The mighty Achilles and a tortoise agree to a race of 100 meters. To make things fair, Achilles gives the tortoise a 50-meter head start. When the starting gun fires, both Achilles and the tortoise head to the finish line. One might think Achilles wins easily—but don’t draw such a hasty conclusion. Before he can pass the tortoise, Achilles must first cover the 50-meter separation. However, in that time, the tortoise will have moved some distance down the track. Achilles must now cover this distance to pass the tortoise, but once again, the tortoise will move farther toward the finish line. And Achilles must cover this distance before passing. . . . So goes Zeno’s paradox and my first recollection of encountering the idea of infinity.

Infinity is more than a mathematical mystery to be solved. It can help us understand the Christian view of the nature of God.

**Covering Infinite “Half Distances”**

In high school, I read Douglas Hofstadter’s *Gödel, Escher, Bach: An Eternal Golden Braid** *where he introduces infinity using Zeno’s paradox. Greek philosopher Zeno (fifth century BC) used the paradox to argue that motion was an illusion that may become more apparent when recasting the paradox. For either Achilles or the tortoise to cover the distance to the finish line, they must first cover half the distance, but that requires moving half of that distance, which requires covering half of *that* distance . . . and so on. Each of these “half distances” represents a step that Achilles or the tortoise must make to move, but there are an infinite number of “half distances” they must make to move at all. Since an infinite number of steps cannot happen, neither Achilles nor the tortoise ever make it off the start line.

Hofstadter’s discussion of infinity stuck in my mind. In fact, during a math class in my senior year, I chose the topic of infinity for my research project. During my research, I quickly discovered that the topic of infinity is far more complex than something being too large to count and, more importantly, that infinite quantities behave far differently than finite ones.

Currently, my interest in infinity relates to philosophical topics (can actual infinities exist?), mathematical issues (are all infinities the same?), and theological issues (is God an actual infinite?) as well as how most scientists approach infinities. Before addressing these topics and Zeno’s paradox in more detail, I’ll provide some background on infinity.

**Infinity: A Basic Definition**

A quick dictionary definition probably matches how most people would define infinite—limitless or endless in space, extent, or size. The words unbounded, without limit, countless, or never-ending serve as synonyms for infinite. When using these words, we must always remember an important distinction. Infinite or its synonyms don’t simply mean that it is beyond our ability to actually count, but rather that it is *impossible* to count all the members. This distinction highlights an important difference between two types of numbers.

People routinely use two types of numbers, cardinal and ordinal. A cardinal number describes the number of elements in a set. You can have 5 apples in your bag, 2 cars in your garage, 352 action figures on your wall, or 1 huge fish that got away. Cardinal numbers provide a way to characterize the size of a set. Ordinal numbers give the ordering of a member. Wednesday is the third day of the work week, I am the second son of my parents, and either Connecticut or North Dakota will be the 49th state I will have visited. For finite sets, cardinal numbers and ordinal numbers have little distinction. A bag of 5 apples will always have a fifth apple, but not a sixth apple. Stated another way, size and order are largely interchangeable for finite sets.

For infinite sets, however, size and order become rather distinct. Whereas the number of elements determines the cardinality (or size) and the largest ordinal of a finite set, an infinite set has no largest ordinal or largest number of elements. Yet, infinite sets still have a distinct size. The set of natural numbers (1, 2, 3, 4, etc., and sometimes including zero) is the “smallest” infinite set. It’s small in the sense that it consists of positive integers only and not all the decimal values in between those whole numbers. As should be evident, this infinite set has no largest value, nor a last element. However, we can talk about the cardinality of this set which mathematicians define as À0 (pronounced “aleph null”). Similarly, mathematicians define w (the lowercase omega symbol) as the set of all finite ordinals. Thus, the first ordinal beyond the natural numbers is w+1.

