Dear Jen,
Why do you assume that the set of natural numbers is infinite?
You are a Christian, so you believe in the beginning and the end of time. Let’s say that all possible computational power went about counting natural numbers. At the end of time, there would be the largest number, and no other larger number would ever be counted after it. Who’s to say that such a number – let’s call it the Ω number – is not the largest possible number? The end of the line?
Don’t say anything about Knuth’s up-arrow notation. Those arrow notations are not numbers but steps or functions to arrive at numbers. But even if they were, the same end of time problem would arise.
The best rejoinder is to say that these computational limits don’t matter because God is the actual infinite and can calculate and conceive any number or set, including so-called infinite sets. Maybe so. That’s a mystery man cannot grasp, and it doesn’t answer the first question. God’s numbers are not natural numbers.
The infinitude of God insists on the finitude of man and his numbers.
Utter failure in other projects and no sign of the surrealists. Enjoy your cave.
vale bene,
D