A doctrine like the Trinity is unique to Christianity. Paul alludes to this in places like Philippians 2:6-8 or Romans 9:5. Jesus himself cites Psalm 110:1 in Mark 12:35-37 to indicate that he's not just a man. Supposed parallels between pagan "triads" is nothing similar anymore than a trio such as the Three Musketeers are - their nature is completely independent of one another, and are common only in a certain general sense.

The idea that one God can be more than one person can appear strange and outright absurd. Some consider it unofficial tritheism - a mixture of Greek polytheistic paganism and Jewish monotheism. Christian theologians have been discussing the Trinity for the last 2000 years. Justin Martyr [Dial. 128-9] compares the idea to a fire that has lit many places but is the same substance. Similarly, John of Damascus uses the example of mirrors reflecting the sun: different lights, one source.

I believe using math, specifically Set Theory, there is a way to understand the Trinity as consistent. I also think there are examples of it that can be given. In addition, I think that it isn't only Christians who have to explain and believe a situation like the Trinity, but also other theists and non-theists alike!

I. Explanation

First, let me try to justify how one God can be three Persons, all of whom are distinct from one another, yet of the same substance and fully God and in communion. Some good examples are given by Peter Abelard. If we have a bowl of wax with no shape versus a figure made out of the same amount of wax, these are clearly two different "things". Yet they are the same wax - it's the same substance. This is because a difference in identity does not mean a difference in what it's made of. If we have a tower made out of stone, this tower is both "hard" and "stone". However, the same could be said of any small stone, which is not necessarily a tower.

However, these examples illustrate only a part of the issues. I don't feel they incorporate all the aspects we need. For example, the tower can be both "stone" and "hard," but the tower and the little stone are clearly not equal and don't each constitute the whole the way each Person of the Trinity constitutes God.

So let's go to the math. In set theory, an infinite set is a group of elements (numbers), that goes on forever. For example:

{1, 1, 1, 1...} is an infinite set of 1's.
{-2, -2, -2...} of -2's, etc.
{1, 2, 3, 4...} is the infinite set of all natural numbers.

Now, you can play around with these sets. You can add or subtract or multiply them. So {1, 1, 1, 1...} * 2 = {2, 2, 2, 2...}. What happens if we subtract 1 from {1, 1, 1, 1...}? Well we get {0, 1, 1, 1...}. But 0 is not a relevant element - it doesn't signify a number, but the absence of one. This is why it's never written down unless it serves to illustrate the relationship of another actual number. So the number "23" can be written down as "000023" (as is the case in organizing files which have or expect a lot more files), but that's pointless. The same goes for something like 12.5, which technically also equals 12.50 and 12.500, etc. So {0, 1, 1, 1...} is the same thing as {1, 1, 1, 1...} and subtracting "1" from it gives us the same set because it has an infinite number of 1's.

So how does this relate to our original purpose? Let's take our set of all odd numbers: {1, 3, 5, 7...}. What happens if we subtract another infinite set from it? Let's say {1, 3, 5, 7...} - {1, 1, 1, 1...}. This simply equals {0, 2, 4, 6, 8...}. Notice that this equals exactly our set of all even numbers when we simplify it by removing the irrelevant "0" at the front. So what does this mean - that the set of all odd numbers is "greater" than the one of all even? No, because if we did the same to the even one, we'd get the odd as well: {2, 4, 6, 8...} - {1, 1, 1, 1...} = {1, 3, 5, 7...}. The same happens if we add {1, 1, 1, 1...} to either. So they are clearly not smaller or greater than one another.

This is where this directly relates to the problem of logically understanding the Trinity. Let's look at all natural numbers: {1, 2, 3, 4...}. The two subsets of all odds and evens are contained by it. However, what would happen if we divided our set of all evens by 2? {2, 4, 6, 8...} / 2 = {1, 2, 3, 4...}! We can do the same to the odds by subtracting {1, 1, 1, 1...}, which we saw gave us the even set and then we can divide by 2 again.

