Punting to the infinity of God
Sometimes, people settle the issue of the incarnation or the trinity by appealing to God's infinite nature. They say the incarnation is possible because God is infinite. I don't see how being infinite solves the problem of the incarnation. I don't see what they have to do with each other. My impression is that the reasoning goes something like this: The incarnation is something we don't understand. Infinity is also something we don't understand. Therefore, if God is infinite, then lots of things can be true of God that we don't understand.
When people say that God is infinite, I must admit that I don't know what they mean. There are a few possible things people might mean by saying God is infinite.
Well, God is not a physical being, so he doesn't literally have size. The Bible talks in anthropomorphic languages when it says that a Temple was built for him to dwell in, but that the whole heavens cannot contain him. I get the impression that these are figures of speech. But even if it's true, what does size have to do with the problem of the incarnation? I don't get it.
If time has always existed, and God has always existed in it, then God has existed for an infinite amount of time. But if God doesn't exist in time, or if God created time, then he hasn't really existed for an infinite amount of time. There isn't an infinite amount of time for God to have existed in, if God created time. If time itself is finite, then nobody could have existed for an infinite amount of time. But even if God has existed for an infinite amount of time, I don't see what that has to do with the problem of the incarnation or how it solves it.
God is said to have infinite love, infinite power, infinite knowledge, and infinite goodness. Just about anything that can be said about God, it is commonly said he has it infinitely. But saying, for example, that God has infinite love is ambiguous. Are we referring to the depth of his love or to the scope of his love, or both? It seems to me that it's not possible for God's love to be infinite in scope, because there aren't an infinite number of beings for God to love. If his love is infinite, then it's infinite in depth, not scope.
But how does infinite love solve the problem of the incarnation? Perhaps infinite power solves it. One might argue that if God has infinite power, then he has the power to engage in logical impossibility. But power has nothing to do with logic. Our inability to do draw four-sided triangles, for example, has nothing to do with a lack of power.
Sometimes I think that when we say God is infinitely good, holy, or just, we don't mean "infinite" in a literal sense. We're just using it qualitatively, not quantitatively. We mean that God is wholly, completely, and perfectly good, holy, and just. I don't know what it would mean for God to be infinitely good in a literal sense. A circle isn't infinitely round; rather, it's completely or perfectly round. In the same way, God is perfectly and completely good.
There's only one God, and there are only three persons in the Trinity, so this one is easy to rule out. Besides, after reading about the kalam cosmological argument, it doesn't seem that it's possible for there to be an infinite number of things.
I sometimes think the only reason people punt to "infinity" to solve the problem of the incarnation and the trinity is because infinity is one of those things that's hard to understand. Since it's such a mystery to us, we figure anything is possible in the infinite, so we throw all of our logical difficulties in the infinite, and we don't even worry about explaining what we're talking about.