If one is a presentist (i.e., believes that only present objects and/or events exist), then one can accept that it is not logically possible for there to be an actually infinite number of objects and/or events, and still believe it is possible that the universe have had no beginning, but existed infinitely. The reason is because an infinite past is unreal (not actual) if presentism is true. There may be an infinite number of presently true propositions which are past-tensed, but the presentist doesn’t have to be committed to the existence of propositions themselves (as though propositions were part of the furniture of the real world). Indeed, the presentist may have good reasons for rejecting the existence of propositions (for instance if they have good reasons for thinking a deflationary theory of semantics is correct, or if they are worried about set-theoretic paradoxes which may rear their heads if one affirms that propositions really exist as platonic-like entities in the world).
Thus, the infinite past would be a potential, not an actual infinite. This was the view of St. Thomas Aquinas.