Cantor’s Theorem and Jesu

Cantor’s Theorem and Jesu: How Mathematical Notions of Infinity Contribute to the Understanding of Jesus

Now that you have started reading this journal, I may have imprisoned your attention for an infinite amount of time. Let me prove this to you. As you turn this page in two minutes, you will only have half of my argument left to ponder. One minute later, you will have gazed at half of the remaining words, leaving one-quarter of the material unread. Thirty seconds more, and only one-eighth of the words will be unfamiliar to your eyes. This process will continue indefinitely, and very quickly you will merely need to read 1/16th, 1/32nd, and 1/64th of the words within these pages before you can continue on with your life. Nevertheless, you will always take some amount of time to finish the half of the article that remains unread. You will require milliseconds to read half of the final word, microseconds to read the quarter of the word remaining, and femtoseconds to read the eighth remaining. There will always be an infinitesimal part of the word yet to be perceived by your eye that will take infinitely long to finish.

Such paradoxes, authored by the Greek philosopher Zeno, baffled several of the most celebrated thinkers of the ancient world. Left frustrated, mathematicians and philosophers alike set to disprove Zeno’s arguments, and the formalized study of infinity was born in the 5th century B.C. Although the logic behind Zeno’s paradoxes was disproven with the foundation of calculus in the 14th century, the concept of infinity became the subject of significant scholarship. In fact, early church writers attributed the notion of infinity to God, which became “a cornerstone in Christian Theology.”[2] Yet in 1891, a German Lutheran by the name of Georg Cantor radically transformed the world’s understanding of infinity in a simple yet elegant proof. Cantor verified that there are multiple levels of infinity.

In his proof, Cantor discovered the first of the transfinite numbers. Such sequences of numbers, although indisputably infinite, are less in magnitude than an alternate form of infinity, absolute infinity. Theologians and mathematicians alike were outraged, claiming that Cantor’s ideas were beyond blasphemous. In fact, the Church argued that Cantor supported pantheism, claiming his proof implied that “layers of infinity” could be nestled within the world.[3] Yet Cantor defended himself by reserving the notion of absolute infinity (represented by Ω) for God alone, [4] while arguing that “the transfinite numbers are eternally existing realities in the mind of God.”[5]

In the following two hundred years since this proof, numerous books, journals, and articles have been published attempting to rationalize how God relates to transfinite and infinite numbers. However, no such scholarship has theorized where Jesus belongs in regards to the infinite. Yet, analysis of Cantor’s Theorem provides a sophisticated metaphor for understanding a fundamental question in Christology: how can Jesus be fully human yet completely divine? Hence, I will outline the remainder of this argument as any mathematician would in the form of a proof.

Proof:

Part 1 – The Incarnation and Divinity

The opening line of the Nicene Creed unequivocally emphasizes that Jesus is “begotten, not made, being of one substance with the father.” Central to the Creed is the claim that Jesus is a distinct person from the God the Father, the first person of the trinity. Yet the question must be asked, “how is it possible for God to become incarnate in a human being without sacrificing divinity?” Although such a manifestation of divinity in human form appears to be illogical, certain properties of transfinite sequences analogously support such a claim.

Imagine that all of the numbers in the real number system R (which consists of all numbers that can be written in a decimal form) are placed in a giant cauldron. Next, we seek to “beget” Jesus from God by simply removing the transfinite sequence of natural numbers (N=1.0, 2.0, 3.0,…) from the mixture. Jesus, akin to N, is unmistakably infinite (and thus divine). Likewise, although we have removed an infinite amount of numbers from the mixture, all of the non-integer numbers remain in the cauldron, so God himself remains infinite (and divine) as well. Yet this appears to be a contradiction. How can ∞ – ∞ = ∞ ? Such a statement can only be understood in terms of Cantor’s transfinite numbers. In order for both God and Jesus to be fully divine (infinite) after the incarnation, it is essential for God to exist as a different transfinite order than Jesus. Mathematically, this is only plausible way for God to maintain divinity while manifesting in the distinct yet divine human form of Jesus.

Although God the Father and Jesus certainly exist as different transfinite orders, this in no way implies that the Son is inferior to the Father. In fact, the mutual infiniteness unifies Jesus with God the Father, allowing both to exist as fully divine entities. Yet though they are united, their unique transfinity allows God the Father and Jesus to remain distinct. This mathematical argument is in accordance with the theological claim that God the Father and Jesus exist as “true God from true God” while remaining separate entities.[6] Hence, the properties of transfinite numbers allow us to understand that in the incarnation, Jesus maintained his divinity while remaining equal yet distinct from God the Father.

Part II – Humanity without Sacrificing Divinity Adding up all that is Jesus.

