Is the axiom of infinity in ZFC equivalent to the existence of a complete ordered field? Suppose we delete the axiom of infinity in ZFC, which states that there exists a set of all natural numbers, and instead put in the axiom that there exists a complete ordered field. Would we still be able to derive the axiom of infinity? What I am basically asking is whether the axiom of infinity is equivalent to the existence of a complete ordered field, modulo (ZFC - Infinity).
 A: Of course, if we have a complete ordered field $F$ we can define a copy of $\Bbb N$ inside it: $0$ and $1$ are the ones from the field, and we can define $\Bbb Z$ as the smallest subring that contains $1$ and $\Bbb N$ are just the elements $\ge 0$ in that $\Bbb Z$. I don't see any issue with that and it's well-known that from $\Bbb N$ we can construct a complete ordered field. So assuming one or the other is equivalent. But $\Bbb N$ is more in line with a minimality ideal: assume the simplest object as an axiom (Occam's razor like). And we have the tradition of starting with Peano's axioms in foundational theory as well. So it's never done in "your" order.
A: It's possible to define the natural numbers in ZF even without the axiom of infinity as in
A set $n$ is a natural number if

*

*Every non-empty subset of $n$ has an $\in$-minimal member.

*$n$ is either $\emptyset$ or a successor of one of its members.

*Every member of $n$ is either $\emptyset$ or a successor of a member of $n$.

(Successor is $Sx=x\cup\{x\}$).
From this it can be proved that natural numbers are von Neumann ordinals, and the class $\mathbb{N}$ of natural numbers is transitive and well-ordered by $\in$.   The absence of the axiom of infinity means that $\mathbb{N}$ may be a proper class (i.e. $
\mathbb{N}=\mathbb{ON}$ in a model).
Furthermore, it can be proven by induction that any natural number $n$ is Dedekind-finite.
Therefore - Every Dedekind-infinite set must be a infinite (i.e. not in bijection with a natural number.)
An ordered field is always Dedekind infinite because you can map the subset $x>0$ to $x+1$ and leave all other members of the field stationary.  The existence of a Dedekind infinite set is the essence of the axiom of infinity - because from such a set and proper injective function $f$ into itself you can construct a sequence by recursion $x_0,f(x_0),f(f(x_0)),\dots$ (starting from some $x_0\in\operatorname{Dom}(f)\setminus\operatorname{Im}(f)$) and by applying replacement in reverse deduce that $\mathbb{N}$ is mapped bijectively from a subclass of a set and is therefore a set.
