JST-SCUCantor’s Transfinites and Divine Infinity               BY FADY EL CHIDIAC, S.J.
Table of Contents   1               T ABLE OF C ONTENTS                    INTRODUCTION, 2TRANSFINITES AND ABSOLUTE INFINI...
Introduction        2                                                 I NTRODUCTION                                     Gr...
Introduction     3       This paper joins the previous interests in divine infinity, while it diverges from an atheist’spr...
Transfinites and Absolute Infinity in Cantor’s Theory   4                 T RANSFINITES AND ABSOLUTE I NFINITY IN C ANTOR ...
Transfinites and Absolute Infinity in Cantor’s Theory    5number 4. Then, he introduces the most creative mathematical con...
Transfinites and Absolute Infinity in Cantor’s Theory   6       The construction continues. The number ω+2 is the sequence...
Transfinites and Absolute Infinity in Cantor’s Theory    7any given transfinite there exists another transfinite which con...
Transfinites and Absolute Infinity in Cantor’s Theory   8           Indeed, infinite ordinals transcend finites, which is ...
Transfinites and Absolute Infinity in Cantor’s Theory          9numbers are equinumerous 14. Hence, all of them share ℵ0 a...
Transfinites and Absolute Infinity in Cantor’s Theory      10number of ordinals, such as ℵ0 is the cardinal of all countab...
Transfinites and Absolute Infinity in Cantor’s Theory   11must be at least and at most an ordinal-like. Rucker treats it a...
Transfinites and Absolute Infinity in Cantor’s Theory          12          To see how the reflection principle caused this...
Transfinites and Absolute Infinity in Cantor’s Theory         13In a larger view, positing an absolute infinite and assumi...
Mathematical Infinity and Theology   14                         M ATHEMATICAL I NFINITY AND THEOLOGY      Rare are the con...
Mathematical Infinity and Theology         15                         R ESPONSE TO S OME T HEOLOGICAL R ESERVATIONS       ...
Mathematical Infinity and Theology     16admitted here. If the set theorists preferred to avoid the expression ‘set-like,’...
Mathematical Infinity and Theology    17reflection principle should be supplied with additional elements for theological i...
Mathematical Infinity and Theology          18appeal to Thomas Aquinas’s concept of divine infinity 33. I will show that t...
Mathematical Infinity and Theology          19          The cataphatic side of Aquinas’s ‘infinity’ is the eminence it exp...
Mathematical Infinity and Theology     20          As for the knowledge of God, Aquinas distances his theology from the ap...
Mathematical Infinity and Theology           21          Last but not least, for Aquinas, the divine infinity is to the wo...
Mathematical Infinity and Theology          22On the contrary, In ST, Q.7, a.2, Aquinas, the medieval theologian, counters...
Conclusion      23                                           C ONCLUSION          To sum up, Aquinas’s ‘divine infinity’ a...
Conclusion     24the finite. Insofar as no further theological reservations arise, big omega is to be appreciated bytheolo...
Bibliography   25                                            B IBLIOGRAPHYAquinas, Thomas. The Summa Contra Gentiles of Sa...
Bibliography     26Russell, Robert J. Cosmology from Alpha to Omega: The Creative mutual Interaction of Theology andScienc...
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Cantor’s transfinites and divine infinity by fady el chidiac sj

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Cantor’s Transfinites and Divine Infinity by Fady El Chidiac SJ

Although the notion of infinity is not commonly present in Christian worship, several theologians strived to incorporate it with the Christian faith. Among the first theologians who argued for the infinity of God are Gregory of Nyssa and John of Damascus1. Later, in the Middle Age, Thomas Aquinas “was fighting enemies in three or even four corners”2 to systematically establish divine infinity. For some reason, most of the modern theologians did not embrace this notion in their God-talk. Nonetheless, some contemporary theologians and philosophers show an interest in ‘infinity’ to restore an alleged defect in Aquinas systematic theology3, to augment Pannenberg’s systematic theology4, to soften the apparent contradictions between some cosmological theories and theology5, or to argue against the cosmological argument of the existence of God

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Cantor’s transfinites and divine infinity by fady el chidiac sj

  1. 1. JST-SCUCantor’s Transfinites and Divine Infinity BY FADY EL CHIDIAC, S.J.
  2. 2. Table of Contents 1 T ABLE OF C ONTENTS INTRODUCTION, 2TRANSFINITES AND ABSOLUTE INFINITY IN CANTOR’S THEORY, 4 TRANSFINITES, 4 ABSOLUTE INFINITY, 10 MATHEMATICAL INFINITY AND THEOLOGY, 14 RESPONSE TO SOME THEOLOGICAL RESERVATIONS, 15 CANTOR AND AQUINAS: BIG OMEGA AND DIVINE INFINITY, 18 CONCLUSION, 23 BIBLIOGRAPHY, 25
  3. 3. Introduction 2 I NTRODUCTION Great is the Lord, and greatly to be praised: and of Her greatness there is no end. (Ps. 144:3) Although the notion of infinity is not commonly present in Christian worship, severaltheologians strived to incorporate it with the Christian faith. Among the first theologians whoargued for the infinity of God are Gregory of Nyssa and John of Damascus 1. Later, in theMiddle Age, Thomas Aquinas “was fighting enemies in three or even four corners” 2 tosystematically establish divine infinity. For some reason, most of the modern theologians did notembrace this notion in their God-talk. Nonetheless, some contemporary theologians andphilosophers show an interest in ‘infinity’ to restore an alleged defect in Aquinas systematictheology 3, to augment Pannenberg’s systematic theology 4, to soften the apparent contradictionsbetween some cosmological theories and theology 5, or to argue against the cosmologicalargument of the existence of God 6.1 For an overview, cf. http://en.wikipedia.org/wiki/Gregory_of_Nyssa#Infinity andhttp://en.wikipedia.org/wiki/John_Damascene (accessed 12/11/2009).2 Robert Burns, “Divine Infinity in Thomas Aquinas,” The Heythrop Journal 39 (1998): 66.3 Leslie Armour, “Re-thinking the Infinite,” in Wisdom’s Apprentice, ed. Peter A. Kwasniewski (Washington D.C.:The Catholic University of America Press, 2007). Elsewhere, I argue that Armour’s attempt is untenable, althoughhis goal is of a great interest for an interactive view of science and theology. My point is that Armour’s projectcannot be grounded on Aquinas’s theology, unfortunately.4 Robert Russell, Hegelian Infinity in Pannenberg’s Doctrine of the Divine Attributes: New Insights from Cantor’sMathematics. (Forthcoming) This work of Russell inspired my current paper.5 Robert Russell, “The God Who Infinitely Transcends Infinity: Insights from Cosmology and Mathematics,” inCosmology from Alpha to Omega: The Creative Mutual Interaction of Theology and Science (Minneapolis: FortressPress, 2008): 56-76.6 Graham Oppy, Philosophical Perspectives on Infinity (New York: Cambridge University Press, 2006). This bookis the precursor of a forthcoming promised book arguing against the cosmological argument. Oppy’s philosophicalcareer is based on defeating theism. He published Arguing about Gods and Ontological Arguments and Belief inGod.