The Christological claim that Jesus is both human and divine at the same time is analogous to the claim that Jesus is both finite and infinite. On the surface, the latter appears to be the mathematician’s worst nightmare: a contradiction. Just as infinity is by definition opposite of the finite numbers, certain characteristics of humanity are directly opposite of divinity.[7] This understanding of God’s relation to the physical universe, formally known as via negativa, claims that the infinite God utterly contrasts with the finite world and can only be defined by what God is not.[8] Ancient Jews used the Hebrew word Eyn-sof to describe how God’s infinite greatness could not be contained on Earth.[9] How then is it possible for Jesus, infinite and divine, to exist within the defined boundaries of Earth? Again, the properties of Cantor’s transfinite numbers allow us to understand how an infinite sequence can manifest itself in a finite way.

Let us again imagine that Jesus is a transfinite set, this time comparing Christ to the sequence of numbers defined by 1/2n (thus the elements of this set are 1/2, 1/4, 1/8, 1/16 and so on). There are an infinite number of elements defined by 1/2n so in our analogy Jesus is indeed infinite and thus divine. Yet adding this sequence of elements, defined by the notation:

Cantor's Theorem sequence1

actually yields the finite result of one (see figure 1 below).

Cantor's Theorem diagram

Again, the mathematical logic seems perplexing: adding an infinite amount of nonzero quantities somehow yields a finite result. Yet such limits, characteristic of certain transfinite sets, allow the concept of infinity to manifest in the finite. Hence in our analogy, Jesus, certainly a “divine” set of numbers, actually is capable of revealing himself in a finite, completely human form in the physical world.

Since Jesus was able to exist as a human being, he provided humans with a way to conceptualize the Father. As God’s absolute infinity (Ω) is “closely associated with the notion of having no limit or boundary in terms of human conceptual understanding,” it is impossible for God to manifest himself in a finite manner.[10] Yet Jesus represents a notion of infinity that humans can comprehend. Jesus becomes finite, in the form of the number 1, and attains a purely human identity. Yet as we examine the core of Jesus’ identity, we are able to see that his finiteness consists of an infinite amount of elements. Under his humanity lies a profound divinity, so rich that his character can be understood through a transfinite number as both infinite and finite – God and human.

Part III: Humanity + Divinity = א

As Cantor believed his work to be more significant theologically than mathematically, he used the Hebrew symbol א (aleph) to represent the magnitude of his transfinite sets.[11] Although Cantor himself did not equate Jesus with א, certainly his theorem can help the modern Christian rationalize the Christological nature of Jesus’ humanity and divinity. Jesus (א) became incarnate in a different magnitude of transfinity than God the Father himself, which allowed both figures to maintain equal yet distinct levels of divinity (infinity). Yet because Jesus is transfinite and not absolutely infinite (Ω), he could to fully embody humanity. Such transfinity, in its elegant mathematics, allows us to understand that Jesus was fully human while maintaining his complete divinity. This completes the proof.

 

1 Eli Maor, To infinity and Beyond: A Cultural History of the Infinite, (Boston: Birkhauser Mathematics, 1991) 10.

2 Robert Russell, “The God Who Infinitely Transcends Infinity: Insights from Cosmology and Mathematics,” (Minneapolis: Fortress Press, 2008): 56.

3 Joseph Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, (Princeton: Princeton University Press, 1990): 144.

4 Anne Newstead, “Cantor on Infinity in Nature, Number, and the Divine Mind,” American Catholic Philosophical Quarterly 83:4 (2009): 535.

5 Dauben, 245.

6 Wolfhart Pannenberg, Jesus – God and Man, (Westminster: John Knox Press, 1982): 115.

7 Kathyrn Tanner, Jesus Humanity, and the Trinity: A Brief Systematic Theology, (Minneapolis: Fortress Press, 2001): 9.

8 Michael Heller, Infinity: New Research Frontiers (Cambridge: Cambridge University Press, 2011): 279.

9 Sandra Valabregue-Perry, “The Concept of Infinity (Eyn-sof) and the Rise of Theosophical Kabbalah,” The Jewish Quarterly Review 102:3 (2012): 408.

10 Wilf Malcolm, “Thinking About God and Infinity, Can Mathematics Contribute?” Stimulus 18:2 (2010): 35.

11 Ibid.

Figure 1: The sum of the sequence

 

Cantor's Theorem sequence2

 

can easily be visualized in the square. If we add half of the square’s area to a quarter of the square’s area to an eighth of the square’s area, and so on, our infinite sequence of additions will approach a limit and manifest as a finite sum, 1. The logic behind such limits disproved Zeno’s paradox given in the opening paragraph, allowing you to actually finish this paper!

Drew Voigt ’14 is from Bloomington, Minnesota. He is studying mathematics with a concentration in Biomolecular science, and he is a member of the men’s soccer team.

 

Thumbnail image illustrates Zeno’s Paradox.

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