Let $A$ be a structure in the language $L$ and $c_1,c_2...,c_n$ new constants.
- Show that $A$ can be extended to a structure in the language $L' = L $ $\cup$ $\{c_1,c_2...,c_n\}$.
I know that for every $L$-structure $A$, the language can be extended by new constants for elements of m. The language extension $L'$ consists of L together with new constants cm for each element $ m \in M$, and $A$ becomes an $L'$-structure $A_M$ by $c_mM = m$
but how can i show that formally ? any ideas?
A (non-empty) structure can always be extended to a larger language with more constant symbols. All we need to do is interpret the new constants, and they can be interpreted as any elements of the original structure you wish to get a structure in the larger language. For instance if the structure is $\mathbb N,$ with some interpretations for the symbols in the original language, we can just interpret the new symbols as all being $0,$ or all being $27,$ or we can make $c_1^{\mathbb N}=10,$ $c_2^{\mathbb N}=162$... literally any assignment is fine.
Where things aren't quite so simple as this is when we add new axioms involving the constant symbols and demand the new structure satisfy these axioms. For instance, if our base structure is $A=(\mathbb N, \le)$ and we add a single new constant $c,$ to the language, and we consider the theory $Th(\mathbb N, \le )$ along with the axioms $\{n\le c:n\in\mathbb N\}$, then we cannot find an assignment for $c$ such that $(\mathbb N, \le , c^{\mathbb N})$ satisfies this theory. Any assignment makes a perfectly good structure, just not a model of the theory. On the other hand, if we instead only add a finite number of axioms $\{n\le c:n\le k\},$ then $(\mathbb N, \le , c^{\mathbb N})$ will satisfy this theory provided we choose $c^{\mathbb N}\ge k.$