A physical whole is an extensional totality: its identity is constituted by the actual, determinate constituents that make it up. Unlike an abstract mathematical set, which can be defined intensionally by a rule or a formula (e.g., “the set of all natural numbers”), a physical whole is not a rule. It is the concrete sum of everything that physically exists. Its identity is not given by a logical definition; it is given by the completed collection of all actual physical things.
For an extensional totality to satisfy the Law of Identity (A = A), it must be structurally complete. “Structurally complete” means that the collection of its constituents forms a closed, exhaustive set. There is no further constituent to add; the whole is fully constituted by what it actually contains. An extensional totality that is structurally incomplete is not a determinate whole; it is an open‑ended aggregation that has not yet arrived at a definitive state. Only a structurally complete extensional whole can be this whole and no other.
Now consider an actually infinite physical whole. To be a completed extensional totality, it would have to actually contain infinitely many distinct, determinate physical constituents. But an infinite collection is, by definition, endless. To be “complete,” a collection must have no further member to add; it must be finished. To be “endless” is to have no final member; it is to be intrinsically unfinishable. “Complete” and “endless” are mutually exclusive properties: one asserts closure, the other asserts the absence of closure. To claim that an infinite physical whole is a completed extensional totality is to assert that an endless collection has an end – a direct violation of the Law of Non‑Contradiction, which is a direct corollary of the Law of Identity (A = A).
Abstract mathematical sets escape this contradiction because they are not extensional totalities. They are intensional rules. The set of natural numbers is not a completed list of all numbers; it is a definition that generates them. No mathematician claims that the set of natural numbers is a finished enumeration of concrete objects. The identity of an intensional set resides in its defining rule, which is perfectly determinate without requiring the actual completion of an endless sequence. But the physical whole is not a generation rule; it is the actual stuff of reality. If that stuff were infinite, reality would have to be both finished (to be a determinate whole) and unfinished (because infinite). That is impossible.
Therefore, the physical whole cannot be actually infinite. It must be finite: a completed, closed, extensional totality with a definite, specific number of physical constituents, fully determinate and fully self‑identical.
The Bekenstein Bound I ≤ 2πRE / (ℏcln2) independently confirms that any finite physical region has a finite informational capacity. The Planck scale defines a minimum resolvable length, confirming that physical reality is not infinitely divisible. Hilbert’s Hotel paradox demonstrates the logical absurdity of treating an actual infinite as a completed physical entity. These results are not the deductive proof; they are the external verification that the logic is sound.
To reject Theorem 2 is to assert that a physical set can be infinite – lacking a determinate magnitude – and still possess a unique identity, which is a direct A ∧¬ A contradiction. The processor crashes.