
## 3.1 Events And Their Occurrences

The following definition will give us the classification of all models of T*univ*.

Definition 1. Let Muniv denote the class of all possible models for Tuniv*. Then any* model M ∈ Muniv consists of the following sets E, EO, and O such that 1. the universe M of M is E ∪EO∪O
2. E, EO, and O are pairwise disjoint 3. E is a partitioning of the set EO.

4. E = eventM
5. EO = event occurrenceM
6. O = observationM
7. occurrenceM = {(x,e) ∈ EO×E : x ∈ e}
The correctness of our definition follows from the following theorem.

Theorem 3 (Satisfiability Theorem for Tuniv). If the class Muniv is defined as in Definition 1, then for any model M ∈ Muniv, we have M |= Tuniv.

Proof. The fact that M satisfies axioms (2.1) and (2.2) follows from the condition that E, EO, and O are pairwise disjoint. The fact that M satisfies axioms (2.3) and (2.5)
follows from our construction that E is a partitioning of the set EO and the interpretation of occurrence as the membership relation between EO and E. ⊓⊔
Theorem 4 (Axiomatizability Theorem for Tuniv). Any model of Tuniv is isomorphic to a structure of Muniv.

Proof. Let M be a model of T*univ*. We will show that M satisfies the conditions of the structures in M*univ* from Definition 1.

Since M |= (2.1) , we know that any element of the universe M of M belongs to one of the following sets eventM , event occurrenceM and observationM . Since M |=
(2.2), all of these sets eventM , event occurrenceM and observationM are pairwise disjoint. Hence, the conditions (1), (2), (4)–(6) are satisfied.

Since M satisfied axioms (2.3)–(2.4), we know that occurrenceM a function occurrenceM : event occurrenceM → eventM
Hence, given the set EO = event occurrenceM , we can define the set E as E
df= {{x : f(x) = e} : e ∈ eventM }
Since occurrenceM is a function, it can be easily checked that E defines a partitioning of EO. Thus, the condition (3) and (7) are also satisfied. ⊓⊔