# Finite Model Theory/FO EFM

The method for employing Ehrenfeucht-Fraisse-Games for (in-)expressibility-proofs is given by the following:

**Theorem**

Let P be a property of finite σ-structures. Then the following are equivalent

- P is not expressible in FO
- for every k there exist two finite σ-structures and , such that the following are both satisfied
- has P and does not have P

**Remarks**

- Thus using the EFM works roughly as follows:
- choose a k
- construct two structures - one with the property, one without - that are big enough s.t. the duplicator wins the k-ary EFG
- show that this can be expanded with k

- So, a non-expressible property (i.e. the effort to check it) must be somehow 'expandable' with k

**Examples**

- To begin pick two linear orders say A ={1, 2, 3, 4} and B ={1, 2, 3, 4, 5}. For a two-move Ehrenfeucht game D is to win, obviously. This gives us two structures that satisfy the above conditions for k = 2 and the Property having even cardinality (that A has and B doesn't). Now we have to expand this over all k . From the above example we adopt that in a linear order of cardinality or higher D has a winning strategy. Thus we choose the cardinalities depending on k as |A| = and |B| = +1. So we have found an even A and an odd B for every k, where D has a winning strategy. Thus (by the corollary) having even/odd cardinality is a property that can not be expressed in FO for finite σ-structures of linear order.