tautological: Let us suppose, for the sake of argument, that the brain's capacity is finite. The question now touches on the Platonic/Formalist debate, which despite the death blow to Hilbert's Program dealt by Gödel, is not dead; only the classical Formalism is. Today's Formalism is concerned with how to divide mathematical structures into syntax and semantics. Taking a formalist stance, then, we can say that the words or symbols "infinite" and "finite" are either:
(a) syntactically, finite symbols for syntactical statements such as the symbol N in the axiom of infinity in ZF or its negation, respectively, or a statement such as "A set A such that there exists a bijection between the A and a proper subset of A", or
(b) semantically, the interpretation in a model of such symbols.
As long as there is no provable contradiction from these usages, then if they are useful, we keep them. From the viewpoint of a Platonist who does not believe that infinite quantities actually exist (whatever "exist" means), "infinite" may be thought of in the same way as school children are unfortunately often taught to think of complex numbers or infinitesimals: as something that is simply an imaginary go-between between a problem stated in real numbers and the solution stated in real numbers. There are also Platonists who believe that infinite quantities exist, who would then just tell you not to get the finger that points to the moon mixed up with the moon: the human brain could be finite, but it could point (via model theory) to something that was infinite. Finally, from the viewpoint of a staunch Formalist, even the interpretations are just symbols to be pushed around, and the symbols need not actually point at anything.
So, you pick your mixture of formalist and platonist (I don't think anyone is a pure formalist or a pure platonist any more), and go from there. A good starting point is to purge the question of the physics aspects which Stephen Tashi objected to by delving a bit into the theory of finite models and restating your question in those terms.
