@Joel Thanks! Your statement "...stronger and stronger theories may prove higher and higher lower bounds..." and mhum's comment that non-countable should be rejected, point to an analogy. In Calculus class, I had a similar problem understanding the $\epsilon$, $\delta$ definition of the limit of a real valued function - which now seems intuitively obvious. Given an $\epsilon > 0$, I should be able to come up with a computably axiomatized theory that proves a lower bound for BB(n) to within that $\epsilon$

@mhum The Wikipedia article on Intuitionism certainly helps. Finitism rejects the existence of countably infinite, Intuitionism rejects the existence of uncountably infinite; I was looking for a constructive theory that accepts the uncountably infinite but rejects the non-computable. But, thanks!

Gerhard : What bothers me is that I cannot understand the meaning of existence outside of computability. If you could point to an actual weak axiomatic system with its limited scope, I'd be very interested to know some simple statements that cannot be expressed in them, but intuitively, I should be able to. i.e what prompts the bootstrapping of properties that render axiomatic systems victims of Godel's incompleteness theorems.