When I learned in November 2006 that Prometheus Press would soon publish The Kalam Cosmological Argument for God [TKCAG] by Mark R. Nowacki, I immediately placed an order with Amazon Books. I was therefore very pleased to have finally received my copy in the first week of August of this year despite several unexplained deferments of the publication date. I had been looking forward to reading TKCAG because of my own keen interest in the KCA, as is evidenced by my article on that subject (“The Kalam Cosmological Argument: The Question of the Metaphysical Possibility of an Infinite Set of Real Entities”) that appeared in Philo 5 (2002): 196-215 (thereafter electronically published [2003, updated 2005] by the Secular Web, as well as by two follow-up articles on that website[1]). I am now in the course of writing a review article of Nowacki’s concededly interesting book for eventual submission to some philosophical journal. However, as an interim measure, I write this blog to call the reader’s attention to a serious defect in Nowacki’s book—one that concerns me personally.

The reader will indulgently understand my eagerness in first checking whether Nowacki cited and discussed my articles in his book.[2] I discovered that he cited (only) my Philo article and referred to it (TKCAG, p. 122) in his statement of an objection to KCA:

What the KCA thought experiments directed against the actual infinite teach us is that any attempt to straightforwardly apply Cantor’s theory to the real world is misguided. Rather, when discussing the possibility of an infinite multitude of real entities, we should adopt a version of Bernard Bolzano’s theory of the infinite, wherein both the principle of correspondence and Euclid’s principle hold true. Once a Bolzano-inspired account of the infinite is adopted, the absurd consequences drawn in the various KCA thought experiments are dissolved and an infinite multitude of real things is shown to be possible.90

The statement of this objection is immediately followed by Nowacki’s Response.[3] But first I quote his endnote 90 (TKCAG, p. 152): “Although Guminski does not acknowledge an intellectual debt to Bolzano, this is the central objection raised in [Guminski’s Philo article].”[4] Nowacki clearly insinuates that I was aware that I have an intellectual debt to Bolzano based allegedly upon the ground that my objection to the KCA is “a Bolzano-inspired account of the infinite,” and that I nevertheless failed to acknowledge this debt in my article. Let me assure the reader that not only I was (and still am) unaware of any intellectual debt to Bolzano but that I deny that my thesis set forth in my Philo article was in any way inspired by Bolzano. Quite apart from Nowacki’s implied charge of a breach of duty on my part, the very fact that he professes to see my article as being a “Bolzano-inspired account” discloses how egregiously he misunderstands my position. And the result is that his value of his book is vitiated to the extent that he systematically fails to accurately take into account my thesis of how the Cantorian theory of the actual infinite can apply to the real world without generating counterintuitive absurdities.

Before describing what Nowacki means by a “Bolzano-inspired account of the infinite,” I shall first summarize my account of the matter. According to the Cantorian theory of transfinite arithmetic, the cardinal number of the set of natural numbers cannot itself be a natural number (as there is no highest natural number) but rather it is a transfinite number, i.e., אo (read alelph-zero–the lowest transfinite number). The set of natural numbers is a denumerable infinite, and so also is any other actual mathematical infinite corresponds one-to-one with the set of natural numbers. So, for example, the set of negative numbers (i.e., (….,-5,-4,-3,-2,-1}) corresponds one-to-one with the set of natural numbers by virtue of a rule (i.e., z=-n). But, mirabile dictu, the set of all even natural numbers also corresponds one-to-one with the set of natural numbers, of which the former is a proper subset, by virtue of a rule (i.e., e=2n). Two actual infinites are said to be equipollent when their members, by virtue of some rule, are in one-to-one correspondence.[5] The principle of correspondence is that two sets have one and the same cardinality if and only if they are equipollent.[6] Moreover, if two mathematical infinite sets A and B are each equipollent to mathematical infinite set C then A and B are mutually equipollent. Thus the principle of correspondence obtains in the domain of pure mathematics according to Cantorian theory.

