Infinity in Religion An Analysis of the Work of Georg Cantor

1. Infinity in Religion An Analysis of the Work of Georg Cantor Danielle Collins, Cooper Byrne, and Harry Wise

2. Background • Human ability to understand the infinite • Le Blanc • Can we apply the same concepts to both the finite and the infinite? • Is there something about the infinite that prevents attribution to it of concepts that we can attribute to the finite?

3. The Kabbalah • Ten Sefirot • 10 attributes of the Ein Sof • “The entity is so large, so supreme, so far beyond description, that it is given the only name the Kabbalists could possibly describe it: Ein Sof. The two words mean infinity. God is infinite.” (Aczel, 34) • "The idea of infinity could contain finite parts, but that the whole, infinity itself , was immeasurably greater than its parts” (Aczel, 34)

4. God • God has infinite knowledge, power, and benevolence • God is a mystery • Nobody knows exactly what God is • God is inexhaustible • One can never find an end to God’s forgiveness, one cannot sin enough to lose God’s forgiveness • God has infinite goodness • Cannot be greater • If God did not eliminate all evil, he would be finite • God - Completeness • Aristotle says that if something is complete, it is missing nothing. Nothing is outside of it. • Infinity is a never ending process so there is always something left out • Humans can only grasp infinity in a finite manner

5. History • Infinity and God date back to Greeks and Pythagoreans • Importance of numbers • “Number is all” (Voss) • Relationships of numbers yield “The harmony of the spheres” (Voss) • Anaximander - Cosmological Theory • Perion: ‘limited, clearly defined, having simple form’ (Heller) • Aperion: ‘unlimited, indescribable, chaotic’ (Heller)

6. Progression of the Perception of Infinity -Infinity was seen by the ancient world as a threat to order, people feared apeiron qualities --God as a perfect being could not possibly possess apeiron qualities -The religious world was mostly responsible for the shift in thought -Infinity would soon become a concept filled with mysticism and wonder that infinity could encompass the highest good and source of the world -Philosopher Plotinus was first to view infinity in a positive religious light (He connected God with attribute of potential infinity, set it forth as something a human could strive for)

7. Descartes Ontological Proof of God God is unlimited and unbounded... -God is a perfect being -It is more perfect to exist than not to exist/ existence is perfection -Therefore, God must exist Humankind has an innate view of God as an infinite being, therefore, God must have an infinite reality Something cannot arise out of nothing, therefore, God has instilled in us the perception of his existence

8. Aristotle • Potential Infinity • Can always be extended • Always coming into being, never fully present • Recursive Procedure: Initial and successor rule (Steinhart, 262) • Example: Zeus begins at 0 and continues counting 0, 1, 2, 3, 4 forever. He is immortal so he will always continue counting • Always another possible number • Does not define last number

9. Actual Infinity • Greeks unable to grasp the infinite • Prefer geometry as opposed to algebra • “The square root of two can be constructed geometrically in just a few steps, but to define it algebraically takes some understanding of an infinite procedure • Actual Infinity • Completed totality with infinitely many members • A series that is wholly present at some moment in time (Steinhart, 262) • Last number is defined • Example: Set of all positive integers • Initial = 0, n+1 • 0, 1, 2, 3, 4 … n+1

10. Actual Infinity • Aristotle denies the existence of any actual infinities • Gregory of Rimini (1300s) • What we as humans see as potentially infinite, God sees as actually infinite • “This means that an agent whose creative power is unlimited can produce actually infinite series of objects… God creates an initial angel in ½ a minute; God then creates every successor angel twice as fast; at 1 minute, God has created an actual infinity of angels. (Steinhart, 262)

11. Predominant Thought • Two separate spheres of thought • Many of the brightest minds forged connections with theological matters in the divine realm, however, many professors in the mathematical realm “generally welcome the fact that religion in no way impinges on their field” (Kneale) • Examples • Numbers in the Bible (666, 153) • Copernicus: Liturgical calendar • Initial thoughts on potential and actual infinity lasted six hundred years, until the nineteenth century when Georg Cantor revolutionized the idea

12. Georg Cantor: Infinity and God “I entertain no doubts as to the truths of the transfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science”

13. Cantor and Absolute Infinity • Cantor brought the notion of absolute infinity into mathematics in a wholly new way • Felt a profound religious connection, that God had helped him discover the truths of his work • God has provided him with a sense of infinity • Formalism: The view that mathematics does not need to be grounded in physical reality • The term “absolute infinity” became identified with God

14. Cantor and Absolute Infinity ABSOLUTE INFINITY: -Transcends the infinity of the transfinite numbers -Connected with God -Amongst its mathematical properties lies the Reflection Principle -Consists of two different kinds of infinity: 1)transfinite--potential infinity 2)absolute--actual infinity (a concept beyond human capacity for understanding)

15. Cantor and the Reflection Principle -If P is any describable property enjoyed by the Absolute, then there must be something smaller than the Absolute that also has property P. There must be an ordinary set that is infinite as well Connection to religion: -The human mind, no matter how advanced or progressive, cannot attain the concept of God but only what is beneath God This principle also shines light on the Absolute itself, the properties of the Absolute Infinite must be shared with morals through the properties of the transfinites “What God has chosen to disclose to us, God’s existence as Creator, Gods’ goodness, love and beauty - is a veil behind with the reality of God is endlessly hidden precisely as it is endlessly revealed” (Heller, 11)

