Trinitarian Arithmetics

The trinity consists of God the Father, God the Son, and God the Holy Spirit, but there are not three Gods, but one God. This is the essence of Athanasian Creed. But it drives me crazy when someone tries to simplify it using the arithmetic “one plus one plus one equals one” and then proceed to claim that human mind cannot possibly understand why God can’t do arithmetics. There are several arithmetically correct analogies that we can use, so let’s not settle with the lack of creativity here.

I think that mere addition is way too elementary to describe the internal relationships of the Father, Son and the Holy Spirit trinity. I propose we take a look at more profound operators, such as multiplication.

One, multiply by one, multiply by one, equals one. Perfect!

It turns out that exponentiation also works. In case the reader needs a reminder how it works, 2 raised to the power of 3, written as 2³, means multiplying 2 three times, so 2 × 2 × 2 = 8. It is also possible to stack exponentiation, like 2 raised to the power of 2 raised to the power of 2, which equals to 2 raised to the power of 4, which is 16. Exponentiation makes numbers grow faster.

It turns out to be convenient for us as well: one, raised to the power of one, raised to the power of one, equals one. One Father, raised to the power of One Son, raised to the power of One Holy Spirit, equals One God.

But the true mysteries of God cannot be fully appreciated until we realize that our God is an infinite God, consisting of one infinite Father, one infinite Son, and one infinite Holy Spirit. Most people who study calculus would use the symbol ∞ to denote infinity; however, calculus is a sloppy way to understand the nuances of infinity because infinity is only used to describe the vague notion of divergence, but it is not a number, and it has no actual definition.

To understand infinity better, we need to appeal to the more rigorous mathematics known as set theory, which axiomatically builds up a number system called ordinal numbers through set membership, e.g. 0 = {} the empty set, 1 = {0} which is {{}} a set containing the empty set, 2 = {0, 1} which is {{}, {{}}}, 3 = {0, 1, 2}, etc. The idea is that x < y (x less than y) is defined by x ∈ y (x is a member of y). We build larger numbers x + 1 inductively by taking the union set x ∪ {x} which is the content of x plus the set containing a single item x.

“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.

Georg Cantor, who invented set theory, appropriately used the symbol ω (Greek letter “omega”) to denote the set of all finite ordinal numbers, such that x < ω for all ordinal number x. A concrete example what it may represent is the set of all natural numbers, written as ℕ. Because ordinality is defined by set membership, ω is itself a well-defined ordinal number. Intuitively, it is the least infinite ordinal defined in a very specific way, so don't confuse it with ∞ which has no definition.

It turns out that we can keep building ω + 1 = ω ∪ {ω}, ω + 2, ..., ω + ω which is ω ⋅ 2. As ω is analogous to the set of natural numbers ℕ = {0, 1, 2, ...}, ω ⋅ 2 is analogous to the set of all integers ℤ = {..., -2, -1, 0, 1, 2, ...}. It is interesting to note that ℕ and ℤ are the same size because we can come up with a one-to-one mapping.

From ℕ 0 1 2 3 4 5 6 7 8 ...
To ℤ 0 1 -1 2 -2 3 -3 4 -4 ...

So even though we added more onto the infinite set ω, we didn't end up making it larger!

We can continue to build ω ⋅ 2, ω ⋅ 3, ..., ω ⋅ ω, and ω ⋅ ω is analogous to the set of all rational numbers ℚ = ℕ × ℕ because rational numbers, written in the fraction form of a/b, is defined as ℚ = { a/b ∣ for all a ∈ ℕ and b ∈ ℕ}. It turns out that ℚ and ℕ also have the same size because we can enumerate them diagonally to come up with a one-to-one mapping, as described by the pairing function.

We can continue to build ω², ω³, ..., ωω. At this point, it is difficult to come up with a pairing function, but we can apply the argument that we can build one if we really tried using transfinite induction. Afterwards, we can continue to build ωωω, ..., ωω etc. Cantor actually went on much more than that, but we'll stop here.

Now, after developing the necessary toolkit to study infinity, we can now understand the mathematical relationship behind the Trinity.

  • One infinite Father, plus one infinite Son, plus one infinite Holy Spirit, equals one infinite God. ω + ω + ω = ω.
  • One infinite Father, multiply by one infinite Son, multiply by one infinite Holy Spirit, still equals one infinite God. ω ⋅ ω ⋅ ω = ω.
  • One infinite Father, raised to the power of one infinite Son, raised to the power of one infinite Holy Spirit, still equals one infinite God. ωωω = ω.

And it somehow all mathematically works out.

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