A rather large quantity. And a concept we may have encountered, and possibly struggled with, in an occasional math course.
But why bother talking about it? Infinity hardly seems relevant to the practical matters of our normal day, or even our abnormal days.
Well, possibly, but infinity does pose a high intellectual intrigue. So a few minutes with infinity should provide a strong mental challenge and a diversion from the tribulations of our normal day. At least enough to warrant a few minutes consideration.
And dismissing infinity as irrelevant misses at least one relevant aspect of the concept.
Believer or not, searcher for faith or not, detester of the concept or not, God, whether as an object of faith, or an ultimate question, or an irrational delusion, God looms as unavoidable. God either serves as guidance for our life, or poses questions bedeviling our minds, or lingers as an outmoded concept born of ancient history in pre-scientific times.
And a major tenet in most theologies, and in philosophy in general, points fundamentally to an infinite God – infinite in existence, infinite in knowledge, infinite in power, infinite in perfection.
So as a passing, but intriguing, diversion, and as an attribute of a spiritual figure deeply imbedded in our culture and our psyche, infinity does provide a subject worth a few minutes of our time.
So let’s begin.
How Big is Infinity?
Strange question, right. Infinity stands as the biggest quantity possible.
But let’s drill down a bit. We should apply some rigor to examining infinity’s size.
Consider integers, the numbers one, two, three and up, and also minus one, minus two, minus three and down. We can divide integers into odd and even. Common knowledge.
But let’s consider a not-so-obvious question, a question you might have encountered. Which is larger, all integers, or just even integers? The quick answer would say the group of all integers exceeds the group of even integers. We can see two integers for every even integer.
If we have studied this question previously, however, we know that answer is wrong.
Neither infinity is larger; the infinity of all integers equals the infinity of just even integers. We can demonstrate this by a matching. Specifically, two groups rank equal in size if we can match each member of one group with a member of the other group, one-to-one, with no members left over unmatched in either group.
Let’s attempt a matching here. For simplicity, we will take just positive integers and positive even integers. To start the match, take one from the set of all positive integers and match that with two from the set of all positive even integers, take two from the set of all positive integers and match that with four from the set of even positive integers, and so on.
At first reaction, we might intuit that this matching would exhaust the even integers first, with members of the set of all integers remaining, unmatched. But that reflexive thought stems from our overwhelming experience of finite, bounded sets. In a one-to-one matching of the rice kernels in a two pound bag with those of a one pound bag, both finite sets, we well expect the one pound bag to run out of rice kernels before the two pound bag.
But infinity operates differently. An infinite set never runs out. Thus even though a one-to-one matching of all integers verses even integers runs up the even integers side quicker, the even integers never run out. Infinity presents us features counter-intuitive to our daily experience filled with finite sets.
And so with fractions. The infinite set of all fractions does not exceed the infinite set of all integers. This really throws a counter-intuitive curve, since we can not readily devise a one-to-one matching. Would not the fractions between zero and one loom so numerous that no matching can be created? But that would be wrong.
To see why, let me suggest a web search, on the following phrase, “bijection rational numbers natural numbers.” Rational numbers, i.e. ratios, are the fractions, and natural numbers are the integers. The matching proceeds with 45 degree marches down and back up a grid of the rational, i.e. fractional, numbers.
A Bigger Infinity
We might now conclude that infinity stands undefeated, and that no set, however constructed, would escape the rigor of one-to-one matching.
If you have studied this question before, you know that does not hold. The set of real numbers, i.e. numbers with digits to the right of a decimal point, exceeds the set of all integers.
Wait though. If we exercise enough cleverness, might we find a matching of real numbers with integers?
No. A proof, well examined, exists that we can not so find a match. We can thank the mathematician Georg Cantor and mathematicians following him for the rigorous development of how infinity works.
Now the proof. Take the first integer, one, and match that with the real number 0.0111111… where the digits of one extend rightward forever. That falls well within the properties of real numbers, that no limit exists to the number of digits in the decimal portion.
