Theorem 2) There are abelian $p$-groups of every infinite ordinal length, where the length $\ell(G)$ of a group $G$ is the least ordinal $\sigma$ such that $p^\sigma G=0$. See Laszlo Fuchs, Infinite Abelian GroupsInfinite Abelian Groups, vol 2: p. 58 for the definition and p. 85 for the construction of these generalized Prufer groups.
The bad news is that theTakeuti's theory Th(On) is a theory in a large language, which starts with $a=b$, $a<b$, $(a,b)$ (ordered pair), and then goes on to include $+$, $\times$ and all primitive recursive functions of ordinals. So to interpret this Th(On) in ETCG, we would at a minimum need to find formulas $\phi_\le, \phi_{\wedge}$ in ETCG such that:
Claim 3) We can characterize the abelian groups in the language of ETCG. (I expect there are mistakes in the below, but I also expect that the mistakes can be corrected.)
$1$ is the unique terminal object in the category
a morphism is constant if it factors through $1$.
$G$ is almost free iff it is not 1, and for every $H$ other than $1$, there is a non-constant map from $G$ to $H$.
$\mathbb{Z}$ is the unique almost free group with monos into all other almost free groups.
$G$ has two elements iff there are exactly two maps from $\mathbb{Z}$ to $G$.
$G$ has eight elements iff there are exactly eight maps from $\mathbb{Z}$ to $G$.
$H$ is a subgroup of $G$ iff there is a mono from $H$ to $G$.
$G/H=K$ iff there is a mono and an epi
$H$ is a normal subgroup of $G$ iff $G/H=K$ for some $K$.
$G$ is cyclic iff it is $\mathbb{Z}/H$ for some $H$.
$Q$ is the unique 8-element group which is not cyclic, but containswhich has a two-element subgroup $S$ that is contained in every subgroupwhose mono into $R$$Q$ factors through any subgroup of $Q$ other than $1$.
$G$ is abelian iff all of its subgroups are normal, and $G$$Q$ is not the direct producta subgroup of $Q$ with another group$G$.