  4. 4. Introduction 3 This paper joins the previous interests in divine infinity, while it diverges from an atheist’sproject, such as Oppy’s. It attempts to show that Georg Cantor’s mathematical concept ofinfinity and a Christian concept of divine infinity are relevant to one another. Furthermore, theycan engage a mutual interaction. The steps undertaken here are simple. After an overview of Cantor’s set theory in a firstchapter, I engage a discussion in which I soften some theological reservations about themathematical absolute infinity, and then I show the correspondences between the Cantorian‘absolute infinity’ and the Thomist ‘divine infinity’ 7 and the support they can offer to oneanother. As a basic motivation, this paper intends to be an incipient work exploring whetherfurther development of the correlation between a mathematical concept of infinity and divineinfinity is worthwhile.7 I have chosen Thomas Aquinas as theological counter-part to Cantor because the latter mentions the former in hiscorrespondences with theologians about religion and science.
  5. 5. Transfinites and Absolute Infinity in Cantor’s Theory 4 T RANSFINITES AND ABSOLUTE I NFINITY IN C ANTOR ’ S T HEORY In the late nineteenth and early twentieth centuries, mathematicians were delighted to finda possible foundation for mathematics in Cantor’s set theory. Whenever Cantor’s theory ismentioned, David Hilbert’s adage is cited: “No one shall expel us from the paradise Cantor hadcreated.” 8 Cantor unfolded a mathematical paradise through the establishment of transfinitenumbers. Subsequent mathematicians and philosophers discover that the foundations ofmathematics are still far from being unveiled. However, this deception does not impedemathematicians from pursuing Cantor’s project, mainly continually unfolding transfinitenumbers. Even though Cantor’s theory is still a research in progress since around 1875, it hasreached some unshakable constructions. For the purpose of this paper, it suffices to illustrate theconsistency of Cantor’s theory by treating the transfinites first, and then the ‘absolute infinity.’ T RANSFINITES Transfinites are categories included within the set theory. The first striking feature in thetheory of transfinites, as launched by Cantor, is that sets are considered to be numbers. Sets areusually the ensemble of numbers, and here with Cantor numbers become also sets. Beginnersmay receive this reluctantly, but defining numbers as sets is a well entrenched mathematicalconception. Some problems arise from this conception when absolute infinity is addressed. Thisissue will be discussed in the next section. Cantor starts with natural numbers. He defines eachnatural number as the set of natural numbers preceding it, such as the set {0, 1, 2, 3} is the8 The citation is found in all the literature listed in the bibliography of this paper. In Oppy’s Philosophicalperspectives on infinity, the chapter on transfinites is entitled “Cantor’s Paradise.”
  6. 6. Transfinites and Absolute Infinity in Cantor’s Theory 5number 4. Then, he introduces the most creative mathematical conception ever: the wholeinterminable set of natural numbers is a number, symbolized by ω. Notice that this mathematicalconception is basically a philosophical intuition rooted in understanding the many as a unit. Thisvision, applied to natural numbers as previously noted, is now extended to endless sets. Thus,the first transfinite number is established, ω = {0, 1, 2, 3…}. This is the key to the so calledCantor’s paradise. And from it, mathematicians construct myriads of transfinite numbers asfollows. For reasons of clarity, some mathematicians talk about transfinites as sequences as well assets. I adopt the notion of sequence here, for it conveys ordered elements, which will help us tounderstand the construction of other transfinites. The transfinite number ω+1 is the sequence ofnatural numbers at the end of which is added the number 1. The number ω+1 is represented asthe sequence 0, 1, 2, 3 …, 1. Adding a number at the end of an endless sequence is, in fact,another stumble block for common sense and traditional philosophy. Nonetheless, it is wellgrounded mathematically 9 by distinguishing between adding the number 1 at the beginning ofthe sequence and at its end. Here, 1+ω does not equate ω+1. The number 1+ω means adding 1at the left limit of the natural numbers, that is, 1+ω = {1, 0, 1, 2 …}. Adding a number from thelimited side of an endless sequence does not affect its endlessness. It means just pushing thenumbers one step ahead, whereas this is not possible when the number is added at the end of theendless sequence. The phrase ‘the end of an endless sequence’ is rendered plausible once theendless sequence is seen as one unit, here as ω. This is like saying ω+1 = {ω, 1}.9 Cantor argues for that against mainly Aristotle in his On infinite: Linear Point-Manifolds. A nice summary ofCantor’s argument is found in Jean Rioux, “Cantor’s Transfinite Numbers and Traditional Objections to ActualInfinity,” The Thomist 64 (2000): 109-119.
