I apologize if the title of this post is in any way incorrect, please feel free to point it out.
Let $\Gamma$ be the set of sentences of the language of arithmetic that are true in the standard model of arithmetic. Let $c$ and $d$ be two distinct constants that do not occur in $\Gamma$.
Let $\Delta$ be the set of sentences $\{ \exists x(2*x=c)\}\cup\{c\not = 2n : n\in\mathbb{N}\}$
Let $\Sigma$ be the set of sentences $\{\exists x(2*x+1=d\}\cup\{d\not = 2n+1 : n\in\mathbb{N}\}$
Show that there is a countable model of $\Gamma\cup\Delta\cup\Sigma$.
Let $\mathcal{M}$ be a countable model for $\Gamma\cup\Delta\cup\Sigma$ and let $c^\mathcal{M}$, $d^\mathcal{M}$ be the denotations of $c,d$ in $\mathcal{M}$. Is $c^\mathcal{M} + d^\mathcal{M}$ odd or even in $\mathcal{M}$? To be precise, is the sentence $\exists x(2x=c+d)$ true in $\mathcal{M}$ or is the sentence $\exists(2x+1=c+d)$ true in $\mathcal{M}$ ?
I honestly am not sure how to attack this problem. Is it possible for someone to offer insight on how to begin this proof? Maybe once I have inspiration I can piece the rest of it together. There's a hint: $\Gamma$ contains all the sentences that are true of the natural numbers.