**Different Infinities**

Finite sets easily have different cardinalities, but it might appear that all infinities have the same cardinality. However, that is not the case. For example, consider the natural numbers compared to the real numbers. The set of natural numbers includes all the positive integers and is an infinite set. However, the real numbers include all the positive (and negative) integers as well as all the decimal numbers in between the integers. One simple way to think about this recognizes that between each integer, there exists an infinite number of real numbers. For example, there’s an infinite number of decimals between 1 and 2. While the formal description includes more precise language, the net result is that the cardinality of the real numbers is larger than the cardinality of the natural numbers. To use the proper symbols, the cardinality of the natural numbers is À0, where the cardinality of the real numbers is À1. Stated another way, the size of the set of real numbers is larger than the size of the set of natural numbers. Mathematically speaking, there is no “largest” infinite set, so for every natural number *a*, there is a corresponding set with cardinality Àa and ordinal wa.

Here’s one last point on differing infinities. Any infinite set with cardinality, À0, is called *countably* infinite since every member of the set can be put in a one-to-one correspondence with the natural numbers. Any infinite set that cannot be put in such a one-to-one correspondence is called *uncountably* infinite, and its level of uncountability is closely related to its cardinal number. One might be interested in an example but the only potentially useful examples are mathematical because our finiteness means we don’t truly experience infinite sets. With that caveat, the set of natural numbers has cardinality À0. The set of real numbers has cardinality À1. (Technically, the real numbers have the cardinality of the continuum, but there is reason to think that this is the same as À1.) The set of all sets of natural numbers ({0},{1},{2}…{0,1},{0,2},{0,3}…{1,2},{1,3}…{0,1,2}…) has cardinality À2.

**Infinite and Finite Differences**

These descriptions emphasize the language similarities used when talking about infinite and finite sets, but you may have discerned that the two types of sets behave differently. For finite sets, we intuitively understand how to add, subtract, multiply, and divide sets and we have a well-developed “set theory” dealing with all these operations. Similarly, with infinite sets, we also have a well-developed set theory, but those familiar operations diverge from our intuition.

For example, when adding to nonzero finite sets (like a basket of 5 apples to a basket of 3 apples), the resulting set is always larger. However, when adding any set (whether infinite or finite) to an infinite set (noted by the symbol, ¥), the resulting set is the same size as the largest set. So, where 5 + 3 = 8, ¥ + 5 = ¥, and ¥ + ¥ = ¥. Even more bizarre, subtraction does not exist as a proper function when working with infinite sets because the results are indeterminate. While subtracting a finite value from an infinite set gives an infinite set, subtracting two infinite sets can be either finite or infinite. Similarly, multiplication is well-defined for infinite sets, but division is not. (I will discuss this more clearly in a later blog that considers Hilbert’s hotel).

**Infinities and God**

When I first realized this unusual feature of infinite arithmetic, my thoughts went to the well-loved hymn, “Amazing Grace.” The first verse describes how God’s endless grace saved the wretch, but it’s a later verse that caught my attention:

When we’ve been there ten thousand years

Bright shining as the sun

We’ve no less days to sing God’s praise

Than when we first begun

I always wondered how to reconcile the fact that 10,000 years have passed, but we still have the same number of days left to sing God’s praise. Understanding how the addition of infinite sets works resolved my difficulty. If I have an infinite (or endless) number of days to sing God’s praise, then adding 10,000 years (or 3,650,000 days) to that infinite number does *not* change the number of days.

Any finite thing will always remain finite and any infinite thing will forever remain infinite. Nothing ever grows from finite to become infinite. Nothing infinite can ever be divided into a finite number of sets, each with a finite number of elements. Any infinite set will always include either an infinite subset or an infinite number of subsets. This set of information about finites and infinites provides a basis for addressing Zeno’s paradox, discussing whether actual infinities exist and whether God might be infinite, and a whole list of other fascinating topics related to infinities. I’ll develop these topics in future posts, but for now a basic understanding of infinity helps us to consider what it means that the God of the Bible has no beginning or end and there was never a time that he didn’t exist.

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