velocity-tags from a body by subjecting myself to an acceleration. But if some phenomenon passes me traveling at that natural speed limit, I know that Reality has such a structure that I cannot add more velocity-tags to it through any amount of self-acceleration: if that proposition does not stand true to Reality, then I can add new velocity-tags to the phenomenon endlessly and that fact would make the velocity-tags elements of an infinite set.
Having attempted to add velocity-tags to the phenomenon by accelerating in a direction opposite the phenomenon=s motion, can I reverse my acceleration and subtract velocity-tags from the phenomenon? If I can, then when I return to the inertial reference frame whence I began my imaginary journey I will see the phenomenon moving at a speed less than the natural speed limit, which means that I will have given the phenomenon a net change of motion without ever touching it (talk about spooky actions at a distance!). I say that, because Reality exists independent of me, such a thing will not happen.
In my effort to add velocity to the phenomenon I visited, however temporarily, many inertial frames of reference and in each of them the phenomenon moved at the natural speed limit. In my effort to subtract velocity from the phenomenon I revisit those frames and must find that observers in those frames have detected no change in the motion of the phenomenon; therefore, I must also detect no change in the motion of the phenomenon. That conclusion must stand true to Reality because the independence of Reality from its observers necessitates that I (or any other observer) can only change the characteristics of a phenomenon through direct interaction with it. Thus, we infer that anything that moves at the natural speed limit moves at that same speed in all inertial frames, in any direction, even if our meter-per-second velocity-tags differ from each other (and if we occupy different inertial frames, they will). Of course we recognize the natural speed limit as the speed of light and that statement as one version of Einstein=s second postulate of Relativity.
The fact that the inertial reference frames that make up space and time also comprise an infinite set might seem to moot that analysis. After all, we can subdivide space and time endlessly, as we do when we put the set of the real numbers onto a line to label its points, so we can certainly subdivide their ratio endlessly. If we represent the relative velocities of inertial frames on a line, then between any two inertial frames corresponding to points on that line an infinite set of inertial frames exists. But space and time don=t come under a conservation law (space couldn=t expand if it did), so we have no expectation that the finite-value theorem applies to them in themselves.
Velocity as a factor in a conserved quantity differs from velocity-in-itself, though we take the former to mark the latter. We need to ask, not of velocity-in-itself, but of the velocity of a body whether we can subdivide it endlessly. Can we make the changes of velocity that we impose on a body endlessly small? Changing a body=s velocity comes from changing that body=s linear momentum, which necessitates enacting at least a minimum unit of action (Planck=s constant, as noted above). Thus, since we calculate action as the product of a change of linear momentum and distance over which the action takes place, we can only make an infinitely small change in a body=s linear momentum if we have an infinitely large distance over which to enact that change.
Linear momentum, the product of a body=s mass and its velocity, comes under a conservation law, so we cannot make an infinitely small change in any body=s linear momentum (to do so would be to make the tags representing the minimum change of linear momentum elements of an infinite set). Likewise, we cannot make an infinitely large mass, so we cannot make an infinitely small change in any body=s velocity, which answers the question above. As a bonus we discover that we cannot have an infinitely large distance in space, because we cannot have an infinitely small linear momentum; thus, space must have only finite extent. And likewise again, because we cannot have an infinitely large linear momentum, so we cannot have an infinitely small distance involved in any interaction. Thus the finite-value theorem constrains the ways in which space can manifest itself to us through phenomena that involve conserved quantities.
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