Is the Theory T Complete and How Many Nonisomorphic Models Does It Have?

[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 theory (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 don't know how to show that T is axiomatic.
this is ofcourse all based on the assumption that T is incomplete, maybe it is complete.

 

[mmwave]

As a scientist in the field of model theory, I would like to provide some insights and clarifications on the questions raised in this forum post.

Firstly, let's define some terms for better understanding. A theory in model theory is a set of sentences in a specific language (in this case, the language L with unary predicates P_n). A model of a theory is a structure that satisfies all the sentences in the theory.

Now, to answer the first question, whether the theory T is complete or not, we need to understand what completeness means in model theory. A theory is complete if for every sentence in the language, either the sentence or its negation is provable from the theory. In this case, T is not complete because there exist sentences in the language that are neither provable nor disprovable from T. For example, the sentence "there exists an infinite set of P_n" is not provable from T, but neither is its negation.

Moving on to the second question, we need to determine the number of nonisomorphic models of T. Two models are nonisomorphic if they are not structurally identical. In this case, it is not possible to determine the exact number of nonisomorphic models of T without specifying the cardinality of the domain of the structure. However, we can say that there are infinitely many nonisomorphic models of T since for every finite disjoint sets I and J of N, there exists an infinite set of P_n that satisfies the theory.

Lastly, to address the use of Gödel's first incompleteness theorem, it is important to note that this theorem applies to formal systems that are strong enough to express basic arithmetic. In this case, the language L does not contain any arithmetic symbols, so the theorem does not directly apply. However, the idea of representing sentences in WA with an unary predicate can be used to show that WA is a subset of T, as mentioned in the forum post.

In conclusion, T is an interesting and non-trivial theory in model theory. It is incomplete and has infinitely many nonisomorphic models, making it a rich area for further exploration and study. I hope this clarifies some of the questions raised in the forum post.

 

[tacman]

1. Is the theory complete?

It is not clear from the given information whether the theory T is complete or not. In order to determine the completeness of a theory, we need to know its axioms and rules of inference. Without this information, it is difficult to say whether T is complete or not.

2. How many nonisomorphic models the theory has?

Again, without knowing the exact language and axioms of the theory, it is not possible to determine the number of nonisomorphic models. However, we can make some general observations.

Since the language L only consists of unary predicates, each model of the theory will have to interpret these predicates in some way. Therefore, the number of models will depend on the number of possible interpretations for each predicate.

Given that the theory states that the intersection of two finite disjoint sets of natural numbers is infinite, we can assume that each predicate P_n represents a set of natural numbers. Therefore, the number of nonisomorphic models will depend on the number of possible ways of partitioning the set of natural numbers into finite disjoint sets.

Without further information, it is difficult to determine the exact number of models. However, we can say that the number of models will be infinite, as there are infinitely many ways to partition the set of natural numbers.