According to the standard view, with which I agree, were an actual infinite of concrete entities or events instantiated in reality, then this infinite must be denumerably so. However, in my view, the principle of correspondence does not fully obtain with respect to real infinites (i.e., an actual infinite of concrete entities or events instantiated in reality). That is to say: any real infinite (e.g., our hypothetical set of infinitely many humans each with two and only two hands) is equipollent with the set of all natural numbers (and therefore with any other denumerable mathematical infinite); but surely the set of infinitely many humans is not equipollent with the set of infinitely many human hands—although the latter set is also equipollent with the set of all natural numbers and therefore with any other denumerable mathematical infinite. Accordingly, two real infinites that are not equipollent with each other nevertheless have the same cardinality because each is denumerably infinite.

In his Response (TKCAG, pp. 122-23), Nowacki asserts that Bolzano’s theory of the infinite is such that both “the principle of correspondence and Euclid’s principle hold true.” But Nowacki’s principle of correspondence appears to be simply a definition of equivalence (read, if you please, equipollence). On the other hand, I use the term principle of correspondence to refer to the proposition, sounding in pure mathematics, that two infinite sets have one and the same cardinality if and only if they are equipollent (i.e., equivalent according to Nowacki). Now, subject to correction, I understand that Bolzano held that two mathematical infinite sets (e.g, the sets of all natural numbers and that of all even natural numbers) do not have one and the same cardinality even though they are equipollent.[7] But that, as the reader can well see, is not my position since both sets have the same cardinality by virtue of being equipollent. Again, in my view, although two real infinites necessarily have the same cardinality if they are equipollent, there may be two real infinites (e.g., the set of infinitely many humans each with two and only two hands and the set of their infinitely many hands) that are not equipollent even though they have the same cardinality.[8]

What I have proposed is that the instantiation of Cantorian theory in the real world need only involve the po

stulate that any real infinite of concrete entities or events necessarily is equipollent with the set of natural numbers and thus has the cardinality of that set (i.e., אo). Doing so serves, as I argue in my article, to preclude counterintuitive absurdities. However, the hypothetical instantiation of Cantorian theory involving the postulate that real infinites are to be deemed as having all the properties of mathematical infinite sets inevitably leads to the generation of counterintuitive absurdities—including those pertaining to Nowacki’s hyperlump thought experiments insofar as these are based upon transfinite mathematical considerations.

As the discerning reader readily sees, mathematical platonists need not justly be apprehensive about the full application of the principle of correspondence within the domain of pure mathematics notwithstanding that two real infinites have the same cardinality even were they not to be equipollent.

Well, it is not my intention to presently refute Nowacki upon any substantive issues concerning the merits of the KCA. I just want to point out how he has entirely missed his mark by having confounded my account of the infinite with any supposedly Bolzano-inspired account of the same. But I am nevertheless grateful for this opportunity to affirm my thesis that the instantiation in the real world of the Cantorian theory of transfinite arithmetic does not entail that the common cardinality of two real infinites necessarily precludes their nonequipollence.

