Attempting to prove that the Collatz conjecture is unprovable

For those of you unfamiliar with the problem, see here: http://en.wikipedia.org/wiki/Collatz_conjecture

Latest tack on the problem: trying to prove it can't be proven true.

My thoughts are as such. What does it mean to prove a statement for all elements in a set? Let's say you've got a statement K that you want to prove for all elements in a set A. Now, let's say that you can pick any element x in that set and through some method establish that K is true about x. Does this mean you have a proof?

No, it does not. Proofs must be finitely written and must generate results in finite time. If A is an infinite set and in your "proof" every element of A requires at least one extra bit of information to be written, then you can't say it's a proof because it is not finite.

Similarly, if you can establish that K is true about every element in the infinite set A, but every element requires at least one extra operation to establish its truth, then you don't have proof.

So, I would establish these as necessary for the existence of proof of a statement K over all elements of a set A:

1. There exist sets A1, ... , An such that the union of these sets is A. There is a finite number of these sets, and they are all definable in finite space. It is possible to show that the union of these sets is A in finite time.

2. For each i from 1 to n, all elements of Ai have a property Li which implies K. Li can be checked for the entire set Ai in finite time.

The Collatz conjecture seems to fail at the intersection between steps 1 and 2. I've tried multiple ways of defining sets of integers that imply the conjecture, but the requirement of finitude in space and time prevents a "proof" from existing. If there is a finite number of sets which cover all integers, then each set takes an infinite amount of time to establish the conjecture on. Conversely, if I can find a set sharing an easily checked property which implies the conjecture, that set is non-dense in the integers, and so it would require an infinite number of such sets to cover the integers.

My current view of the conjecture is that it is true but unprovable in finite space and/or time.