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

JST-SCU

Cantor’s Transfinites and Divine Infinity

BY FADY EL CHIDIAC, S.J.

1 Table of Contents

TABLE OF CONTENTS

INTRODUCTION, 2

TRANSFINITES 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

2 Introduction

INTRODUCTION

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, 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 God6

1 For an overview, cf.

.

http://en.wikipedia.org/wiki/Gregory_of_Nyssa#Infinity and http://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, although his goal is of a great interest for an interactive view of science and theology. My point is that Armour’s project cannot be grounded on Aquinas’s theology, unfortunately. 4 Robert Russell, Hegelian Infinity in Pannenberg’s Doctrine of the Divine Attributes: New Insights from Cantor’s Mathematics. (Forthcoming) This work of Russell inspired my current paper. 5 Robert Russell, “The God Who Infinitely Transcends Infinity: Insights from Cosmology and Mathematics,” in Cosmology from Alpha to Omega: The Creative Mutual Interaction of Theology and Science (Minneapolis: Fortress Press, 2008): 56-76. 6 Graham Oppy, Philosophical Perspectives on Infinity (New York: Cambridge University Press, 2006). This book is the precursor of a forthcoming promised book arguing against the cosmological argument. Oppy’s philosophical career is based on defeating theism. He published Arguing about Gods and Ontological Arguments and Belief in God.

http://en.wikipedia.org/wiki/Gregory_of_Nyssa#Infinity

http://en.wikipedia.org/wiki/John_Damascene

3 Introduction

This paper joins the previous interests in divine infinity, while it diverges from an atheist’s

project, such as Oppy’s. It attempts to show that Georg Cantor’s mathematical concept of

infinity and a Christian concept of divine infinity are relevant to one another. Furthermore, they

can engage a mutual interaction.

The steps undertaken here are simple. After an overview of Cantor’s set theory in a first

chapter, I engage a discussion in which I soften some theological reservations about the

mathematical 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 one

another. As a basic motivation, this paper intends to be an incipient work exploring whether

further development of the correlation between a mathematical concept of infinity and divine

infinity is worthwhile.

7 I have chosen Thomas Aquinas as theological counter-part to Cantor because the latter mentions the former in his correspondences with theologians about religion and science.

4 Transfinites and Absolute Infinity in Cantor’s Theory

TRANSFINITES AND ABSOLUTE INFINITY IN CANTOR’S THEORY

In the late nineteenth and early twentieth centuries, mathematicians were delighted to find

a possible foundation for mathematics in Cantor’s set theory. Whenever Cantor’s theory is

mentioned, David Hilbert’s adage is cited: “No one shall expel us from the paradise Cantor had

created.”8

TRANSFINITES

Cantor unfolded a mathematical paradise through the establishment of transfinite

numbers. Subsequent mathematicians and philosophers discover that the foundations of

mathematics are still far from being unveiled. However, this deception does not impede

mathematicians from pursuing Cantor’s project, mainly continually unfolding transfinite

numbers. Even though Cantor’s theory is still a research in progress since around 1875, it has

reached some unshakable constructions. For the purpose of this paper, it suffices to illustrate the

consistency of Cantor’s theory by treating the transfinites first, and then the ‘absolute infinity.’

Transfinites are categories included within the set theory. The first striking feature in the

theory of transfinites, as launched by Cantor, is that sets are considered to be numbers. Sets are

usually the ensemble of numbers, and here with Cantor numbers become also sets. Beginners

may receive this reluctantly, but defining numbers as sets is a well entrenched mathematical

conception. Some problems arise from this conception when absolute infinity is addressed. This

issue will be discussed in the next section. Cantor starts with natural numbers. He defines each

natural number as the set of natural numbers preceding it, such as the set 0, 1, 2, 3 is the

8 The citation is found in all the literature listed in the bibliography of this paper. In Oppy’s Philosophical perspectives on infinity, the chapter on transfinites is entitled “Cantor’s Paradise.”


number 4. Then, he introduces the most creative mathematical conception ever: the whole

interminable set of natural numbers is a number, symbolized by ω. Notice that this mathematical

conception is basically a philosophical intuition rooted in understanding the many as a unit. This

vision, 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 called

Cantor’s paradise. And from it, mathematicians construct myriads of transfinite numbers as

follows.