16. Cantor and the Reflection Principle • Reflects the tension between comprehensibility and incomprehensibility (Heller, 284) • People would like to assume God (as a figure of absolute infinity) is a source of wisdom and knowledge far beyond the scope of human grasp • However, people also want to affirm that they know their Creator and Redeemer • Poses an innate contradiction • “The God who is incomprehensible is the God who makes Godself known to the world” (Heller, 285) • God is both knowable and unknowable, and in terms of the theory of divine revelation, God is hidden in God’s own self-disclosure (Heller, 285)

17. Cantor’s Set Theory and God Cardinality: The cardinality of the transfinites and of the finites represents part of the difference between the infinite (ie the transfinites) and the finite Ordinality: The ordinality of the transfinites and of the finites represents an aspect of the similarity between the infinite (ie the transfinites) and the finite while still acknowledging their difference, as shown by the representations of their differing forms of cardinality

18. Cantor’s Rational Research 1) Rational discernment of infinity -Cantor created a new sort of number, defining a transfinite set (aleph sub 0), as the first in a transfinite set {1,2,3…}, created arithmetic for the transfinite numbers and the concept of the cardinal number 2) Real antinomies -Idea of the totality of the cardinal numbers, the set of all cardinal numbers results in a contradiction 3) Resolving the antinomies -Claim that the Absolute Infinity is God, God as “the creative source of all quantities existing in the world, and an intuitive insight of God is possible” (Heller, 41)

19. Cardinality and Ordinality Ramifications -Suggestive relationship between the finite and the transfinites with different cardinalities and similar ordinalities -Perhaps God’s holiness “not only opposes the profane world, but ‘embraces it, bringing it into fellowship with the holy God’” (Heller, 288) -Could be that God’s holiness is fully infinite, opposing the troubled world, yet still entering, penetrating, and transforming it

20. Infinity and Proof of God

21. Cantor and the Transfinites Transfinite Numbers: Express the “extensive domain of the possible in God’s knowledge (Cantor, 405) -This is the sort of infinity that humans can grasp, unlike Absolute Infinity -Viewed by Cantor as proof of the majesty of God -Instead of weakening the difference between ‘infinite’ God and finite man, the transfinites illustrate God’s impressiveness, supports the Christian tradition of the separation between the finite and the infinite

22. Cantor Infinity and Proof of God Cantor reasons that since God is of the highest perfection, one can conclude that it is possible for him to create a transfinitum ordinatum (cardinality of transfinites, a method of ordering the sizes of infinity) -From his pure goodness and omnipotence, we can conclude that a transfinitum ordinatum has been created -Transfinite numbers are an expression of God’s infinite capacity and knowledge -Not all infinities are of equinumerous size (Transfinite is increasable, Absolute infinity is not)

23. Absolute Infinity and the Connection to God Cantor felt that trying to mathematize the Absolute was a mistake (All numerable things already existed in the realm of the finite and the transfinite) The Absolute and God are beyond determinability • The Absolute is the category of everything mathematizalbe, but it is not the mathematizable itself • “Each transfinite number represents all finite forms which have that number, and thus either sequence of numbers represents simultaneously all sets and the Absolute” (Halle, 43) • → The Absolute embodies the realm of all sets, the collection of them

24. Absolute Infinity and the Connection to God • “The Absolute acts as a kind of universe for set theory” (Hallet, 44) • The transfinites indicate the Absolute because they have to rely on some sort of completed, total domain (ie the Absolute) • Cantor justified his conception of the transfinites existence by expressing them as ideas of God’s divinity

25. Religious Infinity and the Hope for Humanity • Despite the infinity of God, there is hope for us as individuals • A singular person counts as much as a numerous amount • Even humble careers and tasks are equivalent to more grand tasks, can be extended into the area of human rights • Goodness of God is unlimited

26. Bibliography 1. Aczel, A. D. The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. Washington Square Press, 2001. Print. 2. Le Blanc, Jill. "Infinity in Theology and Mathematics." Religious Studies 29.1 (1993): 51-62. Print. 3. Crumplin, Mary-Ann. "Descartes: God as the Idea of Infinity." International Journal of Systematic Theology 10.1 (2008): 3-20. Print. 4. Heller, Michael, and Woodin , W. Hugh. Infinity : New Research Frontiers. Cambridge: Cambridge University Press, 2011. Web. 5. Henry, G. C. Logos: Mathematics and Christian Theology. Bucknell University Press, 1976. Print.

27. Bibliography 6. Johnson, Dale M. "Georg Cantor: His Mathematics and Philosophy of the Infinite ." The British Journal for the History of Science 14.1 (1981): pp. 101-103. Print. 7. O'Brien, D. "Matter and Infinity in the Presocratic Schools and Plato." The British Journal for the Philosophy of Science 20.2 (1969): pp. 163-167. Print. 8. Shapiro, Stewart. "Theology and the Actual Infinite: Burley and Cantor." Theology and Science 9.1 (2011): 101-8. Print. 9. Steinhart, Eric C. "A Mathematical Model of Divine Infinity." Theology and Science 7.3 (2009): 261-74. Print. 10. Hallett, Michael. “Cantorian Set Theory and Limitation of Size.”