Take the second integer, two, and match that with real number, 0.1011111… where the digit one repeats to the right forever. Take three and match that with 0.1101111… again with the digit one repeating to the right forever. Proceed similarly with each integer. In this way, by placing a zero in the slot corresponding to the right decimal position equal to the integer being matched, we match every integer with a unique real number.
Now we can construct a real number not in the matching, via a process called diagonalization.
Start with the integer one, and pick a digit not in the first position to the right of the decimal of the matched real number. Let’s pick 2, as that differs from the zero in the first right position in the real number we just matched with one.
The first position of our (potentially) unmatched real number contains a 2 just to the right of the decimal.
Now consider the integer two, and pick a digit not in the second right position of the matched real number. Let’s pick 3. Put that digit in the second position right of decimal of the real number we look to construct. That real number now starts with.23 We continue the sequence. We march through the integers, and in the position with the zero in the matched real number, we put alternately 2 and 3 in the corresponding position of the real number that we look to be unmatched.
We proceed by this process, which marches diagonally down the positions of the matched real numbers. In this example, we create the real number 0.2323232… with 2 and 3 alternating forever. That by construction does not lie in the real numbers we matched to integers, since our constructed real number 0.23232.. contains a digit not present in any matched real number.
Of importance, this diagonalization process works regardless of any matching we attempt. We can always construct a real number by sequentially picking a digit not in each real number of the attempted match.
Why in rough terms does this work? Real numbers, in an informal sense, present a double challenge. Real numbers first extend upward in size infinitely, to larger and larger quantities, and extend downward infinitely, splitting numbers to smaller and smaller distinctions, infinitely. This double extension allows real numbers to outrun the integers, and even fractions.
A Bigger Infinity
We have not finished with the sizes of infinity.
To explore these increasing sizes, we must introduce power sets. So far in this discussion, our sets have consisted of numbers. The set of integers comprised a set of all natural or counting numbers, the set of fractions comprised a set of all numbers resulting from the division of two integers, the set of complex numbers (not discussed here, but used as an example) comprise numbers containing the square root of negative one.
Sets can contain other things, of course. We can construct the set of cities that have won professional sports championships, or the set of individuals that have climbed Mount Everest. Sets can contain sets, for example the set of the two member sets that comprise an integer number and its square. This set equates to (1,1),(2,4),(3,9),… .
Sets can be subsets of sets. The set of cities that have won championships in four or more professional sports represents a subset of the those that have won championships in any one of the sports. The set of integers that are integer cubes (say 8 or 27 or 64) represents a subset of the set of all integers.
The Power Set is the set of all subsets of a set. In other words, take the members of a set, and then construct all the various unique combinations, of any length, of those members.
For example, for the set (1,2,3) eight subsets exist. One is the empty set, the set with nothing. (Yes a set containing nothing comprises a valid set.) The other subsets list out as follows: 1,2,3,1,2,(1,3},(2,3},(1,2,3}. The power set of the set (1,2,3) contains those eight members. Note (3,2) does not count as a subset, since (3,2) simply flips the members of the (2,3) subset. Rearranging set members does not count as unique for power sets.
Power sets grow rapidly in size. The power set of the first four integers contains 16 members; of the first five integers, 32 members; the first ten, 1,024 members. If so inclined, one could list out these subsets in say Excel. Don’t try that for one hundred integers. The spreadsheet would run a billion, billion, trillion cells, or ten to the power of thirty.
We can see the next step. Take the power set of the (infinite set) of integers. If the power set of the first 100 integers looms big, the power set of all integers must loom really big. How big? How many member reside in the power set of all integers?
An infinity greater then the infinity of the integers.
Let’s demonstrate by attempting to match the set of integers with its power set.