For reasons of clarity, some mathematicians talk about transfinites as sequences as well as

sets. I adopt the notion of sequence here, for it conveys ordered elements, which will help us to

understand the construction of other transfinites. The transfinite number ω+1 is the sequence of

natural numbers at the end of which is added the number 1. The number ω+1 is represented as

the 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 well

grounded mathematically9

9 Cantor argues for that against mainly Aristotle in his On infinite: Linear Point-Manifolds. A nice summary of Cantor’s argument is found in Jean Rioux, “Cantor’s Transfinite Numbers and Traditional Objections to Actual Infinity,” The Thomist 64 (2000): 109-119.

by distinguishing between adding the number 1 at the beginning of

the sequence and at its end. Here, 1+ω does not equate ω+1. The number 1+ω means adding 1

at the left limit of the natural numbers, that is, 1+ω = 1, 0, 1, 2 …. Adding a number from the

limited side of an endless sequence does not affect its endlessness. It means just pushing the

numbers one step ahead, whereas this is not possible when the number is added at the end of the

endless sequence. The phrase ‘the end of an endless sequence’ is rendered plausible once the

endless sequence is seen as one unit, here as ω. This is like saying ω+1 = ω, 1.


The construction continues. The number ω+2 is the sequence of natural numbers at the

end of which is added the sequence of 1 and 2. And, so on until we get ω+ω, noted ω·2. Here

too, we can have ω·3, ω·4 … ω·ω. The latter can be symbolized as ω2. Consider the following

representation10

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 includes

all precedent numbers. Hence, one can define an order through transfinites, such as

of ω2, which repeats ω times the sequence of natural numbers. In the figure,

each stick stands for a number.

ω < ω+ω < ωω < ωωω , where the sign < means ‘is included in.’

The Zermelo–Fraenkel set theory with the axiom of choice (abbreviated as ZFC), which is, so

far, the most successful axiomatic theory laying the foundation of mathematics, proves that for

10 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.

1 2 3 4 5…

1 2 3 4 …

1 2… 1 2… …

http://en.wikipedia.org/wiki/File:Omega_squared.png


any given transfinite there exists another transfinite which contains the former. Hence, one can

always construct a bigger transfinite number.

Transfinite numbers are more complex than ordinary numbers, since they designate an

internal order (for instance, ω+1≠1+ω) and a set of numbers. Cantor applies this distinction to

all numbers. He ascribes an ordinal to each number, designating the order of sequences

constituting 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 of

transfinites. They do not follow the same arithmetic rules, such as the commutative addition. To

see the difference, recall that ω+1≠1+ω (the order counts for transfinites), whereas addition is

commutative when applied to finite numbers. Radicalizing the distinction between finites and

transfinites, Bertrand Russell discards Cantor’s theory altogether, for transfinites involve “no

reference to the special peculiarities of quantity”11. He argues that unlike finite numbers

transfinites are irrelevant to the questions of ‘how much’ or ‘how many’. Following Bertrand

Russell, Jean Rioux concludes that transfinites seem to be “something more akin to quality or

relation”12

11 Bertrand Russell, Introduction to Mathematical Philosophy (New York: Norton & Company Publishers): 188. Bertrand Russell addresses other critiques to Cantor’s theory, such as actual infinite magnitudes are not one-side limited like the set of integers. He appeals to the continuous set of real numbers. That question will be defeated once 2ℵ0 = ℵ1 is proved. The equation means that the set of real numbers is equivalent to the ordinal 𝜀𝜀0.

rather than quantity. Hence, if numbers are confined to quantities, transfinites will no

more be numbers. Bertrand Russell radically challenges any commonality between transfinites

and finite numbers.

