Yes although I don't think it's particularly important for the natural numbers. While countability is not absolute, being countable is upwards absolute, so if y

Yes although I don't think it's particularly important for the natural numbers. While countability is not absolute, being countable is upwards absolute, so if you expand your universe enough you eventually get the natural numbers and never lose them so to speak. The more interesting theorem here is Tennenbaum's Theorem, which gives strong philosophical ammunition to realists that there is only one true set of natural numbers (and is a good argument why my hypothetical arithmetical statement might never exist). It's also another good reason why PA is so special (e.g. Robinson arithmetic is enough for Godel's incompleteness theorems but it fails Tennenbaum's Theorem).