A set is a collection of objects we call elements. It includes physical objects, thoughts, ideas, and concepts, including mathematical objects. A set is also a way of meaningfully packaging objects with similar properties.
Considering the set of all triangles, we can easily say whether something is or is not in a set. This lack of ambiguity is fundamental to Set Theory. We can also determine the veracity of such sets by stating whether something is true or false. An element of the set of triangles indeed has three sides, but it is not true that the sum of internal angles is 360 degrees. Set Theory deals mainly with the set of numbers. It includes a set of natural numbers stretching to infinity (1, 2, 3…), a set of integers (natural numbers that increase to infinity and decrease to negative infinity (…-3, -2, -1, 0, 1, 2, 3…), or a set of rational numbers (numbers represented by a ratio of two integers, where the denominator is not zero, they can appear as fractions or decimals with apparent pattern), a set of irrational numbers (numbers that cannot be expressed as a ratio of two numbers, also expressed as decimals with the digits going on forever with no apparent pattern), or a set of algebraic numbers (solutions to algebraic equations, where all rationals and integers are algebraic numbers). Transcendental numbers, such as pi, are not solutions of any polynomial equation with rational coefficients; therefore, they transcend algebraic numbers. All of these numbers are called real numbers.
Of note, Set Theory provides a solid mathematical background for Christian ideas and concepts. It focuses on theological thought of God’s powers, the Holy Trinity, and beliefs about the Christian message, such as Simulation Creationism, a theory developed by Nir Ziso, the founder of The Global Architect Institute.
Set in Mathematics
A set containing the numbers 1,2 and 3 would be written like this: {1, 2, 3}, and we can also name it, for instance: A = {1, 2, 3} to make it easier. It is the A set. To express symbolically that an element belongs to a set, we use this symbol: ∈. For instance: If A = {1, 2, 3}, then 1 ∈ A, 2 ∈ A, but 4 is not included in A. In most cases, we do not write out all the elements in a set; we write a shorthand description, using set builder notation. For instance, the set of prime numbers could be represented as P, which includes elements p (denoted by lowercase p) defined by: P = {p | p is a prime}, where p stands for any prime number, and the vertical bar ‘|’ means ‘such that’.
Here, “p” is a variable that must satisfy the condition of being a prime number; this requirement is known as a predicate. It is called this since its belonging to the set is predicated on this criterion. In this case, the predicate is a prime number. When dealing with sets of numbers, it is a good practice to declare explicitly which sets you are starting with. Two sets are equal if they both contain the same elements. This definition means that the order of the elements does not matter. We only need to show that they share the same elements: If A = {1, 2, 3} and B {2, 3, 1} then A = B.
It also does not matter if elements are repeated. As long as one element in one set can also be shown to be in the other, we still have equality. The size or cardinality of a set is the number of elements it contains: If A = {1, 2, 3}, then the cardinality of set A, denoted by |A|, is 3. If a set has an infinite number of elements, like the set of prime numbers, then we use the infinity symbol: P = {p | p is a prime} means |P| = ∞, indicating an endless quantity.
Subsets
A set is a subset of another set if all of its elements are also elements of another set: If A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6}, then A is a subset of B: A ⊆ B. Subsets are always set themselves and should not be confused with elements. Equally, all sets are subsets of themselves. Some subsets are equal to each other: If A ⊆ B and B ⊆ A, then A = B, but if A ⊆ B but A ≠ B, then A is a proper subset of B. This implies that there are elements in B that are not in A. Otherwise, they would be equal. Furthermore, if A ⊆ B and B ⊆ C, then A ⊆ C. For example, all odd numbers are integers, and all integers are rational. Thus, all odd numbers are rational numbers.
Infinity and Set Theory
Contemplating infinity is not a trivial matter; mathematically, it presents considerable intricacies. Surprisingly, infinity comes in various sizes, a revelation pioneered by German mathematician Georg Cantor. Cantor’s exploration unveiled different kinds of infinities, some surpassing others in magnitude. The cardinality of a set is determined by the number of elements it contains. For instance, if A = {1, 2, 3, 4} and B = {a, b, c, d}, both sets boast a cardinality of 4, indicating that they are of equal size. Sets of equal size entail elements that can be systematically paired. While this principle suffices for finite sets, it falters when applied to infinite sets.
