# Questions on model theory.

1. Jun 17, 2007

### MathematicalPhysicist

question in model theory.
let L={P_n|n \in N}
every P_n is an unary predicate. let's define a theory that says that every two finite disjoint sets I and J of N, such that the intersection (^(i \in I)P_i)^(^(j \in J)~P_j) ("^" means conjunction) is infinite.
1. is the theory complete?
2. how many nonisomorphic models the theory has?

well, i think that the thoery (which i shall denote it by T) is incomplete, one way to show this is to use godel's first incompleteness theorem, so i need to prove that T is axiomatic, and the weak arithematics theory is a subset of it.
to show that WA (weak arithematics) is a subset of T, i think that every closed formula in WA can be written with 2-place predicate "<" and suitable quantifiers and connectives, so i need to show that we can represent them with an unary predicate, not sure how to do it.
i dont know how to show that T is axiomatic.
this is ofcourse all based on the assumption that T is incomplete, maybe it is complete.