  7. 7. Transfinites and Absolute Infinity in Cantor’s Theory 6 The construction continues. The number ω+2 is the sequence of natural numbers at theend of which is added the sequence of 1 and 2. And, so on until we get ω+ω, noted ω·2. Heretoo, we can have ω·3, ω·4 … ω·ω. The latter can be symbolized as ω2. Consider the followingrepresentation 10 of ω2, which repeats ω times the sequence of natural numbers. In the figure,each stick stands for a number. 1 2 3 4 5… 1 234… 1 2… 1 2… …Now too, ω3, ω4… ωω can be conceived. The same construction proceeds further and further.Notice that the rule of inclusion applies to all transfinite numbers: a transfinite number includesall precedent numbers. Hence, one can define an order through transfinites, such as ω < ω+ω < ωω < ωω , where the sign < means ‘is included in.’ ωThe Zermelo–Fraenkel set theory with the axiom of choice (abbreviated as ZFC), which is, sofar, the most successful axiomatic theory laying the foundation of mathematics, proves that for10 The representation is taken from http://en.wikipedia.org/wiki/File:Omega_squared.png (visited on 12/5/2009).The sequences of numbers are mine.
  8. 8. Transfinites and Absolute Infinity in Cantor’s Theory 7any given transfinite there exists another transfinite which contains the former. Hence, one canalways construct a bigger transfinite number. Transfinite numbers are more complex than ordinary numbers, since they designate aninternal order (for instance, ω+1≠1+ω) and a set of numbers. Cantor applies this distinction toall numbers. He ascribes an ordinal to each number, designating the order of sequencesconstituting the latter. For instance, the ordinal ω+5 designates the succession of two sequences:the sequence of natural numbers, ω, followed by the sequence of integers from 1 to 5. Clearly,the ordinal of any finite number is itself; for example, the ordinal the set {0, 1, 2, 3, 4} is 5. Here, differences are noticed between the ordinals of finite numbers and the ordinals oftransfinites. They do not follow the same arithmetic rules, such as the commutative addition. Tosee the difference, recall that ω+1≠1+ω (the order counts for transfinites), whereas addition iscommutative when applied to finite numbers. Radicalizing the distinction between finites andtransfinites, Bertrand Russell discards Cantor’s theory altogether, for transfinites involve “noreference to the special peculiarities of quantity” 11. He argues that unlike finite numberstransfinites are irrelevant to the questions of ‘how much’ or ‘how many’. Following BertrandRussell, Jean Rioux concludes that transfinites seem to be “something more akin to quality orrelation” 12 rather than quantity. Hence, if numbers are confined to quantities, transfinites will nomore be numbers. Bertrand Russell radically challenges any commonality between transfinitesand finite numbers.11 Bertrand Russell, Introduction to Mathematical Philosophy (New York: Norton & Company Publishers): 188.once 2ℵ0 = ℵ1 is proved. The equation means that the set of real numbers is equivalent to the ordinal 0 .Bertrand Russell addresses other critiques to Cantor’s theory, such as actual infinite magnitudes are not one-sidelimited like the set of integers. He appeals to the continuous set of real numbers. That question will be defeated12 Rioux, 121. This critique holds also against the cardinals ℵi.
  9. 9. Transfinites and Absolute Infinity in Cantor’s Theory 8 Indeed, infinite ordinals transcend finites, which is the reason Cantor calls themtransfinites. And, for this reason, some differences must be detected. One must not expect fromtransfinites what one expects from finites, and vice versa. However, commonalities are foundbetween transfinites and finites. Mathematicians succeeded in setting arithmetic rules thatgovern ordinals of both kinds 13. Addition, multiplication, exponentiation, power are enjoyed byfinites as well as transfinites. Addition and multiplication are distributive on the right, but not onthe left. Subtraction and division apply to some extent to transfinites. Hence, several, but notall, usual features of arithmetic rules are reproduced in the case of transfinites. The finitenumber of cultures and countries on the Globe are not agreeing yet on common human rights;how can one expect finites and transfinites to satisfy all arithmetic rules? In addition to ordinals, Cantor ascribes cardinals to numbers, noted as card(), in order toaccount of the number of elements contained in the set. For example, the cardinal of number 7 isthe number of elements contained in the set {0, 1, 2…, 6}, that is, card(7) = 7. Obviously, thecardinal of any finite number is the same as the number, card(n)=n, for any integer n.This is not true for transfinite numbers, from ω further on. The cardinal of ω, i.e., the set ofnatural numbers, is infinite designated by ℵ0. Notice that some infinites are alike, while other infinites differ. On one hand, it can beeasily demonstrated that the sets of natural numbers, odd numbers, even numbers, and rational13 For details, see Waclaw Sierpinski, Cardinal and Ordinal Numbers, second ed. (Warszawa: PWN, 1965).
