TITLE: Models of the theory of real closed fields with extra constants
QUESTION [6 upvotes]: Let $L$ be the language of real closed fields $\{0,1,+,-,\times\}$, and $T$ the theory of real closed fields, i.e. the $L$-sentences that are true in the standard $\mathbb{R}$ model. Let $L'$ be $L$ extended with a countable set of constants. 
If $\Gamma$ is a satisfiable countable set of $L'$ sentences that contains $T$, must it have a model that can be obtained from the standard $\mathbb{R}$ model of $L$ by assigning a real to each of the added constants? 
If yes, is the same true when there are $|\mathbb{R}|$-many added constants? When there are $|\mathbb{R}|$-many added constants and when $\Gamma$ has larger cardinality?
If you create a new theory of a model by augmenting the language with constants, does the new theory contain sentences that says anything "original"? is a related question.

REPLY [5 votes]: The other two answers are absolutely correct; let me give an answer in a slightly different direction. Say that a theory $T$ in the language of ordered fields + constants is $\mathbb{R}$-satisfiable if it has a model which is an expansion of $\mathbb{R}$ with the usual ordered field structure; I'm interested in the model-theoretic properties of $\mathbb{R}$-satisfiability.
One natural question to ask is, "Is there a compactness theorem for $\mathbb{R}$-satisfiability?" The answer is no (and this provides another counterexample to the question you ask). Consider the theory $T$ consisting of the axioms of real closed fields, together with 

$c_i+1<c_{i+1}$ for $i>0$, and 
$c_0>c_i$ for $i>0$.

Then every finite subset of $T$ is $\mathbb{R}$-satisfiable, but $T$ itself is not, since $c_0$ would have to be infinite.

What about "compactness in other cardinalities"? Say that $\mathbb{R}$-satisfiability is $(\kappa, \lambda)$-compact if whenever $\Gamma$ is a set of sentences of cardinality $<\lambda$, and every subset of cardinality $<\kappa$ is $\mathbb{R}$-satisfiable, then $\Gamma$ is $\mathbb{R}$-satisfiable. (So usual compactness is $(\omega, \infty)$-compactness, and countable compactness is $(\omega,\omega_1$)-compactness.)
Is $\mathbb{R}$-satisfiability $(\omega_1, \omega_2)$-satisfiable? No! Given $\omega_1$-many constants $c_\eta$ ($\eta<\omega_1$), consider the theory $S=\{c_\alpha<c_\beta: \alpha<\beta\}$. Every countable subset of $S$ is $\mathbb{R}$-satisfiable, since any countable linear order embeds into $\mathbb{R}$, but $S$ itself is not $\mathbb{R}$-satisfiable: if $c<d$ are reals, then there is a rational $c<q<d$, so the $\mathbb{R}$-satisfiability of $S$ would imply that the rationals are uncountable.
Is $\mathbb{R}$-satisfiability $(\omega_2,\omega_3)$-satisfiable? Not provably in ZFC! Suppose CH holds - that is, $\vert\mathbb{R}\vert=\aleph_1$ - and we have constant symbols $c_\alpha$ for $\alpha<\omega_2$. Then $\{c_\alpha\not=c_\beta: \alpha\not=\beta\}$ is not $\mathbb{R}$-satisfiable, but every subset of size $<\omega_2$ is $\mathbb{R}$-satisfiable. (Similarly, for any cardinal $\kappa$ of uncountable cofinality, it is consistent with ZFC that $\mathbb{R}$-satisfiability is not $(\kappa^+,\kappa^{++})$-compact.
This leaves two interesting questions:

Can we prove in ZFC that $\mathbb{R}$-satisfiability is $(\omega_{\omega+1},\omega_{\omega+2})$-compact?
Is it consistent with ZFC that $\mathbb{R}$-satisfiability is $(\omega_2, \omega_3)$-compact?

Off the top of my head I don't know the answer to either question.
If you're interested in this sort of question, you might enjoy looking at abstract model theory.

Note: the general question of when an equational theory has a model compatible with a particular topological space has been extensively analyzed in universal algebra, particularly by Walter Taylor, extending the classic theorem of algebraic topology that $S^2$ does not admit a group structure.