Language

Last updated Nov 1, 2022

A Language $\mathcal{L}$ is a Tuple $(\mathcal{F}, \mathcal{R}, \mathcal{C}, n_{\mathcal{R}}, n_{\mathcal{F}})$ where

1. $\mathcal{F}$ is a Set of Function Symbols
2. $\mathcal{R}$ is a Set of Relation Symbols
3. $\mathcal{C}$ is a Set of Constant Symbols
4. $n_\mathcal{R} : \mathcal{R} \to \mathbb{N}$ is a Function that maps each Relation Symbol to it’s respective Arity

1. Languages do not actually give any meaning to their Symbols. This comes from a Language Structure
2. When we write out a Language, we often omit the Tuple structure of it and simply write out all symbols in a single Set. It is usually clear enough which ones are Relation Symbols, Function Symbols, or Constant Symbols.

1. Language of Rings: $\mathcal{L}_{R} = {+, -, \cdot, 0, 1}$.
2. Language of Pure Sets: $\mathcal{L} = \emptyset$.
   • I think we take it as given that we can maniuplate Languages in terms of set theory. In that case, this makes sense, because there is no more data we need to describe Pure Sets.
3. The Language of Directed Graphs: $\mathcal{L} = {R}$. Here $R$ is a Relation that represents the Directed Graph.