  10. 10. Transfinites and Absolute Infinity in Cantor’s Theory 9numbers are equinumerous 14. Hence, all of them share ℵ0 as cardinal. Similarly, the cardinal ofany countable set is ℵ0. All transfinites encountered so far in this paper are countable.Therefore, card(ω) = card(ω+1) = card(ω·3) = card(ω2) = card(ωω) = card(ωω ) = ℵ0. ωOn the other hand, Cantor demonstrated that the set of real numbers, ℜ, is uncountable and,hence, exceeds the set of natural numbers. Card(ℜ) is calculated to be 2ℵ0 . Thus, ℵ0 and 2ℵ0indicate two different kinds of numerical infinites 15. “It turns out that just as we can always findmore ordinals, we can always find more cardinals. After ℵ0 come ℵ1, ℵ2, ℵ3, … ℵω, ℵω+1, …ℵ , … ℵℵ1 , … ℵℵ , …, and so on” 16. The ordinals having ℵ0 as cardinals are called the firstclass ordinals, the ones of ℵ1 the second class, and so on. Each class of ordinals has a distinctproperty, such as the property of ‘countable’ that the ordinals of the first class enjoy. Differentinfinite cardinals denote different kinds of infinites. To summarize, in respect to both ordinals and cardinals, transfinite and finite numbersenjoy commonalities as well as divergences. The similarities and differences in arithmetic rulesregarding ordinals are illustrated above. As for cardinals, ℵ0, ℵ1, ℵ2, etc. are ordinals and, thus,enjoy the same arithmetic rules ordinals share with finites. Of course, cardinals of transfinitesare wealthier than the finite cardinals. Each transfinite cardinal corresponds to an infinite14 The claim that there are as many natural numbers, {0, 1, 2, 3, 4…}, as odd numbers, {1, 3, 5…} counters commonsense, because the former includes the even numbers in addition to the odd numbers. It is worth noting that forinfinity not any addition makes difference. Common sense does not always get it right, specially when it comes tothings uncommon to daily life experience, such as infinity.here. ℵ0 and 2ℵ0 are ordinals as well as cardinals. They specify how many real numbers there are with respect to15 Recall Bertrand Russell’s critique: transfinites are irrelevant to the question “how many.” A response can be giventhe number of integers.16 Rudy Rucker, Infinity and the Mind: the Science and Philosophy of the Infinite (Boston, Basel, Stuttgart:Birkhauser): 77.
  11. 11. Transfinites and Absolute Infinity in Cantor’s Theory 10number of ordinals, such as ℵ0 is the cardinal of all countable infinite ordinals, whereas anyfinite cardinal corresponds to one and only one finite ordinal. The following section will discussthe mathematical use of absolute infinity. A BSOLUTE I NFINITY Cantor was eager to establish the existence of an absolute infinity, entitled Ω. However,he discussed this issue more with theologians than with mathematicians 17. His absolute infinityseems to be grounded on his theological belief rather than proved by his mathematical tools.Leaving the theological discussion for later, the interest here is to see how mathematicians dealwith big omega. Although big omega is an indistinct notion, “talking about Ω is an extremelyuseful and productive thing for set theorists to do” 18. How is this? If you want to understand themathematical status of big omega, prepare yourself for a continuous back-and-forth between aclaim and its opposite. To begin with, it is absolutely plausible to posit a set containing all ordinals. Let this setbe Ω. First caution: Ω must not be an ordinal. Otherwise, we face the contradiction that Ωcontains Ω. Furthermore, Ω should not even be a limit to which a series of ordinals tends 19.Thus, Ω seems to be an outsider. Nonetheless, if Ω becomes an alien notion assembling allordinals, Ω will be irrelevant to mathematics. Therefore, without becoming totally an ordinal, Ω17 Unfortunately, Cantor’s correspondences with neo-thomists, where the absolute infinity is discussed, are nottranslated into English. Adam Drozdek provides overviews of some of these correspondences in “Number andinfinity: Thomas and Cantor,” in International Philosophical Quarterly 39 (1999).18 Rucker, 254.19 For either one of the two reasons: cardinal of Ω should be regular (Rucker, 254-255) and the series of ordinalsconverge towards an ordinal (see Sierpinski, 382-390).
  12. 12. Transfinites and Absolute Infinity in Cantor’s Theory 11must be at least and at most an ordinal-like. Rucker treats it as an “imaginary ordinal” 20. A wayout of this dilemma is provided by the reflection principle: For every conceivable property of ordinals P, if Ω has property P, then there is at least one ordinal k Ω that also has property P 21.In this case, Wikipedia gives a nice non-formal interpretation of the reflection principle: Properties of [Ω], the universe of all sets, are ‘reflected’ down to a smaller set 22.By the reflection principle, Ω, the otherwise ordinal, is connected to the ordinals it contains.Although trans-ordinal (transcending ordinals), Ω communicates all its properties to at leastsome ordinals. The reflection principle has proved very fruitful. It allows the generation of a multitude ofcardinals, which means a diverse variety of infinities. First come the inaccessible cardinals, then,hyperinaccessible cardinals 23, “strongly inaccessible cardinals…Mahlo cardinals…the indescribable cardinals…the ineffable cardinals, partition cardinals, Ramsey cardinals, measurable cardinals, strongly compact cardinals, supercompact cardinals, and, finally [at present], the extendible cardinals” 24.Each kind of cardinals plays an important role in mathematics.20 Ibid.21 Rucker, 255-256.22 http://en.wikipedia.org/wiki/Reflection_principle (visited on 12/6/2009).23 Rucker, 258.24 Rucker, 261.
