How to axiomatize "the domain of a model has at most $n$ elements" in a first order language?

Question

For example, to axiomatize $\phi_n$ translating to "the domain has at least $n$ distinct elements", I can write $\exists x_1 ... \exists x_n[\land_{i\geq1}\land_{j>i}\neg(x_i=x_j)]$.

How can I axiomatize $\theta_n$ to mean "the domain has at most $n$ distinnct elements"?

Answer 1

$$\theta_n \equiv (\exists x_1, \ldots, x_n)(\forall x) \, \bigvee_{i=1}^n x = x_i$$

Of course this also asserts there is at least one element in the domain, but that's a usual assumption in model theory, so perhaps it's all right. Otherwise take $\theta_n \vee \: (\forall x) \, x \neq x$.

Answer 2

$$ \forall x_0,\dots,x_n(\bigvee_{i, j \in \{0,..,n\}, i\neq j} x_i=x_j) $$

or, you just negate your solution for "there exist at least $n+1$"