Peano axioms to prove

Question

How can I prove $2+3=5$ and $x\cdot 1=x$ using Peano axioms? I think that $x\cdot 1=x $ can be partially proven using the 2nd axiom of induction but I feel like it's lacking.

Answer 1

Proving $2+3=5$ is fairly straightforward. Recall that $2$ is a shorthand for $S(S(0))$, similarly for $3$ and $5$. Just apply the axiom $x+S(y)=S(x+y)$ repeatedly until you get the wanted result.

The second proof is indeed a proof by induction, denote by $P(n)$ the proposition $x\cdot S(0)=x$. Then prove it holds for $0$, from the axiom $x\cdot 0=x$; then prove that $P(n)\rightarrow P(S(n))$. Then conclude from the induction schema that this is true for all $n$.

Answer 2

Depends on what version of Peano's Axioms you are using. The usual (non-first-order) version does not define addition or multiplication. Using those axioms and the axioms of set theory, you would have to construct those functions -- not easy! -- and then proceed as Asaf has outlined.