Yes, the question was confused. I got distracted thinking about stuff I don’t know about (asking “how” instead of “whether”).
You don’t need “meta-logic”, whatever that might be, to know that 2+2=3 cannot be consistent with Peano arithmetic.
Here’s the “proof”. We know that 2+2=4 in “our” Peano Arithmetic. Suppose that 2+2=3 in “another” Peano Arithmetic in an alternate reality. Then the two Peano Arithemetics are actually different because they have a different set of “trues”. When we say that 2+2=4 in “our” Peano Arithmetic, we mean “our” PA and not the other. Whatever distinguishes the two arithemetics can be incorporated in our PA as an axiom – indeed it was included implicitly in what we meant by PA all along even if we lacked the imagination to explicitly identify it.
Yes, the question was confused. I got distracted thinking about stuff I don’t know about (asking “how” instead of “whether”).
You don’t need “meta-logic”, whatever that might be, to know that 2+2=3 cannot be consistent with Peano arithmetic.
Here’s the “proof”. We know that 2+2=4 in “our” Peano Arithmetic. Suppose that 2+2=3 in “another” Peano Arithmetic in an alternate reality. Then the two Peano Arithemetics are actually different because they have a different set of “trues”. When we say that 2+2=4 in “our” Peano Arithmetic, we mean “our” PA and not the other. Whatever distinguishes the two arithemetics can be incorporated in our PA as an axiom – indeed it was included implicitly in what we meant by PA all along even if we lacked the imagination to explicitly identify it.