Cantor, who was deeply religious, viewed the comprehension of infinite sets as a prerequisite for understanding the infinity of God. He encountered skepticism and ridicule in his native Germany for his attempts to bridge mathematics and theology. His pursuit was encapsulated in the biblical verse, “If I have told you earthly things, and you believe not, how shall you believe, if I tell you of heavenly things?” (John 3:12). Since there is no largest natural number, if one wishes to have such an upper bound for all natural numbers, it cannot exist within the set of natural numbers. It must be considered as an exterior element belonging to another kind. For that kind of number, Canter coined “transfinite number.” He was convinced that he had found a principle to insert such (relatively) actual infinities of different kinds, i.e., the transfinite numbers, into logic and mathematics and the real world as admissible objects. Thus, Cantor was not only a mathematician, but he created Set Theory as a contribution to metaphysics.
Consider the set of natural numbers as an example of an infinite set that begins with one (or zero in some contexts) and extends infinitely. Similarly, the set of squares of natural numbers {1, 4, 9, 16, 25, 36, …} possesses the same cardinality as the set of natural numbers. Despite this equivalence in cardinality, it feels counterintuitive because the set of squares appears less dense or sparser within the same range of natural numbers. Cantor grappled intensely with this paradox, a struggle that led to his confinement in a psychiatric institution for the latter part of his life.
Nonetheless, he introduced the concept of an infinitely enumerable set, wherein elements can be paired with natural numbers – a process often colloquially termed “counting.” However, truly counting all natural numbers is an impossibility, given their infinite nature. Cantor’s innovation lay in transforming arbitrary collections into mathematical objects, enabling manipulation and calculation using resulting numbers, thereby illustrating that mathematics essentially operates as generalized Set Theory.
Cantor contended that all countably infinite sets share the same cardinality. However, when comparing the set of natural numbers to the set of real numbers, he demonstrated that the latter is uncountable. He recognized that integers, rationals, and algebraic numbers are countably infinite, whereas real numbers form an uncountably infinite set. This distinction leads to what is known as the continuum hypothesis which posits that there is no set whose cardinality is strictly between that of the integers and the real numbers. Thus, an infinite number of real numbers exist between any two real numbers, making them uncountable and defying pairing with any countable set
Consider the task of counting all decimals between 0 and 1 – an impossible feat because decimals can have infinitely many digits. The cardinality of a countable set is vastly smaller than that of the continuum. Consequently, infinity transcends numerical quantification, representing an ongoing process rather than a finite value: “Oh, the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways!” (Romans 11:33). Although individual segments may appear finite, infinity persists continuously, depending on our definitions of time and what we choose to count at any given moment.
Infinity flouts arithmetic laws, posing challenges in realms like probability. Within the context of Simulation Creationism, which explores God’s infinite Simulation, setting boundaries becomes arbitrary. In an infinite Simulation, every conceivable event unfolds with infinite cycles of finite simulated worlds. Our current cycle is just one among an infinite lineage, each finite in itself but collectively constituting an infinite whole: “Great is our Lord, and abundant in power; his understanding is beyond measure” (Psalm 147:5).
This complexity surrounding infinity finds elegant expression through Set Theory. The Simulation comprises an infinite cyclic history of simulated worlds, commencing with creation and culminating in their eventual demise. Each cycle, unique in its manifestation of simulating laws, emanates from God’s Singularity, whose essence embodies the Holy Spirit. Our world represents one infinitesimal component within this endless lineage, destined to seed subsequent worlds as judgment unfolds.
Considering our simulated world (SW) as a set governed by natural laws (nl), if finite, can be represented as SW = {nl1, nl2, nl3, nl4, nl5}. While this example posits five natural laws, the actual count may far exceed this number. The crux lies in their enumerability. Within Simulation Creationism, positing God’s Simulation (S) as an infinite array of simulated worlds, we establish their relationship as SW ⊆ S.