12 Rioux, 121. This critique holds also against the cardinals ℵi.


Indeed, infinite ordinals transcend finites, which is the reason Cantor calls them

transfinites. And, for this reason, some differences must be detected. One must not expect from

transfinites what one expects from finites, and vice versa. However, commonalities are found

between transfinites and finites. Mathematicians succeeded in setting arithmetic rules that

govern ordinals of both kinds13

In addition to ordinals, Cantor ascribes cardinals to numbers, noted as card(), in order to

account of the number of elements contained in the set. For example, the cardinal of number 7 is

the number of elements contained in the set 0, 1, 2…, 6, that is, card(7) = 7. Obviously, the

cardinal of any finite number is the same as the number,

. Addition, multiplication, exponentiation, power are enjoyed by

finites as well as transfinites. Addition and multiplication are distributive on the right, but not on

the left. Subtraction and division apply to some extent to transfinites. Hence, several, but not

all, usual features of arithmetic rules are reproduced in the case of transfinites. The finite

number 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?

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 of

natural numbers, is infinite designated by ℵ0.

Notice that some infinites are alike, while other infinites differ. On one hand, it can be

easily demonstrated that the sets of natural numbers, odd numbers, even numbers, and rational

13 For details, see Waclaw Sierpinski, Cardinal and Ordinal Numbers, second ed. (Warszawa: PWN, 1965).


numbers are equinumerous14

card(ω) = card(ω+1) = card(ω·3) = card(ω2) = card(ωω) = card(ωωω ) = ℵ0.

. Hence, all of them share ℵ0 as cardinal. Similarly, the cardinal of

any countable set is ℵ0. All transfinites encountered so far in this paper are countable.

Therefore,

On 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 2ℵ0

indicate two different kinds of numerical infinites15. “It turns out that just as we can always find

more ordinals, we can always find more cardinals. After ℵ0 come ℵ1, ℵ2, ℵ3, … ℵω, ℵω+1, …

ℵ𝜔𝜔𝜔𝜔 , … ℵℵ1 , … ℵℵ𝜔𝜔 , …, and so on”16

To summarize, in respect to both ordinals and cardinals, transfinite and finite numbers

enjoy commonalities as well as divergences. The similarities and differences in arithmetic rules

regarding 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 transfinites

are wealthier than the finite cardinals. Each transfinite cardinal corresponds to an infinite

. The ordinals having ℵ0 as cardinals are called the first

class ordinals, the ones of ℵ1 the second class, and so on. Each class of ordinals has a distinct

property, such as the property of ‘countable’ that the ordinals of the first class enjoy. Different

infinite cardinals denote different kinds of infinites.

14 The claim that there are as many natural numbers, 0, 1, 2, 3, 4…, as odd numbers, 1, 3, 5… counters common sense, because the former includes the even numbers in addition to the odd numbers. It is worth noting that for infinity not any addition makes difference. Common sense does not always get it right, specially when it comes to things uncommon to daily life experience, such as infinity. 15 Recall Bertrand Russell’s critique: transfinites are irrelevant to the question “how many.” A response can be given here. ℵ0 and 2ℵ0 are ordinals as well as cardinals. They specify how many real numbers there are with respect to the number of integers. 16 Rudy Rucker, Infinity and the Mind: the Science and Philosophy of the Infinite (Boston, Basel, Stuttgart: Birkhauser): 77.


number of ordinals, such as ℵ0 is the cardinal of all countable infinite ordinals, whereas any

finite cardinal corresponds to one and only one finite ordinal. The following section will discuss

the mathematical use of absolute infinity.