[1] The two follow-up Secular Web articles are “The Kalam Cosmological Argument Yet Again: The Question of the Metaphysical Possibility of an Infinite Temporal Series” (2003, updated 2005), and “The Kalam Cosmological Argument as Amended: The Question of the Metaphysical Possibility of an Infinite Temporal Series of Finite Duration” (2004, updated 2005). All three articles can be accessed at www.infidels.org/library/modern/arnold_guminski/.
[2] I take here the liberty of noting that Quentin Smith (then editor of Philo), in his email accepting my first article in 2002, remarked that my thesis as to how Cantorian theory applies to the real world “is at least an under-cutting defeater of Craig’s beliefs about real infinites, even an overriding-defeater. More importantly, it introduces a novel metaphysical theory of the relation of transfinite arithmetic to concrete reality.”
[3] The Response reads: “There are a number of reasons why Bolzano’s theory of the infinite did not find general acceptance. A brief discussion of some of the difficulties with Bolzano’s theory may be found in chapter 1, section 4.2.3.3; two additional rejoinders to the objection are as follows. First, a Bolzano-inspired account of the infinite would be rejected by mathematical platonists who, were they to accept it, would be forced to admit that their abstract yet metaphysically real sets do not behave in the way that Cantor’s theory claims they do. For platonists, accepting Bolzano would be tantamount to admitting that Cantorian transfinite mathematics is simply false as an account of the actual infinite. Second, a Bolzano-inspired account of the infinite does not dissolve all of the counterintuitive absurdities that KCA thought experiments have brought to light. For example, adopting Bolzano’s perspective does not dissolve the shape-related difficulties treated in the hyperlump thought experiment (described in chapter 5).” TKCAG, pp. 122-23.
[4] TKCAG, p. 152 (bracketed matter added).
[5] In my Philo article, I explained why I prefer to use the term “equipollent” instead of “equivalent” or “equipotent” to define the relation of one-to-one correspondence. 5 Philo at 198 and 210 n. 15. The term “equivalence” is ambiguous because it could either be synonymous with “equipollent” or because it obtains when two sets have one and the same cardinality. Nowacki remarks (p. 81 n. 41): “More recent writers on set theory are apt to substitute equipollent or equipotent where Cantor and Craig use equivalent.” In his The Kalam Cosmological Argument (New York: Harper & Row Publishers, Inc., 1979), p. 154 n. 7, William Lane Craig observes: “Despite the one-to-one correspondence, Bolzano insisted that two infinites so matched might nevertheless be non-equivalent.” Quite obviously, by “non-equivalent” Craig here means something other than the absence of a one-to-one correspondence between two infinites.
[6] Nowacki appears consistently, and if so mistakenly, to use the term “principle of correspondence” to refer only to the definition of “equivalence” (i.e., “equipollence”). See, e.g., TKCAG, pp. 40-41, 49-51. But clearly Georg Cantor defined the relation of “equivalence” (i.e., “equipollence”) to refer to the situation when “it is possible to put [two aggregates M and N], by some law, such a relation to one another that to every element of each one of the corresponds one and only one element of the other.” Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (tr. & ed. P.E.B. Jourdain) (New York: Dover Publications, Inc., 1955), pp. 86-87. Cantor proceeds to formulate “the theorem that two aggregates M and N have the same cardinal number if, and only if, they are equivalent” (i.e., “equipollent”). Ibid., p. 87. “Thus,” Cantor wrote, “the equivalence of aggregates forms the necessary and sufficient condition for the equality of their cardinal numbers.” (Ibid., pp. 87-88. This, in my opinion, is what should be understood by the term the principle of correspondence.
[7] Nowacki refers to this point in chapter 1, section 4.2.3.3, where he writes: “Bernard Bolzano sketched the rudiments of a theory of the infinite that allowed for different sizes of infinity based on the part/whole relationship instead of the principle of correspondence.106” Nowacki, p. 50. See also TKCAG, p. 93 n. 106 where Nowacki approvingly quotes from A.W. Moore’s The Infinite (London: Routledge, 1995), p. 113): “On Bolzano’s view there just were fewer even natural numbers than natural numbers altogether, irrespective of the fact that they could be paired off.”
[8] The reader should be in mind that the thought experiments about real infinites of simultaneously existing entities or events in this (or some other) physical universe (e.g., the infinitely many rooms in Hilbert’s hotel or the infinitely many books in Craig’s library) prescind from issues relating to the factual possibility of such infinites upon grounds other than those relating to transfinite mathematical considerations. The reason for this anomaly is that proponents of the KCA, such as William Lane Craig, want to show that every real infinite, whether or not of simultaneously existing entities, is metaphysically impossible. If this showing is made, it then follows (according to Craig et al.) that any infinite temporal series of events is metaphysically impossible. However, the reader is invited to consider the case of infinitely many physical universes (none of which being spatially related to an

y other), each of which includes finitely many humans each with two and only two hands. This latter scenario is free of issues of factual possibility except those arising from transfinite mathematical considerations.