  13. 13. Transfinites and Absolute Infinity in Cantor’s Theory 12 To see how the reflection principle caused this abundant collection, we consider here thediscovery of the first kind of this series, namely the inaccessible cardinals 25. In a nutshell, bigomega is a set, thus, it should have a cardinal. Besides, the equation card(Ω) = Ω must beassumed, for if this is not the case, Ω will depend on the cardinal of a smaller ordinal. Thus, bigomega can be spoken of as ordinal and cardinal. And, card(Ω) should be inaccessible because Ωis the greatest ordinal. ZFC has formalized the property of a cardinal being inaccessible. Now,since ‘being inaccessible’ is a property shared by Ω, due to the reflection principle, at least oneordinal enjoys ‘being inaccessible.’ Hence, the set of ordinals enjoying an inaccessible cardinalis not empty. Any formalized property pertaining to Ω, that is a very ambitious property, isreached by some ordinals. By Ω and the reflection principle, what is beyond all is reachable bysome. The effect of the mere existence of such bizarre cardinals on finites exceeds ourexpectations. The case of measurable cardinals is interesting because of this issue. To put it inRucker’s words, The most curious thing about measurable cardinals is that once one knows that they exist, one is forced to the conclusion that there are many more sets of natural numbers than one had previously suspected.Using a metaphor, Rucker continues, It is as if the discovery of some far away galaxy forced us to the conclusion that there are some additional types of microorganisms present in our bodies 26.25 This illustration follows Rucker, 255-258.26 Rucker, 261. “Specifically, if a measurable cardinal exists, then there is a set of integers, called 0#, that is not inGödel’s universe L of constructible sets” (Ibid, 262).
  14. 14. Transfinites and Absolute Infinity in Cantor’s Theory 13In a larger view, positing an absolute infinite and assuming the reflection principle proves theexistence of unexpected transfinite cardinals, which may affect our understanding of finites.Indeed, this interaction between the absolute infinite and the finite through transfinites is justamazing. Transfinites do stand in the middle between finites and absolute infinity. Notice that, in the reflection principle cited above, k may be a finite ordinal. Hence, Ωmay share properties directly with finite ordinals, without a necessary transfinite mediation.Although transfinites enjoys a middle place between finites and absolute infinity, thecommunication between the two latter does not require transfinites mediation 27. One senses heresome theological implications. The following chapter will treat this at length. While turn to thenext chapter, retain these six points: • Transfinites are infinite numbers, defined as sets. • Transfinite ordinals share commonalities as well as differences with finite ordinals. • There is always a greater ordinal to any given ordinal (‘is greater’ means ‘contains’). • Absolute infinity, Ω, is defined as the set of all ordinals, yet Ω is a set-like. • The reflection principle relates Ω to ordinals by sharing with the latter all its conceivable properties. • Due to the reflection principle and the assumption of Ω, several transfinite cardinals were created and proved to be useful to the mathematical inquiry.27 The communication of properties across transfinite and finite ordinals is relevant to theological inquiries indeed.This needs a research by itself. The issue of properties and sets is very wide.
  15. 15. Mathematical Infinity and Theology 14 M ATHEMATICAL I NFINITY AND THEOLOGY Rare are the contemporary Christian theologians who introduce a mathematical concept ofinfinity in their talk about God. Many allege that mathematics leaves no room for theIncomprehensible. The previous development counters such prejudice by having illustrated howmathematics accommodate an unconceivable notion, namely Ω. We are in debt to Cantor forthis great change in modern mathematics. Would this allow an interaction between theology andmathematics? Yes, it allows an interaction indeed. What follows is an exploration of suchinteraction. The transfinites of Cantor’s theory are inadequate to God, for they are not entirelyunlimited. Any transfinite ordinal is limited by a greater one, as shown above(ωω+ωωωωω …). Besides, transfinites are totally determined by their construction. Thus, ωsuch absolutely conceivable properties are inadequate to God, Who is the beyond, the otherness. From the set theory, only the kind of unlimited infinity that Ω bears may pertain to theGod of monotheism. As depicted above, absolute infinity, Ω, transcends transfinite as well asfinite ordinals. Yet, due to the reflection principle, Ω is related to ordinals. This absoluteinfinity, transcendent-yet-in-communion with non-absolute-infinities, is very tempting totheological use. Nonetheless, some theologians may resist the use of mathematical infinity fortwo cautions I address in the following section.
  16. 16. Mathematical Infinity and Theology 15 R ESPONSE TO S OME T HEOLOGICAL R ESERVATIONS The first circumspection is concerned with pantheistic and panentheistic implications 28.Recall that big omega is defined as the set of all ordinals. However, it is not the convergence ofany series of ordinals, which means that no progression of ordinals can reach absolute infinity.Theologically, this suggests that the evolution of the world can never reach a divine existence.Here, Ω is seen rather as a set containing the totality of numbers than as the term of a succession.Such a concept of Ω conveys an image of an encompassing infinity, which some theologiansmight be reluctant to ascribe to the Christian God. Nonetheless, what is stressed by the fact thatΩ is not the upper limit of any series of ordinals is that every ordinal participates in Ω. In thepicture where Ω is a set, any individuality enjoys a participation in the absolute infinity. On thecontrary, in the picture where Ω is an upper limit, only the last terms of the series enjoy animmediate relation with Ω. As set of all ordinals, the concept of absolute infinity sustainspanentheism, but it is closer to the Christian faith than what seems to be its only mathematicalalternative, which is ‘Ω is an upper limit.’ Examined carefully, the mathematical concept ofabsolute infinity does not support pantheism, for Ω is merely a set-like; big omega wouldbecome an ordinal if it were formalized as a set. Furthermore, it is understandable that, in a settheory, it is very difficult to find an image apart from ‘set’ to designate the absolute infinity. Inthe set theory, all entities are sets. The internal limit of the discipline in question is to be28 Through his correspondence with Johannes Cardinal Franzelin, Cantor addressed the pantheist features, of whichhe was not aware in the incipient development of ‘absolute infinity.’ Unfortunately, I did not have access to anEnglish translation of the German text. The reference is Georg Cantor, Mitteilungen zur Lehre vom Transfiniten(1887) in Gesammelte Abhandlungen: 385. This reference is taken from Drozdek, 42, 44. Drozdek comments thatCantor eschews pantheism by positing the uncreatedness of absolute infinity. I do not find this solution appealing,for the first number of integers, from which all natural numbers are induced, can also be seen as uncreated (seeRobert Russell, Hegelian Infinity in Pannenberg’s Doctrine of the Divine Attributes: New Insights from Cantor’sMathematics: 14). Besides, the panentheist implications are left over.
