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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 .

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  • $\begingroup$ That's a really cute question! $\endgroup$ – Asaf Karagila Jan 27 '14 at 23:33
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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 $\Bbb F_2$.

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  • $\begingroup$ I know it is vector space but without "There are at least n elements in the universe" sentence it is not complete also thanks $\endgroup$ – rmznyzgyr Jan 27 '14 at 23:36

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