Match the integer one with a subset having all the integers except one. Match two with a subset having all the integers except two. Do the same for three. All integers now sit matched with a different subset, and, if we think about it, those subsets are infinite in size. How? We have specified that each matching set be all the integers except just one member, and an infinite set minus one member remains infinite.
So we have matched each integer with an infinite-sized subset element within the power set. What remains unmatched? Any subset of integers a finite size. Thus our matching shows the power set of integers greater in size than just the integers.
And On and On
Without demonstration, the power set of integers equals, in size, the number of real numbers. I say without demonstration, since the proof involves a fair bit of math.
But let’s move upward. If we postulated the power set of the set of integers, we can postulate the power set of real numbers. And yes just like the power set of integers contains more members than the set of integers itself, the power set of real numbers contains more members than the set of real numbers.
We can envision this through a rough consideration of number lines, just an image we can grasp. Take a number line of real numbers. That number line extends in both directions, and the points on the line represent the real numbers.
We can mark-off our normal three-dimensional world by taking three number lines and crossing them at right angles. These three crossed lines create axes that mark off the familiar height, width and depth of our daily experience.
But now cross not just three real number lines, but an infinite number of real number lines. We can not readily visualize more than three dimensions, much less infinitely many, but mathematically an infinite dimensional space stands as valid. This crossing gives us an infinite number of infinities. While not precise, our imaging an infinite number of infinitely extending real number lines provides a view of the power set of real numbers.
We can continue. We can take ever larger power sets, infinitely. Our mind may fail grasping this, but the math remains solid. For every infinite set we can create, we can create a larger one by taking that sets power set. No limit exists to how many ever larger infinities we can create.
Back to the Finite
But now let’s go the other way. Making the infinite finite.
Consider this famous paradox. If we give a turtle a head start, we appear to never be able to catch up. For when we get to where the turtle previously resided, the turtle has moved on. And when we arrive at that new turtle position, the turtle has moved further. The turtle will always arrive at a new position ahead of us, as we move to catch up to its previous position. And this goes on infinitely. You can’t catch up.
But, go try this in real life. Maybe not with a turtle, but say a toddler. We will assume, for most cases, you run faster than the toddler (if not consider an infant in crawling stage.) You catch up. No problem. Every time. Despite the toddler or infant moving ahead as you arrive at their last position, you catch up.
How do we resolve the paradox? How in real life do we catch up, when in descriptive form we always seem, infinitely, to be behind one step.
We do so by realizing that an infinite sequence can reach a finite limit.
So while with power sets we expanded the infinite to larger and larger sets, we will now take an infinitely long sequence and chop the sequence down to the finite.
Consider the time to catch up. Assume we move twice as fast as the turtle/toddler/infant. Give the pursued a two second head start. We need one second to reach that head start spot. The turtle/toddler/infant moves ahead in this one second, a distance that we can cover in one-half second. In that half second, the turtle/toddler/infant moves ahead a distance we can cover in one-quarter second.
Our total time to catch up, if we ever do, equals the sum of those fractional seconds, which decrease by a half for each segment of the race. As an equation, this infinite sum of fractions looks as follows:
Time = 1 + 1/2 + 1/4 + 1/8 +…
That sequence extends forever. How can we total this sequence, since it extends infinitely? We deploy a bit of cleverness. Multiply this sequence by one half on both sides. Some of you may likely have seen before. Multiplying by one-half gives the following.
½ * Time = ½ * (1 + 1/2+ 1/4+ 1/8 +… or
½ * Time = 1/2+ 1/4+ 1/8 + 1/16…
Not much help, at least not yet, as we no more know the sum of the this one-half equation than the original equation. But substitute the one-half equation back into the original equation. In the original equation, the string of fractions starting at ½ and going right, equals the string of fractions in the ½ * Time equation.