  17. 17. Mathematical Infinity and Theology 16admitted here. If the set theorists preferred to avoid the expression ‘set-like,’ they would not findalternative genera entrenched within the theory. Thus, even the expression ‘Ω is a set-like’should not be taken as the complete description of the nature of absolute infinity described by theset theory. The description that best befits the mathematical absolute infinity is ‘external-yet-connected’ entity. Panentheist and other theological inquiries may find this notion of infinityappealing. The other caution regards the reflection principle. The worry here is about emptyingabsolute infinity of any specific property. If, by the reflection principle, infinity shares all itsproperties with ordinals, what would be left as particular to infinity? Infinity devoid ofspecificity does not disturb mathematical functioning, whereas it provokes theologicalreservations. Certainly, as Rucker notes, “although ‘Ω is the class of all ordinals’ is true, we donot expect it to be true of any ordinal less than Ω” 29. The property ‘the class of all ordinals,’which can be interpreted as the property of being uncreated, is obviously specific to Ω.However, leaving divine infinity with only one property, which is ultimately merely its ownexistence, is not satisfying. The idea endorsed here is not an image of a stingy God. On thecontrary, the reflection principle befits the communication of divine properties that Christiantheologians unanimously allege. The Christian God is so generous and loving as to send us HisSon. The point here is the gratuitousness and freedom of God, which the reflection principle isseemingly not considering. God created the world, sent us His Son, and is acting in the world byfree-will, not by necessity. If whatever is enjoyed by divinity should be shared with the world,God would be communicating his properties by necessity. This issue seems to result from thelimit of mathematical language, which cannot be otherwise than deterministic. Thus, the29 Rucker, 256.
  18. 18. Mathematical Infinity and Theology 17reflection principle should be supplied with additional elements for theological interpretation.The key solution for this problem is the notion of ‘conceivable property’ as used in the reflectionprinciple. The properties shared by Ω are those which are conceivable. As understood byRucker, “‘conceivable property’…is supposed to mean a property that is expressible in terms ofsets and language of some kind” 30. Thus, the reflection principle does not apply to unconceivableproperties, which cannot be expressed in any language. Anyway, such properties are irrelevantto mathematics. Without offending the reflection principle as currently it stands, a theologiancan posit the possible existence of specific properties enjoyed exclusively by God. Since thespecific divine attributes are not experienced either intellectually or existentially by finite beingsbecause God does not share these properties with them, they can be expressed neither in a formallanguage nor in a natural language. The non-shared divine properties are simply terraeincognitae for finite beings. Thus, the specific divine properties, which stand as a guarantee ofdivine free-will, do not fall under the scope of the reflection principle. The previous two circumspections pertain to the absolute dependence of infinity on thefinite, which Christian theologians refute 31. Wiping out the theological reservations 32 fromCantor’s concept of infinity without distorting the mathematical aspect, we can now appreciatethe correspondences between Ω and ‘divine infinity’ without any worries. For this purpose, I30 Rucker, 256.31 Even theologians, such as Hartshorne, who follow process philosophy in assuming a dependence of infinity on thefinite, preserve certain specificity in divinity, namely prehending an infinite number of possibilities. Thus, theprimordial nature of God has its self-existing, by which it contributes to the process of creativity. See CharlesHartshorne, Aquinas to Whitehead (Milwaukee: Marquette University Publications, 1976).32 I surmise that no further theological cautions should hinder the theological use of Ω.
  19. 19. Mathematical Infinity and Theology 18appeal to Thomas Aquinas’s concept of divine infinity 33. I will show that these two concepts ofinfinity resemble each other with regard to their intrinsic meaning and their relation to the finite. C ANTOR AND A QUINAS : B IG O MEGA AND D IVINE I NFINITY The Thomistic divine infinity essentially expresses negation and eminence. Joining theapophatic tradition, Aquinas conceives of infinity as ‘not finite.’ He writes, In God, the infinite is understood only in a negative way, because there is no terminus or limit to His perfection (Summa Contra Gentiles –hereafter, SCG– I, 43.3).In his Summa Theologiea (hereafter, ST), the purpose of the negative understanding of infinity ismade clear: to show God as “distinguished from all other beings, and all others to be apart fromHim” (ST I, Q.7, a.1). This feature is found exactly in Ω, which cannot be conceived as anordinal. Would the existence of transfinites in the set theory diminish the distinctness ofAquinas’s divine infinity? Aquinas strives to confine infinity to God by ruling out the absoluteinfinity of matter, forms (ST I, Q.7, a.2), magnitude (ST I, Q.7, a.3), and multitude (ST I, Q.7,a.4). Nonetheless, he acknowledges the existence of relative infinities, such as matter, which isonly potentially infinite (ST I, Q.7, a.2). Even a non-divine actual infinity does not underminehis argument insofar as it is only relatively infinite. Now, transfinites are not entirelyunbounded, since they are contained within an always greater ordinal. Thus, they are merelyrelatively infinite 34.33 One reason for this choice is that some thinkers allege that Cantor’s theory counters the Thomist notion of divineinfinity, see Jean Rioux (2000).34 For more details, see Jean Rioux, 109-120.