ABSOLUTE INFINITY

Cantor was eager to establish the existence of an absolute infinity, entitled Ω. However,

he discussed this issue more with theologians than with mathematicians17. His absolute infinity

seems 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 deal

with big omega. Although big omega is an indistinct notion, “talking about Ω is an extremely

useful and productive thing for set theorists to do”18

To begin with, it is absolutely plausible to posit a set containing all ordinals. Let this set

be Ω. First caution: Ω must not be an ordinal. Otherwise, we face the contradiction that Ω

contains Ω. Furthermore, Ω should not even be a limit to which a series of ordinals tends

. How is this? If you want to understand the

mathematical status of big omega, prepare yourself for a continuous back-and-forth between a

claim and its opposite.

19

17 Unfortunately, Cantor’s correspondences with neo-thomists, where the absolute infinity is discussed, are not translated into English. Adam Drozdek provides overviews of some of these correspondences in “Number and infinity: Thomas and Cantor,” in International Philosophical Quarterly 39 (1999).

.

Thus, Ω seems to be an outsider. Nonetheless, if Ω becomes an alien notion assembling all

ordinals, Ω will be irrelevant to mathematics. Therefore, without becoming totally an ordinal, Ω

18 Rucker, 254. 19 For either one of the two reasons: cardinal of Ω should be regular (Rucker, 254-255) and the series of ordinals converge towards an ordinal (see Sierpinski, 382-390).


must be at least and at most an ordinal-like. Rucker treats it as an “imaginary ordinal”20

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

. A way

out of this dilemma is provided by the reflection principle:

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 set22

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 least

some ordinals.

.

The reflection principle has proved very fruitful. It allows the generation of a multitude of

cardinals, which means a diverse variety of infinities. First come the inaccessible cardinals, then,

hyperinaccessible cardinals23

“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.

http://en.wikipedia.org/wiki/Reflection_principle


To see how the reflection principle caused this abundant collection, we consider here the

discovery of the first kind of this series, namely the inaccessible cardinals25

The effect of the mere existence of such bizarre cardinals on finites exceeds our

expectations. The case of measurable cardinals is interesting because of this issue. To put it in

Rucker’s words,

. In a nutshell, big

omega is a set, thus, it should have a cardinal. Besides, the equation card(Ω) = Ω must be

assumed, for if this is not the case, Ω will depend on the cardinal of a smaller ordinal. Thus, big

omega can be spoken of as ordinal and cardinal. And, card(Ω) should be inaccessible because Ω

is 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 one

ordinal enjoys ‘being inaccessible.’ Hence, the set of ordinals enjoying an inaccessible cardinal

is not empty. Any formalized property pertaining to Ω, that is a very ambitious property, is

reached by some ordinals. By Ω and the reflection principle, what is beyond all is reachable by

some.

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 bodies26

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 in Gödel’s universe L of constructible sets” (Ibid, 262).


In a larger view, positing an absolute infinite and assuming the reflection principle proves the

existence of unexpected transfinite cardinals, which may affect our understanding of finites.

Indeed, this interaction between the absolute infinite and the finite through transfinites is just

amazing. 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, Ω

may share properties directly with finite ordinals, without a necessary transfinite mediation.

Although transfinites enjoys a middle place between finites and absolute infinity, the

communication between the two latter does not require transfinites mediation27

• Transfinites are infinite numbers, defined as sets.

. One senses here

some theological implications. The following chapter will treat this at length. While turn to the

next chapter, retain these six points:

• 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.

14 Mathematical Infinity and Theology

MATHEMATICAL INFINITY AND THEOLOGY

Rare are the contemporary Christian theologians who introduce a mathematical concept of

infinity in their talk about God. Many allege that mathematics leaves no room for the

Incomprehensible. The previous development counters such prejudice by having illustrated how

mathematics accommodate an unconceivable notion, namely Ω. We are in debt to Cantor for

this great change in modern mathematics. Would this allow an interaction between theology and

mathematics? Yes, it allows an interaction indeed. What follows is an exploration of such

interaction.

