Given that $K$ is an ordered field satisfying the least upper bound property and $1$ as the multiplicative identity, the set of natural numbers $\mathbb{N}_K$ in $K$ is defined as: $1 \in \mathbb{N}_K$ and $N + 1 \in \mathbb{N}_K$ if $N \in \mathbb{N}_K$. In this case, $N = 1 + ... + 1 \}$ $N$ times. Show that $\mathbb{N}_K$ satisfies the Peano axioms.
I am new to analysis and was not really taught about the Peano axioms and can't seem to find anything on the web about exactly what I need to prove in this problem. Can someone please briefly list and/or describe the axioms that need to be shown in order to complete the above proof? Any assistance is much appreciated.
Suppose you are allowed to sum numbers, and if you are in the natural numbers, this is quite natural. And define the operation $S(x) = x+1$ to be it, then we have to believe:
$S(x) = S(y) $, meaning that our $x$ equals $y$, so $S$ is injective;
1 is the only element, that doesn't succed any other, then there is no $x$ in natural numbers such that $S(x) = 1$;
Suppose there is a subset of natural numbers with 1 inside it, then if in this subset there is another, then it's because $S(x) \in X$, and so, this subset is the natural numbers set.