  20. 20. Mathematical Infinity and Theology 19 The cataphatic side of Aquinas’s ‘infinity’ is the eminence it expresses. Divine infinity isthe perfection of all perfections. Many contemporary thomists point to the positive aspect ofdivine infinity 35. For instance, David Balas asserts, This esse subsistens…is not only infinite in a negative sense, but implies the infinite fullness of all pure perfections. Nor is esse subsistens a static notion, for esse is the actus omnium actuum and thus, if not limited in any way, contains eminently the actus of vivere, intelligere, and diligere. 36In Aquinas’s words, God is infinite in essence…Hence, no matter how many or how great divine effects be taken into account, the divine essence will always exceed them (SCG II, 26.3).On the mathematical side, eminence can be seen in Ω enjoying simultaneously all the propertiesthat shares and does not share with ordinals. It unitedly enjoys all of them, while each ordinalenjoys only the property that Ω shares with it. While some cardinals are hyperinaccessible, someothers Mahlo cardinals, etc., absolute infinity is all of these. Thus, at once, Ω is inaccessible,indescribable, ineffable, compact, extendible, etc 37. Notice the progression in some propertiessuch as inaccessible, hyperinaccessible, strongly inaccessible, and strongly compact, supercompact 38. This indicates an increase of perfectibility in the movement towards absoluteinfinity: the closer a cardinal approaches Ω, the more eminent its property is. The property ofcompactness is very relevant to Christian theology, for it corresponds to the oneness of God. ForAquinas, all perfections pre-exist as united in God (ST I, Q.13, a.5).35 William Placher, The Domestication of Transcendence (Louisville: Westminster John Knox Press, 1996): 23, 28.John Knasas, “Aquinas, Analogy, and the Divine Infinity,” Doctor Communis 40 (1987).36 David Balas, “A Thomist View on Divine Infinity,” Proceedings of the American Catholic PhilosophicalAssociation 55 (1981): 95.37 One property of the transfinite cardinals is ‘measurability’ (Rucker, 261-262). This property is to be scrutinized,in order to see whether it is a property of Ω and/or befits divine infinity. I hope I will have the opportunity toenlarge and deepen the study herein.38 Rucker, 261.
  21. 21. Mathematical Infinity and Theology 20 As for the knowledge of God, Aquinas distances his theology from the apophatic way. Hecounters explicitly Chrysostom and Dionysius on this issue (ST I, Q.12, a.1). For Aquinas, “itmust be absolutely granted that the blessed see the essence of God” (ibid). Although Godinfinitely transcends the finite, She remains “supremely knowable” (ibid). Obviously, the finiteintellect requires an aid to reach the infinite one. Thomas notes that “it is necessary that somesupernatural disposition should be added to the intellect in order that it may be raised up to sucha great and sublime height” (ST I, Q.12, a.5). The divine aid is the light of illumination. Further,Aquinas argues that “since the created light of glory received into any created intellect cannot beinfinite, it is clearly impossible for any created intellect to know God in an infinite degree” (ST I,Q.12, a.7). Hence, God can be known but never comprehended. It is evident that this is also the case of Ω, for the reflection principle presumes that someproperties of Ω are possessed by some ordinals. There is a better way to say this. Cantordistinguishes between the thing existing in itself and its existence in our intellect. In his words,“is not an aggregate an object outside us, whereas its cardinal number is an abstract picture of itin our mind?” 39 For Cantor, the aggregate is the ordinal, the intelligible part of which is itscardinal. Hence, the equation card(Ω) = Ω implies the intelligibility of Ω. However, theequation does not assert that Ω is comprehended, for the equality of two cardinals means that thecorresponding ordinals are equinumerous. For instance, the equality card(ω+2) = card(ω+7)indicates that the two non-identical ordinals ω+2 and ω+7 have the same number of elements.An equation between a cardinal and an ordinal means nothing more than that the ordinal does notshare the cardinality of a smaller ordinal.39 Georg Cantor, Contributions: 80.
  22. 22. Mathematical Infinity and Theology 21 Last but not least, for Aquinas, the divine infinity is to the world an increasing source ofcreativity. This is expressed in the last part of the following quotation, partly cited above. God is infinite in essence…Hence, no matter how many or how great divine effects be taken into account, the divine essence will always exceed them; it can be the raison d’être of more (SCG II, 26.3).Interpreting Aquinas, John Knasas illustrates the effect of divine infinity on the world throughthe communication of holiness. Different as the sanctity of Teresa [contemplative life] is from Xavier’s [missionary life], it is still the same in both. Sanctity in itself is acknowledged to contain both styles and who knows what myriad others. At the time of Augustine who could have seen a Teresa or a Xavier? Today who can guess what further analogates sanctity will assume. And if the sanctity of Teresa, for example, is awesome to behold, then sanctity in itself must be beyond endurance 40.The infinity of God is the reason for which further novelties can be expected. For Aquinas,divine infinity relates to the finite in terms of creation; God is the first, permanent, ultimatecreator of the world. Is not that true of Ω in the case of the vast collection of transfinitecardinals? Notice that Aquinas cautiously does not say that divine infinity is the raison d’être of aninfinite number of instantiations of divine effects 41. Following Aristotle’s theory of numbers 42,Aquinas would not be able to conceive of an infinite multitude. No species of number is infinite; for every number is multitude measured by one. Hence it is impossible for there to be an actually infinite multitude, either absolute or accidental. (ST I, Q.7,a.4)40 John Knasas, 75.41 Knasas shares the same reservation. He uses the term ‘myriad’ to indicate a big, yet finite number.42 Aquinas mentions his reference to Aristotle on the issue of infinite multitude in III, Physics, lect.8 (352): “numberis a multitude measured by the unit, as is said in the tenth book of the Metaphysics.”