The transfinites of Cantor’s theory are inadequate to God, for they are not entirely

unlimited. Any transfinite ordinal is limited by a greater one, as shown above

(ω<ω+ω<ωω<ωωω<…). Besides, transfinites are totally determined by their construction. Thus,

such 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 the

God of monotheism. As depicted above, absolute infinity, Ω, transcends transfinite as well as

finite ordinals. Yet, due to the reflection principle, Ω is related to ordinals. This absolute

infinity, transcendent-yet-in-communion with non-absolute-infinities, is very tempting to

theological use. Nonetheless, some theologians may resist the use of mathematical infinity for

two cautions I address in the following section.


RESPONSE TO SOME THEOLOGICAL RESERVATIONS

The first circumspection is concerned with pantheistic and panentheistic implications28

28 Through his correspondence with Johannes Cardinal Franzelin, Cantor addressed the pantheist features, of which he was not aware in the incipient development of ‘absolute infinity.’ Unfortunately, I did not have access to an English 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 that Cantor 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 (see Robert Russell, Hegelian Infinity in Pannenberg’s Doctrine of the Divine Attributes: New Insights from Cantor’s Mathematics: 14). Besides, the panentheist implications are left over.

.

Recall that big omega is defined as the set of all ordinals. However, it is not the convergence of

any 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 theologians

might be reluctant to ascribe to the Christian God. Nonetheless, what is stressed by the fact that

Ω is not the upper limit of any series of ordinals is that every ordinal participates in Ω. In the

picture where Ω is a set, any individuality enjoys a participation in the absolute infinity. On the

contrary, in the picture where Ω is an upper limit, only the last terms of the series enjoy an

immediate relation with Ω. As set of all ordinals, the concept of absolute infinity sustains

panentheism, but it is closer to the Christian faith than what seems to be its only mathematical

alternative, which is ‘Ω is an upper limit.’ Examined carefully, the mathematical concept of

absolute infinity does not support pantheism, for Ω is merely a set-like; big omega would

become an ordinal if it were formalized as a set. Furthermore, it is understandable that, in a set

theory, it is very difficult to find an image apart from ‘set’ to designate the absolute infinity. In

the set theory, all entities are sets. The internal limit of the discipline in question is to be


admitted here. If the set theorists preferred to avoid the expression ‘set-like,’ they would not find

alternative 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 the

set 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 infinity

appealing.

The other caution regards the reflection principle. The worry here is about emptying

absolute infinity of any specific property. If, by the reflection principle, infinity shares all its

properties with ordinals, what would be left as particular to infinity? Infinity devoid of

specificity does not disturb mathematical functioning, whereas it provokes theological

reservations. Certainly, as Rucker notes, “although ‘Ω is the class of all ordinals’ is true, we do

not expect it to be true of any ordinal less than Ω”29

29 Rucker, 256.

. 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 own

existence, is not satisfying. The idea endorsed here is not an image of a stingy God. On the

contrary, the reflection principle befits the communication of divine properties that Christian

theologians unanimously allege. The Christian God is so generous and loving as to send us His

Son. The point here is the gratuitousness and freedom of God, which the reflection principle is

seemingly not considering. God created the world, sent us His Son, and is acting in the world by

free-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 the

limit of mathematical language, which cannot be otherwise than deterministic. Thus, the


reflection 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 reflection

principle. The properties shared by Ω are those which are conceivable. As understood by

Rucker, “‘conceivable property’…is supposed to mean a property that is expressible in terms of

sets and language of some kind”30

The previous two circumspections pertain to the absolute dependence of infinity on the

finite, which Christian theologians refute

. Thus, the reflection principle does not apply to unconceivable

properties, which cannot be expressed in any language. Anyway, such properties are irrelevant

to mathematics. Without offending the reflection principle as currently it stands, a theologian

can posit the possible existence of specific properties enjoyed exclusively by God. Since the

specific divine attributes are not experienced either intellectually or existentially by finite beings

because God does not share these properties with them, they can be expressed neither in a formal

language nor in a natural language. The non-shared divine properties are simply terrae

incognitae for finite beings. Thus, the specific divine properties, which stand as a guarantee of

divine free-will, do not fall under the scope of the reflection principle.

