(Lindenbaum) Let $\displaystyle T$ be a decidable consistent theory (in a reasonable language). Show that $\displaystyle T$ can be extended to a complete decidable consistent theory $\displaystyle T^{'}$. Suggestion: Examine in turn each sentence $\displaystyle \sigma$; add either $\displaystyle \sigma$ or $\displaystyle \neg \sigma$ to $\displaystyle T$. But take care to maintain decidability.

I think the corresponding definition of decidability is given in page 62 of the textbook.

Any hint to start this problem?

Thanks.