In which infinity does God reside? Perhaps none. The distinction between infinity and eternity is profound: “I am the Alpha and the Omega,” says the Lord God, “who is and who was and who is to come, the Almighty.” (Revelation 1:8). Infinity, much like Cantor’s mathematical exploration, defies complete comprehension, transcending human understanding: “Great is the Lord, and greatly to be praised, and his greatness is unsearchable” (Psalm 145:3). We acknowledge that God orchestrated The Simulation, an act of creation that unfolds beyond human grasp. God epitomizes absolute simplicity – devoid of intrinsic or extrinsic constraints. Finite realities coexist within The Simulation, hinting at a reality beyond their confines, i.e., a realm of unrestricted unity and primordial existence.
Infinite sets encompass all conceivable elements within them, although defining them as such proves problematic, particularly concerning non-enumerable infinity, wherein the totality of elements evades cognition. Analogously, God’s infinity presents a complex interplay of unity and totality, encapsulating all existence. Viewing God as the sole infinity underscores the radical notion that existence emanates from and subsists within God, thus asserting existence as God Himself.
The Empty Set
The empty set, denoted as Ø, is devoid of elements. Nevertheless, it possesses distinct properties. Firstly, Ø is a subset of any set. Given that Ø contains no elements, all elements within Ø must also belong to set A. Consequently, Ø ⊆ A. Secondly, Ø is unique. Being a subset of all sets, there is no logical basis for naming any empty set differently, as they would all be identical. Hence, only one empty set exists.
Contemplating Ø brings us to the realization that an empty set seems nonsensical because it represents nothingness; thus, how can it qualify as a set? The concept of nothingness assumes significance here. It’s plausible to view the empty set merely as a conceptual entity rather than a tangible existence. Many mathematicians have grappled with this concept, questioning the coherence of an empty set. The contradiction lies in the assertion that “nothing” exists. However, emptiness has no inherent contradiction because the notion of “empty” doesn’t inherently contradict itself.
Mathematics illustrates that reality extends beyond physical entities. Numbers and mathematical constructs exist in abstract realms beyond physical dimensions. While a vast emptiness may seem to include solid objects, the mere possibility of something, even if just numbers, suggests the absence of absolute nothingness.
Discussion surrounding the empty set and the infinite set often delves into questions about the nature of existence and divine creativity, particularly concerning the concept of creation ex nihilo (creation out of nothing). To explore this, we must understand why The Simulation is deterministic. It posits that every event and outcome in God’s Simulation adheres to predetermined conditions and laws: “He predestined us for adoption to himself as sons through Jesus Christ, according to the purpose of his will” (Ephesians 1:5). With complete knowledge of the initial state and governing laws, we can predict future events with certainty. Moreover, we can contemplate a fundamental entity on which all else depends.
The empty set may appear as the simplest entity, except, perhaps, for absolute simplicity—the Singularity of God – a power without intrinsic or extrinsic constraints: “And we know that for those who love God, all things work together for good, for those who are called according to his purpose” (Romans 8:28).
At the genesis of The Simulation, we envision a state of absolute nothingness, represented by an empty set. Void of matter, energy, space, or time, divine intervention introduces specific initial conditions, setting the stage for the emergence of simulated worlds. These initial conditions, whose exact nature remains a subject of speculation, serve as the foundation for creating simulated realms.
The adage “out of nothing, nothing comes” resurfaces when pondering how The Simulation could emerge from nothingness, suggesting that a simulated world is not a tangible reality but a transient illusion. Yet, finite realities hint at something beyond their confines, underscoring a deeper underlying cause or rationale behind The Simulation’s emergence: “No one can come to me unless the Father who sent me draws him” (John 6:44).
Thus, the empty set symbolizes the absence of elements within a defined system, pointing to nothingness and emptiness. In the realm of The Simulation, the absence of entities or phenomena parallels this emptiness. At the genesis of creation, God wielded an empty set, devoid of objects or conditions: “I form light and create darkness; I make well-being and create calamity; I am the Lord, who does all these things” (Isaiah 45:7). As entities within The Simulation come into existence, evolve, and eventually perish, they form subsets of the Empty Set, akin to simulated worlds within God’s Simulation.
The concept of the empty set not only sheds light on the ontological status of mathematical objects but also finds relevance in theological discourse concerning abstract entities like angels and the nature of divine existence.