What does this mean? Is the parent set of two sub sets actually smaller than them? No, it can't be, because it also contains them. Simply put, the terms "subset" and "parent-set" are terms that we use for convenience, because we can't relate to infinity and our bias gets in the way. If we take these three sets to represent our theological issues, we can easily see how the whole problem of the Trinity is resolved. If we substitute the set of all real numbers, namely {1, 2, 3, 4...} with "God," and odd {1, 3, 5, 7...} with "God the Father," and even {2, 4, 6, 8...} with "God the Son," we can see that just as the odd and even infinite sets are equal to each other and to the original set, so also God the Father, God the Son, and God the Holy Spirit can be fully God, yet three different Persons. You can do the same with three subsets, every third number: {1, 4, 7, 10...}, {2, 5, 8, 11...}, {3, 6, 9, 12...}, to show a more direct parallel to the Trinity. For example, divide the third set by 3 to get {1, 2, 3, 4...}, and you can subtract the first by {1, 1, 1, 1...} and the second by {2, 2, 2, 2...}. And the subset of even numbers isn't 1/2 of all natural numbers, it is "fully" (mathematically; not identically) all numbers, just as God the Father/Son/Holy Spirit are each fully God but distinct from one another, but not God, without it being tritheism, modality (where all three are different manifestations of the same Person), or the like. You can do this with any number of sets: so a Binity, Trinity, Quadrinity, Pentinity are all logically possible and justifiable.

This goes to show Abelard's conclusion that similarity in substance is not the same as similarity in form. The sets are all equal, but their elements have different, unique, independent identities and do different things.

II. Objections

II.1. Equality

First of all, if any of the subsets contain any of the others, doesn't that make the other inferior/superior? So all even numbers {2, 4, 6, 8...} - {1, 1, 1, 1...} = all odd {1, 3, 5, 7...}. However, this is our minds fooling us again. We showed both contain each other, and if something is both "smaller" and "bigger" than something else in the same proportion, it's simply equal to it. We showed that by the fact that the parent-set {1, 2, 3, 4...} contains and is contained by both of the subsets.

II.2. Identity

The next question that can easily come to mind is that, if the subsets are contained within one another and with the parent-set, does that mean that they are each other? Does this mean that Jesus is God the Father, and "all" of God? No, because as we noted earlier, there's a difference between identity and substance. A wax sculpture is made out of the same wax as a shapeless bowl of wax, but it's not the same structure. The point of our little mathematical demonstration was to show that neither God the Father, nor Jesus Christ, nor the Holy Spirit are inferior to one another nor to the Godhead. All odd numbers might have as many numbers as all even numbers (and all real numbers), but you will never find a "3" in the set of all even numbers. That element simply does not exist there. In the same way, it wasn't God the Father or the Holy Spirit who was incarnated, but only Jesus Christ.

II.3. Infinity and the Real World

But do we have infinity in the real world? There are many things in math that don't necessarily exist in the real world, such as the square root of -1, i, so perhaps this shows what can't exist. But this is not the case. An ancient Greek called Zeno of Elea proposed countless paradoxes in reality that were unsolved for over 2000 years until Georg Cantor came. In fact, it was for the very real reason of these paradoxes that set theory was invented in the first place!

Let me give an example. Imagine you are standing at one end of a room and want to get to the other. However, you only take a step that reduces the distance by half. So, your first step gets you half way across the room, your next takes you halfway across the remaining half (1/4), and so on. The problem is that there is always distance between you and the remaining distance if you keep dividing it in half. The issue illustrates that if there is always a point between two points in space, how can anyone traverse an infinite amount of distance? Zeno had many other illustrations of this paradox. For example, let's take an archer who shoots an arrow. Would you agree that this arrow exists in a specific space at a given time whether it's in motion or not? Yes? Then how is it in motion if one can assign it such a place? When does the arrow move to the next infinitesimal point, which by definition has zero space between it and the one the arrow occupies? How can one even calibrate different speeds of different arrows? As Aristotle cites him better than my explanation:

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.

