On: Does infinity exist?

[ced]

Yeah, these problems are hard (and fun!) to think about. But I still don't understand how you can go from:

There has to be an infinite number of functions we cannot write programs for!

which is, of course, true, to

So we have managed to show, in a fairly simple way, that there have to exist undecidable problems.

I don't think that you are using "undecidability" in its correct sense, or perhaps I'm missing something. What is your undecidable problem? You wrote

the problem is to decide whether some particular string is in this set

Which set are you talking about? The set of all programs? How does that tie in with your functions defined on integers?

[saurik → ced]

(You can try my description, a sibling to your more-parent comment, to see if that helps.)