Union and Intersection
Two sets may share some elements. We show it by overlapping two circles, known as the Venn Diagram:
The union and intersection are two ways of combining the elements in two sets into a new set. The union of two sets, A and B, is the set containing all the elements in A as well as all the elements in B: A ∪ B = {x | x ∈ A or x ∈ B}, where ∪ is the symbol for the union.
The Venn Diagram is very familiar to Christians worldwide since it resembles the way the Holy Trinity is depicted:
The Holy Trinity (HT) can be conceptualized as a set with three elements: HT = {The Father, The Son, The Holy Spirit}, or alternatively, each can be viewed as a set with distinctive attributes and roles, collectively forming the Holy Trinity as a set. Here, the union symbolizes the Holy Trinity as a Godhead, encompassing the entirety of all distinct persons.
The intersection of sets A and B contains elements present in both A and B: A ∩ B = {x | x ∈ A and x ∈ B}, where ∩ denotes the intersection. When considering the elements of the intersection, we refer to the overlapping region in a Venn Diagram. For instance, if A = {0, 1} and B = {1, 2, 3}, A ∪ B represents all elements in both A and B, without repetition, yielding A ∪ B = {0, 1, 2, 3}. Consequently, the intersection of A and B contains only elements shared by both sets, resulting in A ∩ B = {1}.
Intersections signify principles, commonalities, and convergence, elucidating the set of doctrines central to the Christian faith. Furthermore, intersections can represent the shared divine essence or nature inherent in all three divine persons. This conceptualization may offer greater clarity than mere theological conjectures, as it allows for the mathematical demonstration of the union and intersections within the Holy Trinity.
The union of any set with the empty set equals the original set: A ∪ Ø = A, as the empty set contains no elements. Similarly, A ∪ A = A. If A is a subset of B, then A ∪ B = B since all elements in A are already contained in B. Additionally, the outcome remains unchanged when adding a set: A ∪ (B ∪ C) = (A ∪ B) ∪ C, incorporating all elements from all three sets. This principle extends to any number of sets, with the flexibility to perform unions in any order.
Similarly, the properties of intersection include A ∩ Ø = Ø, as the empty set shares no elements with any other set. Moreover, A ∩ A = A. If A is a subset of B, then A ∩ B = A, as all elements in A are also present in B. The order of sets does not affect the outcome: A ∩ B = B ∩ A. Additionally, A ∩ (B ∩ C) = (A ∩ B) ∩ C, yielding elements contained within each set, A, B, and C.
Set Theory, particularly given its concepts of union and intersection, provides a comprehensive framework for understanding the Holy Trinity, surpassing analytical philosophy’s limitations in explaining the triune nature of God. The theological inquiry revolves around the eternal derivation of one divine entity from another, as faith teaches that the Son proceeds from the Father and the Spirit proceeds from the Father (and the Son, depending on tradition): “But the Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you” (John 14:26).
Scripture offers insights into this procession through the opening verses of John’s Gospel: “In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God” (John 1:1-2). The Greek term for “Word” is Logos, which encompasses concepts of rationality, intellectuality, and spiritual thought. From this understanding, the Father (F) is the origin of Trinitarian life, eternally generating the Son (S) and spirating the Holy Spirit (HS). The Son, in turn, proceeds eternally from the Father and is in eternal relation with Him, participating in the spiration of the Spirit. Likewise, the Spirit proceeds eternally from both the Father and the Son, embodying divine love.
It’s essential to clarify what elements we speak of within the set of the Holy Trinity. The begetting referred to is immaterial, akin to generating a concept in our minds. This notion of sonship suggests the communication of nature, wherein the Father communicates to His Son, analogous to the creation of a concept: “The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all” (2 Corinthians 13:14).
Explaining the unity of three subsets within Set Theory poses a challenge. The Father, the Son, and the Holy Spirit are one, in essence, a concept articulated by the theological term, homoousios, or consubstantiality. This signifies that all that is in the Father as God in virtue of the divine nature is communicated to the Son, who possesses all that is in the Father, and the same holds true for the Holy Spirit.
