Existance of an (in)finite theory having infinite model 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
 A: Hint: it is possible to axiomatize that $P$ is the comparison relation for a linear order (i.e., it plays the role of "$\leq$") without endpoints. Think about how those axioms depend on the presence of the equality relation, and how you might modify them to eliminate that dependency while still ensuring your models are infinite.
A: To make Mike's answer just a little more explicit:
When $P$ is the inequality symbol of a partial order, it interprets equality in the sense that the definable set $\{m \in M \operatorname{ | } P(m,m)\}$ is exactly the diagonal relation $\{m \in M \operatorname{ | } m = m\}$.
In particular, we know that such a $P$ must satisfy: $$\forall a \forall b, P(a,b) \land P(b,a) \rightarrow P(a,a),$$
$$\forall a \forall b \forall c, P(a,b) \land P(b,c) \rightarrow P(a,c),$$
$$\forall a, P(a,a).$$
After writing these out, can you think of just one more axiom we can add to $T$ such that any chain in our partial order must be infinite? (Hint: given an element $a$ in a partial order $M$, the set of all elements $b$ less than $a$ is $a$-definable.)
