I see this sentence in one Logic Note Tutorial.
What arguments are involved in any situation is determined by the meaning of the predicate. Sleeping can only involve one argument, whereas placing naturally involves three. We can distinguish predicates in terms of how many arguments they involve: sleep is a one-place predicate, see is a two-place predicate involving two arguments and place is a three-place predicate.
Suppose P is a Two-place position predicate, and have:
all models of $\varphi$ is infinite.
I try to check the validity of this sentence, but failed. infact I think this is false, but no idea.
Tru with a domain $D = \{ a, b \}$ with only two elements, and try to satisfy the formula [you will not succeed ...].
Then try with a domain $D = \{ a_n \}_{n < \omega}$ , with $n$ whatever, and perform the same check : again the answer will be negative.
In this way, you will convince ourself that the formula is not satisfiable in a finite domain.
Finally, check it with the set $\mathbb N$ of natural numbers, with the usual ordering : $<$, and you will find that the above formula is satisfiable.