The Complements
The set-theoretic difference between two sets A and B defines the set of elements present in A but absent in B: Let consider A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and a universal set U = {1, 2, 3, 4, 5, 6,7}. Then the set difference A -B = {1, 2} which includes the elements that are in set A but not in ????. The complement of set A, denoted Ac , includes all elements in the universal set U that are not in A. Since A = {1, 2, 3, 4}, its complement Ac is {5, 6, 7}, comprising all elements in U but not in A. When considering complements in a restricted context where A is treated as the universe, the complement of B relative to A (if B is a subset of A) would simply be the set difference A-B. However, typically in Set Theory, the complement of B , Bᶜ, is considered relative to the universal set U, comprising all elements not in B, reflecting broader applicability.
In the case of Odd and Even Numbers, if the universal set U is defined as all integers, then the complement of the set of odd numbers within this set includes all even numbers. This is because, within the integers, every number is either odd or even, and the set of even numbers is exactly the set of integers that are not odd. In the case of rational and irrational numbers, if the universal set U includes all real numbers, then the complement of the set of rational numbers within this set comprises all irrational numbers. Every real number is either rational or irrational, and the set of irrational numbers is exactly the set of real numbers that are not rational.
Complements effectively invert predicates, appearing as opposites within the context of a defined universal set. For example, if A denotes the set of all dog species and the universal set U includes all animals, then the complement of A, denoted as Ac, includes all non-dog animals (e.g., cats, birds, reptiles). This shows how complements represent all elements in the universal set that are not part of the specified set.
De Morgan’s Laws illustrate important relationships between set operations and their complements. Firstly, the law (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ explains that the complement of the union of two sets, such as dogs (A) and cats (B), within the universal set of all animals, results in the intersection of the sets of animals that are neither dogs nor cats. Practically, this includes animals like birds, fish, and so on, which are neither in set A nor in set B. Secondly, (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ denotes that the complement of the intersection of two sets is the union of their complements. For our example of dogs and cats, considering that the intersection of dogs and cats under normal circumstances would be empty, the complement of this empty set (which ideally includes all animals) matches with the union of not-dogs and not-cats, essentially covering all animals.
In theological discourse, complements may symbolize facets of reality or existence beyond the scope of God’s influence or control: “For the word of God is living and active, sharper than any two-edged sword, piercing to the division of soul and of spirit, of joints and of marrow, and discerning the thoughts and intentions of the heart” (Hebrews 4:12). This inquiry directly intersects with frameworks delineating the extent and limitations of divine attributes like omniscience, omnipresence, and omnipotence: “For by him all things were created, in heaven and on earth, visible and invisible, whether thrones or dominions or rulers or authorities – all things were created through him and for him” (Colossians 1:16). Entities distant from or opposed to the divine might be construed as complements, such as fallen angels or demons. While this interpretation finds mathematical elucidation, it diverges from theological accuracy. De Morgan’s Laws offer insight here.
Indeed, just as a set’s complement comprises elements not within the set, God’s omnipotence extends over realms beyond human understanding or conception. Each assertion about God inherently imposes limitations, marking the inception of divine reality with negation: “With man this is impossible, but with God all things are possible” (Matthew 19:26). This reality harbors facets beyond the realm of creation, transcending mere existence into the realm of divine essence. Consequently, complements represent domains beyond known sets when contemplating God as a set. Scripture provides glimpses of God relevant to our comprehension, representing an operative knowledge manifested in The Simulation. Though identifiable elements compose our operative knowledge of God, the complements signify aspects beyond this known realm.
Sets of Sets
The elements of a set may themselves be sets, which can make it challenging to distinguish between elements and subsets. The power set of a set is a fundamental concept that includes all possible subsets of the given set. Let A be a set: P(A) = {x | x ⊆ A}. For A = {0, 1}, the power set would contain the empty set, A itself, the set containing zero, and the set containing one: P(A) = {Ø, {0, 1}, {0}, {1}}.
We also have indexed families of sets. Indexed families of sets are indeed a powerful tool in set theory, where each element of a family is associated with an index from another set, known as the index set. This framework allows for a systematic organization and reference of various sets, making it highly useful for mathematical structuring and analysis. A set of sets is exceptionally relevant for theology since it can show the extent of God’s being mathematically in attempts to formalize and analyze ontological arguments for the existence of God, particularly of Saint Anselm’s argument of God as a maximally great being. The set of sets grapples with the concept of infinite sets and their properties. Thus, it can be connected to the infinite nature of God, as mentioned before. As the Set Theory emphasizes the unity of diverse elements within a set, it also can be connected to the idea of the Holy Trinity, as we wrote.