Substituting, we thus obtain:
Time = 1 + ½ * Time
Now subtract ½ * Time from both sides to get
½*Time = 1
Then multiplying both sides by 2 results in
Time (i.e. sum of infinite series) = 2
The time to catch up thus equals two seconds. While mathematically catching up involves an infinite sequence of increasingly smaller fractions, the infinite sequence of those fractions sums to a finite time, i.e. two seconds.
Is this just a special case? No, the sequence of reciprocal positive integer sums represents another infinite series summing to a finite number.
First, what is the sequence of reciprocal positive integer sums? Start with the sequence of positive integer sums. As this name implies, the sequence involves sums of integers, and as a sequence it involves summing increasing numbers of integers. So the sequence starts the first positive integer, one, and sums that to 1. The sequence then takes the first two positive integers, one and two, and sums those giving 3. The sequence then takes the first three positive integers, one, two and three, and sums those giving 6. Doing the additions, the next elements, after 1,2, and 6, equal 10, 15, 21 and so on.
A reciprocal equals dividing a number into one. So we take the reciprocal of our integer sums and then our sequence looks like this:
Sequence = 1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 +…
Unlike the previous sequence for the time to catch up, we see no way to simply multiply the sequence by a number to arrive at a match to the part of the sequence. In the time-to-catch up sequence, multiplying by 1/2 gave a part of the origin sequence. That approach is not available here.
Another approach can be used, though. Take the second element of 1/3. That equals two times (1/2 minus 1/3). We can see that by multiplying out the terms and then finding a common denominator to allow subtraction. Two times (1/2 minus 1/3) equals 1 – 2/3, or 3/3 – 2/3, which gives one third.
Now take the 1/6. That equals two times (1/3 minus 1/4) which is 2/3 – 1/2, or 4/6 – 3/6, which gives 1/6. Take the 1/10. That equals two times (1/4 minus 1/5). And so on, thus the sequence now becomes:
Sequence = 1 + 2*(1/2 – 1/3) + 2*(1/3 – 1/4) + 2*(1/4 – 1/5) +…
Which with a bit of rearrangement becomes
Sequence = 1 + 2*1/2 + 2*(- 1/3 + 1/3) + 2*(- 1/4 +1/4)
We now see that the fractions starting at 1/3 form a pair, one positive and one negative, summing to zero. All those terms starting at 1/3 and going to the right thus sum to zero, leaving the first two terms, i.e. 1 + 2*(1/2) or 2.
Again, we have taken an infinite and produced a finite.
Consider one final infinite sequence, the Basel series. This series comprises the reciprocal not of integer sums but of integer squares.
Unlike the two examples above, the Basel series does not yield to a simple solution. After conceived in the sixteenth century, the series stood unsolved for ninety years. Leonhard Euler finally found the sum, in part by using the infinite sequence for the trigonometric function sin(x). Euler might well stand as the greatest mathematician ever, and certainly of his time, and arguably as the most prolific in terms of publishing.
The curious can lookup the Basel Series for more details. The real curious can look up the rather mind-numbing proof.
God, and Infinity
The Catechism of the Catholic Church, a repository of its central teachings, exclaims the infinite nature of God, and does so multiple times. Paragraph 41 cites God’s infinite perfection, paragraph 43 God’s infinite simplicity, paragraph 270 God’s infinite mercy, paragraph 339 God’s infinite wisdom, and paragraph 1064 God’s infinite love.
The Apostles’ and Nicene creeds, accepted in many common Christian faiths, begin with a decree of God’s almighty, aka infinite, power.
A review of scholarly works in theology will find numerous discourses (attempting) to resolve the tension between God’s infinities (omnipotence, or infinite power; omniscience or infinite knowledge; and omnibenevolence, or infinite mercy) and the ubiquitous presence of evil in our world (how can an all merciful God allow wickedness?) and our clear sense of free will (how can I act freely if God knows my future?)
Clearly, God’s infinity stands as a key concept, and quandary, within religious faith.
