Proper subsets, the size of infinity, and the Trinity

[Yet another unpolished non-story post that was originally a comment on someone else's blog (page 2), I must be getting lazy.]

Ok, so it started like this, there was a discussion of theology, the exact details are not important but you can follow the link above if you're interested.  And someone said that the refutation of a given point would be:

Jesus was God in human form, but because God is greater than humanity, Jesus was not the entirety of God. [...] (math majors would describe it as "the set of Jesus is contained entirely within the set of God, but the set of God is larger than the set of Jesus") 

My response was that the way a math major would say that is, "Jesus is a proper subset of God."  Subset means every element of Jesus is also contained within God, proper subset means that not every element of God is contained within Jesus.

At this point someone else pointed out that, if they're infinite, Jesus being a proper subset of God doesn't actually means that God is larger than Jesus, and at that point I said:

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Given that a set is infinite if a proper subset of it can form a one to one correspondence with it, which makes no intuitive sense but is none the less plain to see when you look at example infinite sets , my personal definition of infinity is "the point at which things stop making sense."

But yes, Jesus being a proper subset of God does not mean God is larger than Jesus even though it does mean God is not limited to Jesus.

For any who don't follow, allow me to give an explanation that probably won't be any easier to follow.

If we were to consider God to be the set of all integers, the Father to be the set of all integers divisible by 3, the Son to be all integers one greater than divisible by three, and the Ghost to to be the set of all integer one less than divisible by three

God = the set of all k such that k is an integer.
The Father = the set of all "3k*1" such that k is an integer
The Son = the set of all "3k+1" such that k is an integer
The Spirit = the set of all "3k-1" such that k is an integer

(it looks better when I can say it in symbols instead of words)*

So The father is 0, 3, 6, 9, and so on.
The son is 1, 4, 7, 10 and so on.
The Spirit is -1, 2, 5, 8, and so on.

It's completely clear that God contains a lot more than just Jesus, It's got these two other subsets that don't overlap with Jesus at all and each of them is as big as Jesus.

It's plain to see that the Father, the Son, and the Spirit are the same size because you can plainly see the one to one correspondence for every element.  If you were to take every element of the Father, preform a simple transformation (add one to the the value of the element) you would be left with every element of the Son.  There would be no duplicate elements of the Son produced, nor would there be left over elements of the Father that had no Son element to change into. There certainly wouldn't be any elements of the Son missing.

That's only possible if they started with the same number of of elements.  Similar correspondences are possible between Son and Spirit and Father and Spirit.  Thus they're all the same size.  And, as noted, they don't overlap.  So what about the size of God, which includes these three non-overlapping things?  Shouldn't God be bigger than them?  No.

Try the same thing.  Take every element of the son, every 3k+1,  and preform a simple transformation.  Subtract 1 (now you've got every element of the Father, but I preferred not to start there)  then divide by three.  Now what do you have?

The son is 1,3,7,10, and so on forever, (and also in the opposite direction, I might add, thus -2, -5, -8, and so on forever), so after this transformation we get 0,1,2,3 and so on forever in on direction, and -1,-2,-3 and so on forever in the other direction.  We get God, all of God, including Father, Son and Spirit.  But this change didn't require us to add any elements, just alter the individual elements, so that means that God is the same size as Jesus in this example.

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*And now I can:

God     = { k | k∈ℤ}Father = { 3k*1  | k∈ℤ}Son      = { 3k+1 | k∈ℤ}Spirit   = { 3k-1  | k∈ℤ}

Looks a lot better, doesn't it?

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Of course, not everyone is on board with the idea God is a set.  Some think God is a class and one person suggested a relation citing, "Infinite thy vast domain, everlasting is thy range."

The point is, by simply looking at how infinities work on the integers, we can see how we can have three non-overlapping infinite things of equal size each of which is likewise of equal to the sum of the three things.

All of which serves no purpose, but does that really matter?