Aristotle seems to refute Zeno on the grounds that time is not indivisible and you can never truly get a "still" arrow that's actually moving, but he entirely misses the point (as he sometimes does when he doesn't agree with someone). Zeno's paradox is actually predicated upon this very refutation. The room example shows this, and it's a little obscured with time in the arrow example.

Some objections to Zeno's Paradoxes have included the idea that space and time are not continuous but discrete (ironically this would then support Zeno's Arrow example). However, space must be continuous due to the Weyl Tile Argument, and so does time which is mathematically equivalent to space as Einstein showed. The Weyl Tile Argument itself has objections, but to me they amount to nothing more than a "metaphysical somehow" without explanation.

II.4. Infinite Sums and Limits

There's a counter-example to the above. There's ways one can add up infinite sums which converge whose limits have different values (a good example can be seen here).

This doesn't invalidate our method, because limits are not the actual result of infinite sums. Moving elements around in an infinite series does change its convergence and limit, but that's only if one is adding or subtracting the elements and doesn't apply to infinite sets which are just elements. Limits aren't the same as equalities exactly because infinity never ends (consider the fact that the limit of 0.999... is 0.999..., whereas in actuality it equals exactly 1). This is shown by the fact that the sum of all natural numbers (1+2+3+4+...) has a limit that converges to -1/12 - not only a fraction, but also a negative number. A famous example that deals with infinity, Hilbert's Hotel, also moves the elements around.

In the link, the author acknowledges Riemann's conclusion that an infinite series that converges has a limit of any positive value, which if anything could be more support for our hypothesis, so convergence and limits are not the same as equality. The Banach Tarski Theorem possibly plays with values like this (legitimately) and may relate to the validity of our observations in Section I, but I reserve judgment (and it certainly doesn't contradict).

II.5. The Existence of Numbers and the Simplicity of God

This obstacle seems a bit more complex, but there is a very simple non-constructive answer. If infinity and the "highest infinity" or cardinality (which contains all infinities) can exist alongside the simple number "1", then this isn't a problem. The fact that there isn't an example of this in nature is because the universe itself doesn't have infinite energy, but the logic is all we need, as it would reflect reality should it have been that way. And if numbers don't exist as Abelard's interpretation of nominalism tells us, this doesn't contradict this answer either because again, the mathematical logic with Cantor's Set Theory has shown us that it's valid.

This means that God can create a creature that's separate from Him, with its "own" powers, who are yet ultimately from Him; but with respect to the creature's environment, they are the creature's - at the same time because of the nature of causality outside of time (when asking ultimately whose power it is). This is perhaps why the Banach-Tarski Theorem is such a paradox, but it helps explain these issues.

The last two objections are actually closely connected. First, doesn't our example fail because God has no "parts" whereas our sets have elements (1's, 2's, etc). If God is the "First Principle" (that is, everything was created by Him), then he can't have parts which means something created Him (!). Second, many philosophers will tell you that numbers don't actually exist in and of themselves! But how can this be? If I have four $1 bills in my pocket, I have $4, all of which are numbers. And no one is disputing that. However, would you have any numbers without those dollar bills? Is a rock still hard if it didn't exist?