Now let’s consider everyday images of an infinite God, images we may have developed ourselves, or heard preached. In terms of God’s infinite mercy, you and I, or say any pious, thinking individual, might conceive the mercy of an infinite God as large as the mercy of an infinite number of people. For God’s infinite creative power, we might picture that power sufficient to create an infinite number of universes, or equivalent to an infinite number of stars. In terms of knowledge, we might view an infinite God’s knowledge as large as an infinite number of computers, or an infinite number of libraries.
But… Those images actually describe a small infinity, an infinity equivalent roughly to the infinity of integers. God’s mercy as equivalent to an infinite number of individual relates his mercy to an infinite number of discrete items, people. We could match the (admittedly infinite) collection of merciful people one-to-one with integers. And God’s creative power as equivalent to the creation of an infinite numbers of universes or power of infinite stars, relates, again, God’s mercy to a set (admittedly infinite) of discrete items. We could do a one-to-one matching with integers. And so on with an infinite number of computers or libraries.
Here is the implication. God as infinite in an integer sense, as a endless, infinite sequence of nonetheless non-infinite, discrete items remains, in a subtle way, touchable, conceivable. God remains like us, or entities around us (universes, computers, stars, books), but just infinitely many more versions of discrete items of which we can see and touch and conceive. God can remain as a Father, Savior, Creator, Preacher, Benefactor, certainly infinitely perfect and infinitely numerous, but nonetheless infinitely perfect versions of tangible items we can touch, conceive, experience, ponder in our everyday lives.
In other words, God resembles items in our world, including us, just in a perfect, endless, infinitely numerous way.
But infinity as a sequence of discrete items, integers, equals the lowest size of infinity. We saw that an infinite number of larger infinities than that of integers looms over us. The infinity of integers descends to such a small infinity that no analogy describes the smallness of the infinity of integers by comparison the infinite hierarchy of infinities.
Consider just the infinity of real numbers. Real numbers of course extend upward just as do integers. But they extend downward, infinitely, to a smallness smaller than we can conceive or experience. We could take the smallest atomic particle, divide that particle a million times a second for every second of the universe, and be no closer to the smallness of the smallest member of real numbers than when we started.
Now take the power set of real numbers. We become lost, we can not readily envision infinite smallness of real numbers, and the power set of real numbers becomes a blur, more than a blur, just a miasma. But God’s infinity looms infinitely larger than the infinity of the power set of real numbers.
A catastrophe strikes, a catastrophe of comprehension and conceivability. We could contemplate a God as an infinite collection of otherwise conceivable discrete items. God looms infinite, but an infinite version of a graspable image, a Father.
Now contemplate a God greater than the infinity of the power set of real numbers. Our mind withers, recoils. We can find no images, fathom no analogies.
Under this expanded infinity, God becomes untouchable, alien, unknown, inconceivable. And our leap of faith leaves the realm of faith in a God infinite in extent and perfection, but an extension and perfection of a finite entity we can conceive, to something cold, mathematical, beyond just mysterious to eerily menacing, abstract, heartless. Our faith lies not in a warm, though infinite Father, but in an entity described best, and possibly only, in the stark, esoteric, forbidding world of the set theory of infinite quantities.
You don’t agree. You think this is not the case. God created man in his image; how can God then recede beyond our conception into a mathematical fog of infinite infinities.
But the logic becomes inescapable, despite our protests. The nature of infinity, as expanded by great mathematicians, combined with the infinity of God, as proclaimed by great theologians, creates an abstract God, distant and harsh. The infinite God becomes a mathematical God, a God described in power sets and number theory, a description which does not offer comfort.
That then identifies, starkly, the leap of faith. We leap into the unknown not to a God imagined as Fatherly and Majestic, but a God inscrutable as math more threatening than any most of us have ever taken.
But is that then where we end up?
Let’s step back. Our conception of an infinite God, within the modern understanding of infinity, becomes alien, abstract. But our discussion of infinity, and the analysis of it by modern mathematicians, included another aspect, that of the infinite converging, occasionally, but critically, to the finite.
