Prohibition of Actual Infinities

Summary of Hilbert's Prohibition of Actual Infinities

In order to expedite the explanation of Hilbert’s prohibition, it will be helpful to draw a distinction between three kinds of infinity which are genuinely distinct from one another and cannot be used as analogies for one another. This will show why Hilbert’s prohibition only applies to C infinities (infinities hypothesized to be within algorithmically finite structures).

Three Kinds of Infinity-

For the sake of convenience, I will term these three kinds of infinity A, B, and C:

1) “A-infinity.” “Infinite” frequently has the meaning of “unrestricted,” (e.g., “infinite power” means “unrestricted power”). It can only be conceived through the “via negativa,” that is, by disallowing or negating any magnitude, characteristic, quality, way of acting, or way of being which could be restricted or introduce restriction into an infinite (unrestricted) power. Therefore, “infinite,” here, is not a mathematical concept. It is the negation of any restriction (or any condition which could introduce restriction) into power, act, or being.

2) “B-infinity.” “Infinite” is also used to signify indefinite progression or indefinite ongoingness. An indefinite progression is never truly actualized. It is one that can (potentially) progress ad infinitum. Examples of this might be an interminably ongoing series, or an ever-expanding Euclidean plane. The series or the plane never reaches infinity; it simply can (potentially) keep on going ad infinitum. Thus, Hilbert calls this kind of infinity a “potential infinity.”

3) “C-infinity.” “Infinite” is sometimes used to signify “infinity actualized within an algorithmically finite structure.” Mathematicians such as Georg Cantor hypothesized a set with an actual infinite number of members (a Cantorian set) which would not be a set with an ever-increasing number of members or an algorithm which could generate a potential infinity of members. Examples might be an existing infinite number line, or an existing infinite spatial manifold, or the achievement of an infinite continuous succession of asymmetrical events (i.e., infinite past history).

The Hilbertian prohibition applies to the C-infinity alone, for, as will be seen, it is not concerned with non-mathematical infinities (i.e., an A-infinity), and it permits indefinitely ongoing (continually potential) successions through algorithmically finite structures (i.e., B-infinities). Before showing Hilbert’s and others’ prohibition of C-infinities, the two permissible kinds of infinities will be discussed.

An A-infinity has long been recognized by the Scholastic tradition.^[4] As noted above, it is not a mathematical infinity (such as infinite sets, infinite number lines, infinite successions, etc.) and it is not applied to algorithmically finite structures (such as spatial magnitudes, temporal magnitudes, fields, forces, etc.). Hence, it does not postulate an infinite Euclidean plane, infinite past time, an infinite number line, infinite space, infinite history, infinite thermometers, infinite density, or an infinite physical force. An A-infinity is simply the recognition of “non-restrictedness” in power. It is, therefore, a negation of any predicate which has restriction or could imply restriction in an infinite power.

As Scholastic philosophers have long recognized, one can only speak about “infinite power” or “infinite being” by negating any restriction (or structure giving rise to a restriction such as a divisible magnitude) to the power itself. Thus, one can say that “infinite power” is not restricted as to form, way of acting, space-time point, or even to spatiality itself (which is a divisible magnitude having intrinsically finite parts).

Such negative statements are not equivalent to “no knowledge” or unintelligibility; for one does know that infinite power does not have a restriction. Yet, at the same time, one cannot positively imagine (through, say, picture-thinking) what such unrestricted power would be. Every image we have is likely to restrict the entity we are conceiving either intrinsically or extrinsically.

Our inability to conceive or imagine this entity does not in any way rule out its possibility, for our inability to conceive of it does not reveal an intrinsic contradiction or “an extrinsic contradiction with some existing reality;” it merely admits the limits of our spatializing, temporalizing, finitizing imagination and conception. Thus, as we shall see, Hilbert’s prohibition of an “actual infinity” does not extend to an A-infinity, for an A-infinity is neither a mathematical infinity nor an application of infinity to an algorithmically finite structure. Interestingly enough, Hilbert’s prohibition of a C-infinity could actually constitute a proof for an A-infinity.

A B-infinity is quite distinct from an A-infinity because it is both a mathematical infinity and an application of infinity to an algorithmically finite structure. Unlike the prohibited C-infinity, the B-infinity applies a mathematical infinity to an algorithmically finite structure in only a potential way. Therefore, the B-infinity only acknowledges the possibility that an algorithmically finite structure could continue to progress indefinitely.

Thus, the B-infinity does not imply that a Cantorian set (with an infinite number of members) actually exists. It implies that a particular algorithm (sufficient to define the set) can continue to generate members indefinitely. Furthermore, it does not hold that an infinite number line actually exists, but rather than one can continue to generate numbers on the line indefinitely. The existence (completion or achievement) of an infinite number line is never advocated, but only the potential to continue to generate numbers according to a particular algorithm indefinitely.

The same holds true for magnitudes such as space (a contemporaneous magnitude) and time (a non-contemporaneous magnitude). A potential infinity implies that a spatial magnitude has the potential to continue expanding indefinitely. Similarly, it holds that a non-contemporaneous succession of events has the potential to continue indefinitely (into the future). It does not imply that an infinite spatial magnitude really exists or that an infinity of continuously successive historical events actually occurred.

