Argument from the impossibility of an actual infinite

One of the traditional arguments for the finitude of the past is based upon the impossibility of the existence of an actual infinite. It may be formulated as follows:

2.11. An actual infinite cannot exist.

2.12. An infinite temporal regress of events is an actual infinite.

2.13. Therefore, an infinite temporal regress of events cannot exist.

In order to assess this argument, we need to have a clear understanding of its key terms. First and foremost among these is "actual infinite." Prior to the revolutionary work of mathematicians Bernard Bolzano (1781-1848), Richard Dedekind (1831-1916), and, especially, Georg Cantor (1845-1918), there was no clear mathematical understanding of the actual infinite (Moore 1990, pt. I). Aristotle had argued at length that no actually infinite magnitude can exist (Physics 3.5.204b1-206a8). The only legitimate sense in which one can speak of the infinite is in terms of potentiality: something may be infinitely divisible or susceptible to infinite addition, but this type of infinity is potential only and can never be fully actualized (Physics 8.8. 263a4-263b3). The concept of a potential infinite is a dynamic notion, and strictly speaking, we must say that the potential infinite is at any particular time finite.

This understanding of the infinite prevailed all the way up to the nineteenth century. But although the majority of philosophers and mathematicians adhered to the conception of the infinite as an ideal limit, dissenting voices could also be heard. Bolzano argued vigorously against the then current definitions of the potential infinite (Bolzano 1950, pp. 81-4). He contended that infinite multitudes can be of different sizes and observed the resultant paradox that although one infinite might be larger than another, the individual elements of the two infinites could nonetheless be matched against each other in a one-to-one correspondence (Bolzano 1950, pp. 95-6).4 It was precisely this paradoxical notion that Dede-kind seized upon in his definition of the infinite: a system is said to be infinite if a part of

4. Despite the one-to-one correspondence, Bolzano insisted that two infinites so matched might nevertheless be nonequivalent.

that system can be put into a one-to-one correspondence with the whole (Dedekind 1963, p. 63). According to Dedekind, the Euclidean maxim that the whole is greater than a part holds only for finite systems.

But it was undoubtedly Cantor who won for the actual infinite the status of mathematical legitimacy that it enjoys today. Cantor called the potential infinite a "variable finite" and attached the sign ^ (called a lemniscate) to it; this signified that it was an "improper infinite" (Cantor 1915, pp. 55-6). The actual infinite he pronounced the "true infinite" and assigned the symbol (aleph zero) to it. This represented the number of all the numbers in the series 1, 2, 3, . . . and was the first infinite or transfinite number, coming after all the finite numbers. According to Cantor, a collection or set is infinite when a part of it is equivalent to the whole (Cantor 1915, p. 108). Utilizing this notion of the actual infinite, Cantor was able to develop a whole system of transfinite arithmetic. "Cantor's . . . theory of transfinite numbers . . . is, I think, the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity," exclaimed the great German mathematician David Hilbert. "No one shall drive us out of the paradise which Cantor has created for us" (Hilbert 1964, pp. 139, 141).

Modern set theory, as a legacy of Cantor, is thus exclusively concerned with the actual as opposed to the potential infinite. According to Cantor, a set is a collection into a whole of definite, distinct objects of our intuition or of our thought; these objects are called elements or members of the set. Fraenkel draws attention to the characteristics definite and distinct as particularly significant (Fraenkel 1961, p. 10). That the members of a set are distinct means that each is different from the others. To say that they are definite means that given a set S, it should be intrinsically settled for any possible object x whether x is a member of S or not. This does not imply actual decidability with the present or even future resources of experience; rather a definition could settle the matter sufficiently, such as the definition for "transcendental" in the set of all transcendental numbers.

Unfortunately, Cantor's notion of a set as any logical collection was soon found to spawn various contradictions or antinomies within the naive set theory that threatened to bring down the whole structure. As a result, most mathematicians have renounced a definition of the general concept of set and chosen instead an axiomatic approach to set theory, by means of which the system is erected upon several given, undefined concepts formulated into axioms. An infinite set in the Zermelo-Fraenkel axiomatic set theory is defined as any set R that has a proper subset that is equivalent to R. A proper subset is a subset that does not exhaust all the members of the original set, that is to say, at least one member of the original set is not also a member of the subset. Two sets are said to be equivalent if the members of one set can be related to the members of the other set in a one-to-one correspondence, that is, so related that a single member of the one set corresponds to a single member of the other set and vice versa. Equivalent sets are regarded as having the same number of members. This convention has recently been dubbed as Hume's Principle5 (on the basis of Hume 1978, bk, I, pt. iii, sec. 1, p. 71). An infinite set, then, is one that is such that the whole set has the same number of members as a proper subset. In contrast to this, a finite set is a set that is such that if n is a positive integer, the set has n members. Because set theory does not utilize the notion of potential infinity, a set containing a potentially infinite number of members is impossible. Such a collection would be one in which the

5. The appellation is due to Boolos (1986-7).

membership is not definite in number but may be increased without limit. It would best be described as indefinite. The crucial difference between an infinite set and an indefinite collection would be that the former is conceived as a determinate whole actually possessing an infinite number of members, while the latter never actually attains infinity, although it increases perpetually. We have, then, three types of collection that we must keep conceptually distinct: finite, infinite, and indefinite.