We thus have within the expanse of the infinite, piece parts, infrequent, but still present, which converge to the finite. We thus possess a concept, an image, a view, with which to envision maybe not God in totality, but a piece of God to be our personal God. That vision parallels, mimics, the convergence of our infinite series to the finite. Within our God, we can envision, within the ineffable infinities, a personal part for each us that emerges from the convergences to the finite.
We should not overreach here. The convergence of a subset of infinite series does not allow a conclusion that infinity as a whole converges. Or that this convergence of some infinite series invalidates the infinite hierarchy of increasing infinities. No.
But rather than overreach we should admit. We should admit, recognize, that within this discussion, within this consideration of God as a touchable verses unfathomably untouchable infinity, we speak not of God. Rather we speak of images, analogies, comparison of fallible human concepts, to God.
And therein lies the likely most important message. We must acknowledge we possess, we talk in turns of, images of God. We do not know the actual God. God spans time timelessly. Mankind lives captured within time. God dwells outside space. Mankind exists bounded by space. God creates. Humans just discover what God creates. Those considerations force us to a realization the humans lack experiences that would give them knowledge of the actual God.
So while modern concepts of infinity call into question some familiar images of God, the modern concepts of infinity at a deep level aid a faith. The modern concepts of infinity, while jarring and obtuse, keep us, in that jarring, from falling into a contentment that we have reached God. The jarring shakes us from any lethargy that our human, fallible images of God mean we have finished our journey towards God.
Infinity never ends. Our travel, or maybe more aptly our wandering, towards a God never ends. The modern exposition of infinity, rather than threatening a faith, reminds us that faith involves not just belief, but a journey.
For the non-believer, the complexities of the infinite may buttress their already strong convictions on the irrationality of a belief in a Diety. For such a non-believer, science, philosophy, math, reason, those provide a sounder basis for truth.
However, the non-believer could not rest contented. They face their own quandaries with the infinite.
History provides a touch point, the ultraviolet catastrophe in the late 19th century. In classical physics, the principle of equipartition dictated that the theoretical object called a black box radiator should possess infinite energy. This pushed classic physics into a crisis. For another equally bed rock principle of physics, conservation of energy, stipulated the impossibility of an infinite energy source. Physics faced a catastrophic contradiction of an infinity.
Max Plank solved the riddle, by postulating energy did not distribute continuously, but rather in discrete steps. His quantum mechanics solved the riddle.
But quantum mechanics generated, and continues to generate, its own quandaries of the infinite. A feature of quantum mechanics, entanglement, predicts (and experiments verify) a type of infinitely fast linkage between paired particles. Two entangled particles, traveling in opposite directions, remain linked such that a measurement of one particle instantaneously dictates the state of the other particle. Infinitely fast linkage. We can write the math for the phenomena, but can not conceptualize the underlying reality. This infinity strains our common sense and equates to no available image.
Another example. Physicists struggle with the riddle of the collapse of the quantum wave function. To solve the riddle, some physicists theorize each quantum event generates a new universe, many, infinite, added universes.
Other infinities abound. Inflation theory predicts, in some versions, infinitely progressing series of Big Bangs. General relativity predicts an object of infinite density at the core of a black hole. Not to be left out, philosophy wrestles with infinite regress, and math with the implications of Geodel’s incompleteness theorem.
The non-believer can profess to not be troubled by these riddles; reason will solve them. But in stating such assurance, does not the non-believer profess a faith? To date, science, math, philosophy – the cornerstones of rationality – have produced new riddles essentially as fast as they have addressed old riddles. If God fails as a truth concept, could not rationality ultimately fail as a truth process. Can rationality escape a fate of continually creating new riddles, and encountering new infinities, never getting beyond nothing better than a pragmatic, interim description, never reaching truth?
Only on a faith can one say yes.
Infinity bedevils us, theist, atheist or agnostic.