Please help me to study the following simple cases:
Let $P$ be a binary predicate symbol. I am trying to find out, if there exists a satisfiable $T$ having infinite models only, for the following cases:
1) T is under {P} language with "="
2) T is under {P} language with "=" and T is finite
3) T is under {P} language without "="
4) T is under {P} language without "=" and T is finite
Where I am:
1) $\varphi_i \equiv \forall x_1 \forall x_2 ... \forall x_i \exists y \land ^i_{j=1} \lnot( x_j=y)$
$T = \{\varphi_i | i>0 \}$ so there exists a T having only infinite models
2) 4) can probably have use of Löwenheim-Skolem theorem, but I don't see how to apply it properly.
3) No adequate idea so far