Gödel's First Incompleteness Theorem By Gödel's First Incompleteness Theorem (in Enderton's "A Mathematical Introduction to Logic", p. 236) if A ⊆ Th R and #A is recursive, then Cn A is not a complete theory. 
Proof: Since A ⊆ Th R, we have Cn A ⊆ Th R. If Cn A is a complete theory, then equality holds. But if Cn A is a complete theory, then #Cn A is recursive (which it is not, since #Th R is not definable.
If Cn A is complete then we have Th R ⊆ Cn A. Is this because, if Cn A is complete then for every φ we have φ or ~φ, and hence a fortiori, Cn A, if complete, WOULD decide whether φ or ~φ belong to Th R?
 A: For preliminaries, see page 155-56 :

We define a theory to be a set of sentences closed under logical implication.
  That is, $T$ is a theory iff $T$ is a set of sentences such that for any
  sentence $σ$ of the language,

$T \vDash σ \implies σ \in T$.


For a class $\mathcal K$ of structures (for the language), define the theory of $\mathcal K$ (written $\mathsf {Th}\mathcal K$) by the equation


$\mathsf {Th}\mathcal K = \{ σ \mid σ \ $ is true in every member of $ \ \mathcal K \}$.


And :

$\mathsf {Cn} \Sigma = \{ σ \mid \Sigma \vDash σ \} = \mathsf {Th Mod} \Sigma$.
For example, set theory is the set of consequences of a certain set of sentences, known, unsurprisingly, as axioms for set theory.
A theory $T$ is said to be complete iff for every sentence $σ$, either $σ \in T$ or $(¬σ) \in T$. For example, for any one structure $\mathfrak A, \mathsf {Th} \{ \mathfrak A \}$ (written, as before, “$\mathsf {Th} \mathfrak A$”) is always a complete theory.

$\mathfrak N$ is the intended structure for the language of number theory, and by number theory we mean the theory of this structure, $\mathsf {Th} \mathfrak N$ [page 182].
Now for page 236 : $A$ is a set of sentences true in $\mathfrak N$ (this means : $A \subseteq \mathsf {Th} \mathfrak N$) and obviously $\mathsf {Cn} A \subseteq \mathsf {Th} \mathfrak N$.
If $\mathsf {Cn} A$ is complete, then we have two complete sets of sentences for the same structure $\mathfrak N$ and they must coincide. 