When we use the word "exist," we mean "be instantiated in the mind-independent world." We are inquiring whether there are extratheoretical correlates to the terms used in our mathematical theories. We thereby hope to differentiate the sense in which existence is denied to the actual infinite in (2.11) from what is often called "mathematical existence." Kasner and Newman strongly differentiate the two when they assert, "'Existence' in the mathematical sense is wholly different from the existence of objects in the physical world" (Kasner & Newman 1940, p. 61). "Mathematical existence" is frequently understood as roughly synonymous with "mathematical legitimacy." Historically, certain mathematical concepts have been viewed with suspicion and, therefore, initially denied legitimacy in mathematics. Most famous of these are the complex numbers, which as multiples of V-1, were dubbed "imaginary" numbers. To say that complex numbers exist in the mathematical sense is simply to say that they are legitimate mathematical notions; they are in that sense as "real" as the real numbers. Even negative numbers and zero had to fight to win mathematical existence. The actual infinite has, similarly, had to struggle for mathematical legitimacy. For many thinkers, a commitment to the mathematical legitimacy of some notion does not bring with it a commitment to the existence of the relevant entity in the non-mathematical sense. For formalist defenders of the actual infinite such as Hilbert, mere logical consistency was sufficient for existence in the mathematical sense. At the same time, Hilbert denied that the actual infinite is anywhere instantiated in reality. Clearly, for such thinkers, there is a differentiation between mathematical existence and existence in the everyday sense of the word. We are not here endorsing two modes of existence but simply alerting readers to the equivocal way in which "existence" is often used in mathematical discussions, lest the denial of existence of the actual infinite in (2.11) be misunderstood to be a denial of the mathematical legitimacy of the actual infinite. A modern mutakallim might deny the mathematical legitimacy of the actual infinite in favor of intu-itionistic or constructivist views of mathematics, but he need not. When Kasner and Newman say, "the infinite certainly does not exist in the same sense that we say, 'There are fish in the sea' " (Kasner & Newman 1940, p. 61), that is the sense of existence that is at issue in (2.11).

These remarks make clear that when it is alleged that an actual infinite "cannot" exist, the modality at issue is not strict logical possibility. Otherwise the presumed strict logical consistency of axiomatic set theory would be enough to guarantee that the existence of an actual infinite is possible. Rather what is at issue here is so-called metaphysical possibility, which has to do with something's being realizable or actualizable. This sort of modality, in terms of which popular possible worlds semantics is typically formulated, is often characterized as broadly logical possibility, but here a word of caution is in order. Insofar as by broadly logical possibility one means merely strict logical possibility augmented by the meaning of terms in the sentence within the scope of the modal operator, such a conception is still too narrow for the purposes of the present argument. Such a conception would enable us to see the necessity of analytic truths in virtue of logic and the meaning of sentential terms used in the expression of these truths (such as "All bachelors are unmarried"), but it will not capture the metaphysical necessity or impossibility of synthetic truths, whether these be known a priori (such as "Everything that has a shape has a size") or a posteriori (such as "This table could not have been made of ice"). Broad logical possibility, then, will not be broad enough for a proper understanding of the argument unless synthetic truths are among those truths that are classed as necessary.

The fact that the argument is framed in terms of metaphysical modality also has an important epistemic consequence. Since metaphysical modality is so much woollier a notion than strict logical modality, there may not be the sort of clean, decisive markers of what is possible or impossible that consistency in first-order logic affords for strict logical modality. Arguments for metaphysical possibility or impossibility typically rely upon intuitions and conceivability arguments, which are obviously much less certain guides than strict logical consistency or inconsistency. The poorly defined nature of metaphysical modality cuts both ways dialectically: on the one hand, arguments for the metaphysical impossibility of some state of affairs will be much more subjective than arguments concerning strict logical impossibility; on the other hand, such arguments cannot be refuted by facile observations to the effect that such states of affairs have not been demonstrated to be strictly logically inconsistent.

Premise (2.12) speaks of a temporal regress of events. By an "event," one means any change. Since any change takes time, there are no instantaneous events so defined. Neither could there be an infinitely slow event, since such an "event" would, in reality, be a changeless state. Therefore, any event will have a finite, nonzero duration. In order that all the events comprised by the temporal regress of past events be of equal duration, one arbitrarily stipulates some event as our standard and, taking as our point of departure the present standard event, we consider any series of such standard events ordered according to the relation earlier than. The question is whether this series of events comprises an actually infinite number of events or not. If not, then since the universe cannot ever have existed in an absolutely quiescent state, the universe must have had a beginning. It is therefore not relevant whether the temporal series had a beginning point (a first temporal instant). The question is whether there was in the past an event occupying a nonzero, finite temporal interval which was absolutely first, that is, not preceded by any equal interval.6

With these explications in mind, let us now turn to an examination of the argument's two premises.

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  • nahand
    Is an actual infinite an actual impossibility?
    5 years ago

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