I met a strange question which gives me a Language L which only has one unary Relation R.(Set R) And it asks me to find an L-Theory which axiomizes the class of L-structure A where both R and A\R are infinite.
Is there really a way to describe infinite by those basic ->, =, etc + this R?
Hint: design a sentence $\phi_n$ for $n = 1, 2, \ldots$ that says that $R$ contains at least $n$ distinct elements. E.g., $$ \phi_3 = \exists x_1, x_2, x_3(x_1 \neq x_2 \land x_1\neq x_3 \land x_2 \neq x_3 \land R(x_1) \land R(x_2) \land R(x_3)) $$
Similarly, design a sentence $\psi_n$ for $n = 1, 2, \ldots$ that says that $A \setminus R$ contains at least $n$ elements.
Now consider the theory $T=\{\phi_1, \psi_1, \phi_2, \psi_2, \ldots\}$. What can you say about $R$ and $A \setminus R$ in a model of $T$.