Language

Language

A (mathematical) language $\mathcal{L}$ for a structure can be represented as follows: $\mathcal{L}=⟨r_i,f_j,c_k{⟩}_{i\in I,j\in J,k\in K}$ a set of relation symbols $\{r_i:i\in I\}$, each of some assigned arity, a set of function symbols $\{f_j:j\in J\}$, each of some assigned arity, a set of constant symbols $\{c_k:k\in K\}$.

Remark

In Language we’ll know the “type” of relation symbols, for example whether it’s a binary relation or it’s ternary relation. Important Distiction: The relation/function/constant symbols in a formal language $\mathcal{L}$, and their interpretation as actual relations/functions/constants in a structure A, should not be confused with each other.

Cardinality of a Language

The cardinality of a language $\mathcal{L}$ is $|\mathcal{L}|:=\max \{{\aleph }_0,|I|,|J|,|K|\}$