The Hilbertian prohibition does not apply to a B-infinity because one is not advocating the existence (actuality) of a mathematical infinity within an algorithmically finite structure. One is only advocating the potential to increase an algorithmically finite structure indefinitely according to a particular algorithm. As we shall see momentarily, infinity applied to the succession of future events will not give rise to a Hilbertian paradox because future events are only potential. An infinity never exists. Future time can only be an indefinitely increasing succession of events; never the existence (actuality) of a mathematical infinity. As will be seen, such is not the case with past time, which explains why infinite past time falls under the Hilbertian prohibition.

A C-infinity, like a B-infinity, is both a mathematical infinity and an application of infinity to an algorithmically finite structure. The important difference, however, between the B and C-infinities is that the C-infinity implies the existence (actuality) of a mathematical infinity within an algorithmically finite structure. As noted above, examples of a C-infinity would be an actual Cantorian set with an actual infinite number of members, or an infinite number line with an actual infinite number of positions, or an actually existing infinite spatial magnitude, or an actual occurrence of an infinite number of events in the past. Thus, if C-infinities could really exist, there could be infinite space, infinity degrees Fahrenheit, infinite mass density, infinite physical force, and infinite past time. These notions seem irresolvably paradoxical prima facie, because the mathematical infinity applied to them utterly destroys their intelligibility as algorithmically finite structures. The proof for this goes beyond prima facie intuition. It extends to the requirements for mathematical intelligibility itself. Thus, as Hilbert shows, a mathematical infinity existing within an algorithmically finite structure undermines the very possibility of finite mathematics, and therefore the very possibility of quantifying those algorithmically finite structures. Therefore, a C-infinity must, in all cases, be illusory.

Now, it was shown above that the succession of past events is a real, non-contemporaneously distended, interactive, asymmetrically related, continuously successive whole. As such, it must be an actual asymmetrical progression. It does not matter that past events no longer exist, because all past events did exist and affected, and were related to, present events as they passed out of existence. Thus, they constitute a real past progression. This is sufficient to qualify “a past succession of events” for Hilbert’s prohibition, because the application of an infinity to it must be a C-infinity (not a B-infinity).

If a C-infinity must in all cases be rejected (because it entails the undermining of finite mathematics and the quantification of algorithmically finite structures), then an infinite past succession of events must also be rejected. This will be shown first by summarizing Hilbert’s (and others’) prohibition of C-infinities and second through a formal proof which illustrates the contradictory and incoherent nature of the C-infinity applied to past time.

It is important not only to distinguish among these three kinds of infinity, but also to avoid analogizing one with the other. Thus, infinite future time cannot be a proper analogy for infinite past time. As can be seen, they are quite distinct (a B-infinity versus a C-infinity, respectively). Furthermore, infinite future time cannot be used as an analogy for infinite power (a B-infinity versus an A-infinity, respectively). The rules for each kind of infinity do not apply meaningfully to the other kinds.

The Mathematical Prohibition of C-Infinities

The above discussion was brought to the attention of philosophers of mathematics by David Hilbert, who attempted to clarify the notion of an infinite numeric series which was thought to exist as a completed totality:

Just as in the limit processes of the infinitesimal calculus, the infinite in the sense of the infinitely large and the infinitely small proved to be merely a figure of speech, so too we must realize that the infinite in the sense of an infinite totality, where we still find it used in deductive methods, is an illusion.^[5]

Hilbert is proposing here that, even though a B-infinity (one with the potential to continue indefinitely without being actual) is mathematically admissible, a C-infinity (the existence of a mathematical infinity within an algorithmically finite structure) is not mathematically admissible because it presents irresolvable paradoxes and contradicts the very axioms of finite mathematics. In recounting the history of the “actual infinite” (Hilbert’s designation of a C-infinity from Georg Cantor’s actual infinite set of numbers) Hilbert notes that the Russel-Zermelo paradox presents so many devastating contradictions that it nearly undermined deductive procedure in mathematics:

These contradictions, the so-called paradoxes of set theory, though at first scattered, became progressively more acute and more serious. In particular, a contradiction discovered by Zermelo and Russell had a downright catastrophic effect when it became known throughout the world of mathematics. Confronted by these paradoxes, Dedekind and Frege completely abandoned their point of view [belief in the coherency of an infinite set as proposed by Cantor] and retreated.^[6]

Hilbert then concludes that the technique of ideal elements (which can imply infinities) cannot be used if they change the fundamental axioms of finite numbers to which they have been applied. Since this does not occur with potential infinities (B-infinities), but always occurs with actual infinities (C-infinities), Hilbert rejects the use of the latter in any way that could apply to the real world (i.e., real magnitudes, real counting, real series, etc.):

In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought – a remarkable harmony between being and thought. … The role that remains for the infinite to play is solely that of an idea – if one means by an idea, in Kant’s terminology, a concept of reason which transcends all experience and which completes the concrete as a totality [a B-infinity]…. ^[7]