2.1.10: Covered Structures for First-order Languages
Recall that a term is closed if it contains no variables.
If \(t\) is a closed term of the language \(\Lang L\) and \(\Struct M\) is a structure for \(\Lang L\) , the value \(\Value{t}{M}\) is defined as follows:
- If \(t\) is just the constant symbol \(c\) , then \(\Value{c}{M} = \Assign{c}{M}\) .
- If \(t\) is of the form \(\Atom{f}{t_1, \ldots, t_n}\) , then \[\Value{t}{M} = \Assign{f}{M}(\Value{t_1}{M}, \ldots, \Value{t_n}{M}).\nonumber\]
A structure is covered if every element of the domain is the value of some closed term.
Let \(\Lang L\) be the language with constant symbols \(\Obj{zero}\) , \(\Obj{one}\) , \(\Obj{two}\) , …, the binary predicate symbol \(<\) , and the binary function symbols \(+\) and \(\times\) . Then a structure \(\Struct M\) for \(\Lang L\) is the one with domain \(\Domain M = \{0, 1, 2, \ldots \}\) and assignments \(\Assign{\Obj{zero}}{M} = 0\) , \(\Assign{\Obj{one}}{M} = 1\) , \(\Assign{\Obj{two}}{M} = 2\) , and so forth. For the binary relation symbol \(<\) , the set \(\Assign{<}{M}\) is the set of all pairs \(\tuple{c_1, c_2} \in \Domain{M}^2\) such that \(c_1\) is less than \(c_2\) : for example, \(\tuple{1, 3} \in \Assign{<}{M}\) but \(\tuple{2, 2} \notin \Assign{<}{M}\) . For the binary function symbol \(+\) , define \(\Assign{+}{M}\) in the usual way—for example, \(\Assign{+}{M}(2,3)\) maps to \(5\) , and similarly for the binary function symbol \(\times\) . Hence, the value of \(\Obj{four}\) is just \(4\) , and the value of \(\times(\Obj{two}, +(\Obj{three},\Obj{zero}))\) (or in infix notation, \(\Obj{two} \times (\Obj{three} + \Obj{zero})\) ) is \[\begin{gathered} \begin{aligned} \Value{\times(\Obj{two}, +(\Obj{three},\Obj{zero})}{M} & =\Assign{\times}{M}(\Value{\Obj{two}}{M}, \Value{+(\Obj{three}, \Obj{zero})}{M})\\ & = \Assign{\times}{M}(\Value{\Obj{two}}{M}, \Assign{+}{M}(\Value{\Obj{three}}{M}, \Value{\Obj{zero}}{M})) \\ & = \Assign{\times}{M}(\Assign{\Obj{two}}{M}, \Assign{+}{M}(\Assign{\Obj{three}}{M}, \Assign{\Obj{zero}}{M})) \\ & = \Assign{\times}{M}(2, \Assign{+}{M}(3, 0)) \\ & = \Assign{\times}{M}(2, 3) \\ & = 6 \end{aligned}\end{gathered}\]
Is \(\Struct N\) , the standard model of arithmetic, covered? Explain.