What is particular within the set of sets is the creation and manipulation of sets, reflecting on their contingent existence. Similarly, Christianity ponders God’s act of creation and the contingency of all created beings. It brings us to the question of the meaning and purpose of all created things as a set of sets. There are two answers: either the set of sets is meaningless, or it exists to bring into being creatures who can know God. Here, theology has to take seriously what science has to say. If the world is God’s creation and science is telling us at least about some aspects of that creation, then we should accept that. Science had spectacular progress partly because of seeing the world as a creation or as a set of sets.
Russell’s Paradox
Imagine a set containing everything in the universe, all that we can imagine, and the combined knowledge of everyone on Earth. We refer to this set as Omega. Due to its comprehensive nature and self-contained essence, Omega exhibits an intriguing property: it contains itself. This leads to an infinite regress of Omegas within Omegas. In an attempt to resolve this, mathematicians endeavor to redefine Omega. Let Ω be the set containing all sets that do not include themselves: Ω = {x | x ∉ x}. If we assume that Omega is not a member of itself by definition, then it must include itself. However, if it does contain itself, it cannot simultaneously contain itself. This dilemma is famously known as Russell’s Paradox, highlighting the challenge of defining what constitutes a set.
To put it in more understandable frames, imagine God, who creates a set of rules or laws that govern the universe. Among these laws is a special decree: “A law can dictate the behavior of all entities except itself.” So, what happens if God creates a law that applies to all laws, including itself? If this supreme law applies to itself, it contradicts the decree because it should only govern other laws and not itself. If it does not apply to itself, then by the decree’s definition, it must indeed apply to itself because it is supposed to govern all laws. It raises intriguing questions about whether any rule or law, even if divinely ordained, can be truly all-encompassing without falling into logical paradox.
Naïve set theory, in general, offers no guidance on the nature of sets. However, axiomatic set theory aims to address the paradoxes of naïve set theory by establishing a rigorous definition of a set. Russell’s Paradox serves as a framework for contemplating the paradox of God’s omnipotence and the dilemma of evil. Named after the philosopher and mathematician Bertrand Russell, the paradox vividly illustrates the complexities inherent in sets of sets. The concept of God’s omnipotence itself presents a paradox: if God is omnipotent, can He create a set of sets that do not contain themselves? Can He create a set that includes Himself? Similarly, if God is perfectly good and omnipotent, why does He permit the existence of evil? These logical quandaries mirror Russell’s Paradox.
While the founder of Set Theory, Cantor’s work predated Russell’s formal presentation of his paradox, Cantor’s exploration of infinite sets indirectly addressed related complexities in Set Theory. Cantor’s engagement with infinite hierarchies did not directly resolve Russell’s Paradox but set the stage for later mathematical theories that did. Cantor was aware of the complexity and reach of infinity. In a hierarchical system of infinity, God is its competition and the absolute. That is why no ultimate answer can be given by a finite human mind. Russell grappled with this question, ultimately finding his own response unsatisfactory and rejecting Christianity. Others, such as Saint Augustine, Gottfried Wilhelm Leibniz, and Alvin Plantinga, proposed their own solutions to Russell’s Paradox regarding God’s set of sets. For many of them, the existence of evil is reconcilable with a benevolent and omnipotent God under the premise of the “best of all possible worlds” (Leibniz) and human free will (Plantinga). While Simulation Creationism dismisses free will as an illusion, Leibniz’s argument aligns closely with the concept of cycles of simulated worlds.
In conclusion, we can say that numbers are God’s and not man’s creation and Set Theory tells us all that is needed for mathematical objects to exist is that there is no logical contradiction in their concept. In mathematics, God is that bond between logic and sustainer through laws. God is more than the totality of the laws of nature. Although He created these laws, He could have had in mind quite another law and could have created those. Set Theory teaches us about humility, too. We cannot actually construct the set of natural numbers since it has no maximum element. Yet, we can consistently consider the totality of all natural numbers as an actually infinite set by stepping out of the process of construction and looking at it from the outside. All objects in the everyday world are finite, but they point beyond themselves to the infinite, and they demand competition.