  23. 23. Mathematical Infinity and Theology 22On the contrary, In ST, Q.7, a.2, Aquinas, the medieval theologian, counters Aristotle by positingtwo kinds of infinity: an absolute one pertaining only to God, and a relative one satisfied bymatter, as potentially infinite 43. Thus, Aquinas would not refute an infinite multitude should thisbe proved as relative. Now, Cantor provides a relative infinite multitude: the transfinites.Hence, Aquinas’s reservation towards an infinite multitude of divine effects vanishes. Byappealing to Cantor’s transfinites, a thomist can ascertain the existence of infinite instantiationsof divine holiness. Concisely, Cantor’s ‘infinity’ does not only conform to Aquinas’s ‘divine infinity,’ butalso tolerates what the latter aspires to, mainly infinite communications of divine properties.43 The Aristotelian infinity, which is seen merely as quantity, is imperfect; thus it cannot apply to God (ST I, Q.7,a.1). Aquinas differs from this position by bringing about a qualitative infinity applying exclusively to God. Thus,all other infinities, such as the potentiality of matter, become relative ones.
  24. 24. Conclusion 23 C ONCLUSION To sum up, Aquinas’s ‘divine infinity’ and Cantor’s ‘Big omega’ harmonize. In the settheory, big omega enjoys an efficient presence within the theory, yet is not grasped by anyconception. The efficiency of Ω is manifested by proving the existence of novel cardinals, asshown in the first chapter. Is not such presence similar to the religious experience of God? It isindeed, as systematized by Thomas Aquinas. The conception of absolute infinity is similar to thedivine as presence extremely transcendent and connected to the finite. And because the divineexistence is connected to the finite from far away, it generates novelties, always furthernovelties, and always further surprising novelties, forever and ever “world without end” 44. Moreover, the current work prevents two reservations about a theological use of absoluteinfinity. First, the definition of Ω as a set conveys that Ω depends entirely on what the setcontains. The theological worry is to be unable to conceive of absolute infinity as self-existing,which is essential to the concept of God as creator. Second, absolute infinity risks a destitutenessof any specificity, because of the reflection principle. The responses to the two cautions arebased on the intrinsic limit of the set theory. As a response to the first reservation, a set theoristdepicts Ω as a set short of better descriptions. In fact, a mathematician is unable to describeabsolute infinity, based merely on the language of the set theory. Set theorists can merely intuitthe absolute infinity as somehow a set-like. Absolute infinity is not grasped, yet is plausiblyposited, by mathematics. As far as the second reservation is concerned, the reflection principledoes not counter an absolute infinity exclusively enjoying properties that are unconceivable by44 Rucker, 78.
  25. 25. Conclusion 24the finite. Insofar as no further theological reservations arise, big omega is to be appreciated bytheologians as a mathematical counterpart of divine infinity. Furthermore, Ω and the Thomist ‘divine infinity’ can mutually support one another. Onthe mathematical side, absolute infinity is not proved, and cannot be proved. A theologicalinfinity which conforms to Ω sustains the latter. On the theological side, transfinites unfetterdivine infinity from a finite communication of properties. This mutual support draws on a smallscale how the mutual interaction of science and theology can be creative for both. It certainlyprompts further developments, where, on the one hand, mathematical and philosophical critiquesto Cantor’s theory are addressed, and, on the other hand, more theologians are introduced intothe discussion.
  26. 26. Bibliography 25 B IBLIOGRAPHYAquinas, Thomas. The Summa Contra Gentiles of Saint Thomas Aquinas: The First Book: LiterallyTranslated by the English Dominican Fathers from the Latest Leonine Edition. London: Burns Oates Washbourne, 1924.––––––––. The Summa Theologica of St. Thomas Aquinas: Second and Revised Edition.Literally translated by Fathers of the English Dominican Province, 1920.http://www.ccel.org/ccel/aquinas/summa.toc.html (accessed on December 10, 2009).Burns, Robert. Divine Infinity in Thomas Aquinas: I. Philosophic0-Theological Background. TheHeythrop Journal 39, no. 1 (1998): 57-69.––––––––––. Divine Infinity in Thomas Aquinas: II. A Critical Analysis. The Heythrop Journal 39, no. 2(1998): 123-139.Dahlstrom, Daniel, David Ozar, and Leo Sweeney. Infinity. Proceedings of the American CatholicPhilosophical Association. Washington, D.C.: The American Catholic Philosophical Association, 1981.Dauben, Joseph. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, New Jersey:Princeton University Press, 1979.Drozdek, Adam. Number and Infinity: Thomas and Cantor. International Philosophical Quarterly 39, no.1 (1999): 35-46.Georg, Cantor. Contributions to the Founding of the Theory of Transfinite Numbers. Translated by PhilipE.B. Jourdain. New York: Dover Publications, 1955.Huntington, Edward. The Continuum and the Other Types of Serial Order. Cambridge: Harvard UniversityPress, 1942.Knasas, John. Aquinas, Analogy, and the Divine Infinity. Doctor Communis XL, no. 1 (1987): 64-84.Oppy, Graham. Philosophical Perspectives on Infinity. New York: Cambridge University Press, 2006.Rioux, Jean. Cantors Transfinite Numbers and Traditional Objections to Actual Infinity. The Thomist64, no. 1 (2000): 101-126.Rucker, Rudy. Infinity and the Mind: The Science and Philosophy of the Infinite. Boston-Basel-Stuttgart:Birkhauser, 1982.Russell, Bertrand. The Principles of Mathematics. New York: W.W. Norton and Company Publishers,1937.
  27. 27. Bibliography 26Russell, Robert J. Cosmology from Alpha to Omega: The Creative mutual Interaction of Theology andScience. Minneapolis: Fortress Press, 2008.Russell, Robert J. Hegelian Infinity in Pannenbergs Doctrine of the Divine Attributes: New Insights fromCantors Mathematics. Forthcoming.Sierpinski, Waclaw. Cardinal and Ordinal Numbers. Translated by Janina Smolska. Wroclawska: PWN-Polish Scientific Publishers, 1965.

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