The answer according to most philosophers today, with whom I agree, is "No." Let's put it this way: does the color red ("redness") exist? We would of course say yes. We have red sweaters, red markers. I even have a red car (a Toyota Yaris). But would the color red exist without those objects (sweaters, markers, cars)? How can it? You might say, "It's not the sweaters themselves intrinsically that make red to exist, but the light's wavelength reflecting off of them that's red." But isn't light a "thing"? Without which the color wouldn't exist? This school of thought is called Nominalism, which says that the color "red," the number "two," and according to some even the past and future, don't exist other than in name only, for our convenience. To give them a physical existence outside the name, would be the fallacy of language. It opposes the philosophical tradition from Aristotle, through Boethius down to Anselm of Laon, of Universalism, which says that "redness" exists whether the objects physically portray it or not. I personally agree with Nominalism (which our objection appeals to). The concept of redness certainly exists. After all in professional chess, one doesn't need to carry out a losing game: if one can see there's no way out, one can quit before checkmate has technically occurred. But the discussion takes place about what actually exists, and a concept without any power to be carried out has no place in a result-oriented framework. There's simply no way to have the color red without an object to manifest the color. Suffice to say, we're giving the second objection the benefit of the doubt.

But whether numbers exist intrinsically or not does not in any way invalidate our logical defense of the Trinity any more than the work of an applied mathematician's or an engineer's plans means he can't make a real bridge. These are merely representations of a truth. And the second objection actually solves the first, that of God's simplicity. If numbers (and thus elements or parts) don't exist, then we're simply talking about a reflection of God's power or magnitude, without any parts or elements. This reflection is accurate with "elements" (the numbers), because they denote something innumerable (infinity). This might seem paradoxical, or downright contradictory, but Zeno's Paradoxes and other aspects of Set Theory (such as Cantor's Diagonal Proof) illustrate that these are real issues (with real solutions) and it's just our mind playing tricks on us. At the end (for those who've had the patience), we'll see that it is in fact the finite that is by far the mysterious thing in reality, not the infinite.

Another question regarding God's simplicity is to suppose that God does have parts if He is three Persons. However, this isn't the sense in which God can't have parts, which would imply there is something more powerful than Him that created Him out of those parts. Nor are they actual parts in the sense that they are smaller (or larger) than the whole, as we showed.

III. Examples like the Trinity

III.1. Cerberus

William Lane Craig gives this as an example of the Trinity. Clearly for the multi-headed dog to make any cohesive move, all three heads would have to simultaneously agree, with perfect, total, and equal control. And yet all of them have minds (and are "persons") of their own, capable of independence in the details.

The problem is that this creature doesn't exist. Technical existence itself isn't the issue. But the fact that we don't have a physical illustration means that this example can easily be mishandled as a sort of imaginary panacea with which any issue can be "solved". For example, how can I make a square circle? Well I will appeal to the mythical square-circle beast of Atlantis.

III.2. Time

A personally favorite example, until somewhat recently, was the analogy of time. The past, present, and future are three parts of time. They are all time themselves. And they can't exist without co-existing. Without the past you can never get to the present or future. Without the future, it means time has ended (and so the past and present have ceased to exist). And without the present, you don't have a reference point to distinguish from the past or the future, and so neither would really exist (and without the present, it means time has also ended anyway).

However, I think there are a few problems with this. The fact that the length of the past is different from that of the present (an instant?) or the future isn't relevant. The analogy doesn't need to go that far - the point is to show how "three or two in one" is possible. But mathematically speaking, neither of these three descriptions are in any way objective. If there is a parallel universe with their own time (and space), to them our time would be neither past, nor present, nor future. Moreover Abelard considered the past and future to be just as non-existent universals as numbers. I'm a little reluctant to agree, but overall I think these are big enough weaknesses to use this example liberally.

III.3. Caesar

This isn't specifically about any of the Roman emperors. It has more to do with a brief office in early 4th century Roman government, called the Tetrarchy (not the Judean one in Jesus' day). Basically, the Roman emperor Diocletian divided the empire into four areas of authority amongst himself and three others. Every one of those four people were "Caesar," and each represented the entire empire in his respective region - one man was basically the entire government/emperor in the respective situation he was in. Of course, this is divided power, or full power in a divided region, but we're dealing with finite amounts.

III.4. Space

I feel this example does the best job of illustrating something similar to the Trinity. An object has three dimensions. It can't exist without either one and all three are space themselves. They are infinite in the sense of not having an indivisible unit and are interdependent in many of the ways and relationships that we explored mathematically in Section I (you need to stack an infinite amount of "lengths" to get a "width," for example). They are real physically and mathematically, and one can't say it's the fallacy of language for calling them length/width/height, or that they are universals and like numbers may not intrinsically exist.

