I am working on the following problem:
Let $ S_{Arithmetic} = \{+, *, 0, 1\}, \mathfrak{M} $ a model for PA (first-order peano axioms) }, and $ \mathbb{N} = (\mathbb{N},+ ^{\mathbb{N}}, *^{\mathbb{N}},0^{\mathbb{N}},1^{\mathbb{N}} )$.
Construct an embedding $f : \mathbb{N} \rightarrow |\mathfrak{M}| $ and show that f is unique.
So, for $f$ to be an embedding, the following has to be true (correct me if I'm wrong):
- $ 0^\mathfrak{M} = f(0^\mathbb{N}) $
- $ 1^\mathfrak{M} = f(1^\mathbb{N}) $
- $ +^\mathfrak{M} (f(a_0),f(a_1)) = f(+^\mathbb{N}(a_0,a_1)) $
- $ *^\mathfrak{M} (f(a_0),f(a_1)) = f(*^\mathbb{N}(a_0,a_1)) $
- $ f $ injective
Now, if I set $f$ to $f(x) := x$, it seems to me that all these properties are satisfied, but I am not sure what to do to prove this assignment and what to do to show that $f$ is unique.
I would be glad if someone could point me in the right direction.