31. Wiping out the theological reservations32

30 Rucker, 256.

from

Cantor’s concept of infinity without distorting the mathematical aspect, we can now appreciate

the correspondences between Ω and ‘divine infinity’ without any worries. For this purpose, I

31 Even theologians, such as Hartshorne, who follow process philosophy in assuming a dependence of infinity on the finite, preserve certain specificity in divinity, namely prehending an infinite number of possibilities. Thus, the primordial nature of God has its self-existing, by which it contributes to the process of creativity. See Charles Hartshorne, Aquinas to Whitehead (Milwaukee: Marquette University Publications, 1976). 32 I surmise that no further theological cautions should hinder the theological use of Ω.


appeal to Thomas Aquinas’s concept of divine infinity33

CANTOR AND AQUINAS: BIG OMEGA AND DIVINE INFINITY

. I will show that these two concepts of

infinity resemble each other with regard to their intrinsic meaning and their relation to the finite.

The Thomistic divine infinity essentially expresses negation and eminence. Joining the

apophatic 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 is

made clear: to show God as “distinguished from all other beings, and all others to be apart from

Him” (ST I, Q.7, a.1). This feature is found exactly in Ω, which cannot be conceived as an

ordinal. Would the existence of transfinites in the set theory diminish the distinctness of

Aquinas’s divine infinity? Aquinas strives to confine infinity to God by ruling out the absolute

infinity 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 is

only potentially infinite (ST I, Q.7, a.2). Even a non-divine actual infinity does not undermine

his argument insofar as it is only relatively infinite. Now, transfinites are not entirely

unbounded, since they are contained within an always greater ordinal. Thus, they are merely

relatively infinite34

33 One reason for this choice is that some thinkers allege that Cantor’s theory counters the Thomist notion of divine infinity, see Jean Rioux (2000).

.

34 For more details, see Jean Rioux, 109-120.


The cataphatic side of Aquinas’s ‘infinity’ is the eminence it expresses. Divine infinity is

the perfection of all perfections. Many contemporary thomists point to the positive aspect of

divine infinity35

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.

. For instance, David Balas asserts,

36

In 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 properties

that shares and does not share with ordinals. It unitedly enjoys all of them, while each ordinal

enjoys only the property that Ω shares with it. While some cardinals are hyperinaccessible, some

others Mahlo cardinals, etc., absolute infinity is all of these. Thus, at once, Ω is inaccessible,

indescribable, ineffable, compact, extendible, etc37. Notice the progression in some properties

such as inaccessible, hyperinaccessible, strongly inaccessible, and strongly compact, super

compact38

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).

. This indicates an increase of perfectibility in the movement towards absolute

infinity: the closer a cardinal approaches Ω, the more eminent its property is. The property of

compactness is very relevant to Christian theology, for it corresponds to the oneness of God. For

Aquinas, all perfections pre-exist as united in God (ST I, Q.13, a.5).

36 David Balas, “A Thomist View on Divine Infinity,” Proceedings of the American Catholic Philosophical Association 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 to enlarge and deepen the study herein. 38 Rucker, 261.


As for the knowledge of God, Aquinas distances his theology from the apophatic way. He

counters explicitly Chrysostom and Dionysius on this issue (ST I, Q.12, a.1). For Aquinas, “it

must be absolutely granted that the blessed see the essence of God” (ibid). Although God

infinitely transcends the finite, She remains “supremely knowable” (ibid). Obviously, the finite

intellect requires an aid to reach the infinite one. Thomas notes that “it is necessary that some

supernatural disposition should be added to the intellect in order that it may be raised up to such

a 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 be

infinite, 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 some

properties of Ω are possessed by some ordinals. There is a better way to say this. Cantor

distinguishes 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 it

in our mind?”39

39 Georg Cantor, Contributions: 80.

For Cantor, the aggregate is the ordinal, the intelligible part of which is its

cardinal. Hence, the equation card(Ω) = Ω implies the intelligibility of Ω. However, the

equation does not assert that Ω is comprehended, for the equality of two cardinals means that the

corresponding 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 not

share the cardinality of a smaller ordinal.


