If the universe is everything that we see then according to current cosmological theories it is bounded. That is it is bounded spatially (as determined by General Relativity). It has been an open question as to whether it is unbounded in the future, current measurements indicate that it is unbounded in the future. But time unlike space we can only probe the present and memory & knowledge of the past. At every present moment what we see is a bounded, but expanding universe to the future and spatially.
But one can speculate - and plenty do - that there are other universes or all that we see is not the whole of the universe. For example the multiverse.
Where should one place the universe of mathematics? Is it in this universe. Is it elsewhere? Surely that is an infinite realm of thought.
Perhaps one can use a Turing Machine to enumerate every possible axiom system - certainly it wouldn't halt.
But why mathematics? Why not biology. The space of possible biological species is infinite too. As is all the possibilities of languages, and scripts and all the poems and essays and jokes that could be written.
Should mathematics have that high role that is traditionally awarded by it by philosophy? As a glimpse of the eternal.
Aristotle denied the possibility that humans can ever grasp a whole infinity, they can see only the potential. Since this has nothing to do with the nature of the universe - it applies whether the universe is steady-state or a big-bang. It rests upon an analysis of what infinite means. He states that the infinite is not an object or a thing. So it can't be identified with the universe.