There appears to be a misunderstanding of the philosophical background within which mathematical theorems reside. Attempts to explain this philosophical background have been met with rebuttals that are indicative of strongly held beliefs which I believe are inconsistent with it, so I'll do my best as a non-philosopher to illustrate why most mathematicians take the position that theorems are no less "real" than anything physical, and that the lack of a physically infinite universe in no way inhibits the conduction of infinite mathematics.
Take a look at the nine Peano axioms that rigorously define the (unequivocally useful) natural numbers 0,1,2...
There is a finite amount of information contained in those axioms, which can be specified in a manner independent of language as done in Whitehead & Russel's Principia, and be given to a computer in such a way that it is able to produce proofs of theorems that result from the axioms completely autonomously, with extensive libraries of such automated theorems being constructed such as the Isabelle Archive of Formal Proofs. However, those axioms describe the infinite set of natural numbers (an "infinite thing"), and can be proven to be able to answer an infinite number of questions about the structure they describe.
The similarities with supposedly more physical objects are sufficient to make the distinction practically meaningless: in order to perceive any undisputedly real object, to verify its existence and sort it into a perceptual class, one serves it a series of queries (in the form of, for instance, touching it, looking at it, turning it over, etc.) whose results are inferred via fallible perception. The result of those queries are a better cognitive model of the object and its properties.
This is not distinguishable from the way in which a mathematician interacts with the natural numbers object: queries take the form of conjectures, and imperfect perception in the form of proofs which we constantly try, but never manage, to extricate from reductionist complaints about dependence on various prejudices.
On a more personally didactic note, I see mathematics as the study of abstraction itself, and that all cognitive models of reality are dependent upon abstractions used to piece them together from sensory perception. As such, mathematics of some kind and quality is a precursor to experiencing the universe in any way, and the academic discipline of mathematics is a twofold effort to make increasingly more complex and useful abstractions and to make explicit the dependencies and statements of all abstractions we use.
Those theorems of information and computation theory you directly asked about are independent of the finiteness of the universe in precisely this way. The questions they represent can be posed not only finitely but very compactly, and their resolution similarly is possible finitely. In summary, in a finite universe, it is still possible to ask and resolve questions about "infinite universes."