Infinite numbers are definable (as in set theory), even though the actual universe may be only finite. Nevertheless we often “prove the existence” of this or that infinite number (set). In this case we might be able to say that infinite number exists relative to the theory that defined it (such as set theory).




Then,



(2) if (1) is satisfied T (necessarily) exists relative to itself.



Now, it would be argued, this relative kind of existence is what we really mean by “existence”. This applies to actual existence: everything that actually exists, does so relative to everything else in actual existence… (Existence forms an equivalence class...)



In this case, (2) is sufficient for actual existence.



What if things either do or do not have “intrinsic” or “absolute” existence? In this case (2) would seem to be insufficient to generate actual existence…



But by the hypothesis, the universe is ultimately mathematical structure. Can parts of a mathematical structure exist relative to each other? Suppose we say



(3) the number 2 exists relative to the number 3



You might say: it may be true, but presupposes absolute existence. But how do you know? If absolute existence is, in fact, relative existence, we would not know the difference.



You might say: this is a false option, and (3) is neither true nor false. But it is either a theorem of the system that



(4)



or else it is not. In the system T we want something like



(5)



which is not to be confused with



(6)