I'm studying the proof of commutativity of addition using only Peano axioms (with the distinguished element being 0 rather than 1), the definition of addition, and x+0=0+x=x. The main idea is induction on natural numbers. I managed to prove it in one way, but the problem is I can't completely understand the other one I saw in class. Actually I doubt whether it is correct.
Here is the proof:(it uses 2 inductions)
Let A be the set A={x∈N|x+y=y+x,∀y∈N}.
We already know 0+y = y+0 ∀y∈N. Therefore 0∈A.
Suppose that x∈A. Let B be the set B={y∈N|ν(x)+y=y+ν(x)}. (v is the successor map in Peano axioms)
Note that 0∈B, since 0∈A.
Suppose that y∈B.
Then we have, using the definition of addition as well as x∈A:
ν(x) +ν(y)
=ν(ν(x) +y) [def of addition]
=ν(y+ν(x)) [assumed y∈B]
=ν(ν(y+x)) [def of addition]
=ν(ν(x+y)) [assumed x∈A]
=ν(x+ν(y)) [def of addition]
=ν(ν(y) +x) ???
=ν(y) +ν(x) [def of addition]
Therefore also ν(y)∈B. By Peano axiom 5(P5), we obtain that B=N. Therefore ν(x)∈A. By P5 again, A=N and addition is commutative.
I cannot understand why we can get the equation with the question marks, because I think it is something that we have not assumed.
It would be really helpful if anyone can explain why it is correct or wrong. Thank you!!