Let $T$ be a theory of Abelian groups where every element has order 2 find complete theory include T

Question

This question is from Marker's book . Let $T$ be a theory of Abelian groups where every element has order 2 . Show that it is not complete . Find $T' \supset T $ a complete theory with the same infinite model as $T$.

I showed it is not complete but i can't find $T'$.I don't want whole answer just give me clue .

Answer 1

HINT:

Note that every such group $G$ is a vector space over $\Bbb F_2$ (the field of two elements). Add to $T$ the sentences "There are at least $n$ elements in the universe", for every $n\in\Bbb N$. Then $T'$ is the theory of an infinite dimensional vector space over $\Bbb F_2$.