Last but not least, for Aquinas, the divine infinity is to the world an increasing source of

creativity. 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 through

the 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 endurance40

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, ultimate

creator of the world. Is not that true of Ω in the case of the vast collection of transfinite

cardinals?

.

Notice that Aquinas cautiously does not say that divine infinity is the raison d’être of an

infinite number of instantiations of divine effects41. Following Aristotle’s theory of numbers42

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)

,

Aquinas would not be able to conceive of an infinite multitude.

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): “number is a multitude measured by the unit, as is said in the tenth book of the Metaphysics.”


On the contrary, In ST, Q.7, a.2, Aquinas, the medieval theologian, counters Aristotle by positing

two kinds of infinity: an absolute one pertaining only to God, and a relative one satisfied by

matter, as potentially infinite43

Concisely, Cantor’s ‘infinity’ does not only conform to Aquinas’s ‘divine infinity,’ but

also tolerates what the latter aspires to, mainly infinite communications of divine properties.

. Thus, Aquinas would not refute an infinite multitude should this

be proved as relative. Now, Cantor provides a relative infinite multitude: the transfinites.

Hence, Aquinas’s reservation towards an infinite multitude of divine effects vanishes. By

appealing to Cantor’s transfinites, a thomist can ascertain the existence of infinite instantiations

of divine holiness.

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.

23 Conclusion

CONCLUSION

To sum up, Aquinas’s ‘divine infinity’ and Cantor’s ‘Big omega’ harmonize. In the set

theory, big omega enjoys an efficient presence within the theory, yet is not grasped by any

conception. The efficiency of Ω is manifested by proving the existence of novel cardinals, as

shown in the first chapter. Is not such presence similar to the religious experience of God? It is

indeed, as systematized by Thomas Aquinas. The conception of absolute infinity is similar to the

divine as presence extremely transcendent and connected to the finite. And because the divine

existence is connected to the finite from far away, it generates novelties, always further

novelties, and always further surprising novelties, forever and ever “world without end”44

Moreover, the current work prevents two reservations about a theological use of absolute

infinity. First, the definition of Ω as a set conveys that Ω depends entirely on what the set

contains. 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 destituteness

of any specificity, because of the reflection principle. The responses to the two cautions are

based on the intrinsic limit of the set theory. As a response to the first reservation, a set theorist

depicts Ω as a set short of better descriptions. In fact, a mathematician is unable to describe

absolute infinity, based merely on the language of the set theory. Set theorists can merely intuit

the absolute infinity as somehow a set-like. Absolute infinity is not grasped, yet is plausibly

posited, by mathematics. As far as the second reservation is concerned, the reflection principle

does not counter an absolute infinity exclusively enjoying properties that are unconceivable by

.

44 Rucker, 78.

24 Conclusion

the finite. Insofar as no further theological reservations arise, big omega is to be appreciated by

theologians as a mathematical counterpart of divine infinity.

Furthermore, Ω and the Thomist ‘divine infinity’ can mutually support one another. On

the mathematical side, absolute infinity is not proved, and cannot be proved. A theological

infinity which conforms to Ω sustains the latter. On the theological side, transfinites unfetter

divine infinity from a finite communication of properties. This mutual support draws on a small

scale how the mutual interaction of science and theology can be creative for both. It certainly

prompts further developments, where, on the one hand, mathematical and philosophical critiques

to Cantor’s theory are addressed, and, on the other hand, more theologians are introduced into

the discussion.

25 Bibliography

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Russell, Robert J. Cosmology from Alpha to Omega: The Creative mutual Interaction of Theology and Science. Minneapolis: Fortress Press, 2008.

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Spiritual

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