I have come across these two questions and I have no idea how to answer them.
(1) Give an example of a FoL= (first order logic with equality) expression that has only models of cardinality ≤ 4.
(2) Give an example of a FoL= expression that has only models of cardinality exactly 3.
What do we mean by "models of cardinality"? How should one approach these problems? This is an introduction to mathematical logic so I am not too refined with jargon. Thanks!
$\forall x_1 \forall x_2 \forall x_3 \forall x_4 \forall x_5 \bigvee_{1 \leq i \neq j \leq 5} x_i = x_j$. Every model which satisfies this formula has cardinality at most $4$, i.e. it has at most $4$ elements. Indeed, this formula means that if you "pick up" any $5$ individuals, $2$ of them are equal.
$(\forall x_1 \forall x_2 \forall x_3 \forall x_4 \bigvee_{1 \leq i \neq j \leq 4} x_i = x_j) \land (\exists y_1 \exists y_2 \exists y_3 \bigwedge_{1 \leq i \neq j \leq 3} y_i \neq y_j)$. Every model which satisfies this formula has cardinality exactly $3$, i.e. it has exactly $3$ elements. Indeed, this formula means that if you "pick up" any $4$ individuals, $2$ of them are equal, and there exist $3$ distinct individuals.