Maybe “earplugs do not model PA,” not the other way around? (Edit: just saw this excellent clarification.)
Number-handling is an older science than Peano arithmetic, and especially older than model theory. The numbers 2 and 3 would “exist” even if PA were shown to have no models. At least, the notation 2 and 3 would still be relevant to things that really exist.
It is very easily verified that 2 + 2 does not equal 3, but not effortlessly verified. It takes a positive amount of effort to verify it, and there is a positive amount risk of having made a mistake while doing so.
Maybe “earplugs do not model PA,” not the other way around? (Edit: just saw this excellent clarification.)
Number-handling is an older science than Peano arithmetic, and especially older than model theory. The numbers 2 and 3 would “exist” even if PA were shown to have no models. At least, the notation 2 and 3 would still be relevant to things that really exist.
It is very easily verified that 2 + 2 does not equal 3, but not effortlessly verified. It takes a positive amount of effort to verify it, and there is a positive amount risk of having made a mistake while doing so.