IV. Trinity's Logical Consistency: Everyone's Problem?

Surprisingly, I feel that for anyone to explain existence the way it is, one has to deal with something very similar to the Trinity. Whether Muslim, Jew, or Hindu, or even atheist, I think the problem comes up one way or another.

IV.1. Monotheism

I will first explain how this is a problem for the monotheist who considers God to be all-powerful (omnipotent). In the 14th century, two theologians named Barlaam of Seminara and Gregory Palamas had an argument regarding the mystical nature of the Christian's connection to God. Gregory Palamas supported the traditional view that some received revelations directly from God. Barlaam instead considered all revelation, miracles, and general experiences from God to be symbolic only. In his view, God didn't even create the world Himself; because of its impurity, God could have nothing to do (not even physically) with its fallen, impure creation. He couldn't even have any kind of contact with it because of this. This is actually a similar argument Muslims often employ as to why they don't think Jesus could be God, which clearly misses the implications Barlaam expounds.

So if God is all-powerful and "infinitely infinite", and according to Muslims or Jews, cannot be Triune, how can he create anything that's differentiated from Him? How can a chair exist without it being a part of God - without it being part of God? Aristotle answered this by saying power does not equal identity (something agreed with earlier). However, the question of identity is inescapable here. Take the Earth - the chair is a part of it. The wood it was made of came from it. It's in the world and a part of it. Without our solution with set theory in Section I, you cannot really explain this problem.

Another way of thinking about this (and supporting Cantor's Diagonal Proof) is, possibly, the fact that there are different increments of time and space in an infinite whole. So, for example, Zeno's Paradox points out that since there is always a midpoint between two spaces, one must traverse an infinite number of midpoints to simply go across a room! Think of it this way: if you tried going from one end of a room to another by going half the distance, would you ever reach the other end? Only after an infinite number of such "half-distance" steps. Yet the room's distance is finite, so how can this be explained? Only through the validity of Cantor's Set Theory, which indirectly validates as logically possible both the Trinity, as well as the ability for an "infinitely infinite" God to create finite and even fallen creations.

IV.2. Polytheism

This problem might seem alleviated by considering multiple, non-omnipotent deities. The gods of Hinduism, indigenous religions, or the dualism of Zoroastrianism seem to solve this and a host of other problems, for which reason Europeans were occasionally attracted to them (Honderich, Ted (ed.), The Oxford Companion to Philosophy, "Zoroastrianism" (1995), p.923). The rest of "everythingness" is some unidentified, primordial force - chaos.

But from where did everything, including these gods come? If they were eternal, why are they this way? If an infinite amount of causes (infinite regress) occurred, or it spontaneously came into existence, that only begs the question to be asked again (more details in IV.3).

IV.3. Naturalism and Other

Pure naturalism (and non-omnipotent Deism/montheism) faces the same problem. Why does the universe contain 10^80 electrons and not 10^79 or 10^81? I'm not arguing that this proves there must have been a God who created everything/something at all. I'm only pointing out that the answer is the same logically consistent answer as how the Trinity can be true and non-contradictory.

If the cause that originated the universe was eternal, why did it create the universe in this way? Randomly and arbitrarily? If it wasn't random, doesn't that imply an intelligence at work? But if there were an infinite number of possible random variations, like Zeno's Paradox, how was one even "picked"? How does one force of nature differentiate itself from another, especially since all of them were part of the same original force at the Big Bang (and as superstring theory tries to understand them as)?

I feel that if all of reality originally comes from nothing, or even an infinite number of causes before it as we show earlier, without Set Theory it's like having a chair sit on itself. Similarly the example given before with a black hole's singularity existing in literally zero space makes it impossible conceptually without the math.