I love philosophy Calvinball, so I would counter by asserting that undetectable implies no possession, an immediate contradiction. Or go further and assert that undetectable implies nonexistence. We all possess an immense undetectable nonexistent sphere. No bounds on assumptions means I can make up anything to annoy the interlocutor.

So, you're right. This shows why we should use formal math, so we can agree on the result yet bicker about the interpretation. Some folks point to Cantor's diagonalization theorem to show that some unnameable things exist, when the theorem doesn't say that at all.