Finite Model Theory is a branch of mathematical logic that deals with the study of the expressive power and complexity of logical languages restricted to finite structures. Although it has proven to be a useful tool in various fields such as database theory, computer science, and philosophy, there are also some objections to this theory. Some of the common objections are:

1. Limited Applicability: Some argue that Finite Model Theory has limited applicability because many real-world problems involve infinite structures or infinite domains, which cannot be captured by the finite model theory.
   
   
2. Overly Simplistic: Critics argue that the Finite Model Theory oversimplifies the complexity of logical languages by restricting them to finite structures, which may not accurately represent the complexities of the real world.
   
   
3. Incomplete: Some argue that Finite Model Theory is an incomplete theory because it does not provide a complete characterization of the complexity of logical languages. It only focuses on the complexity of specific classes of logical languages.
   
   
4. Limitations in Expressive Power: Some argue that the expressive power of logical languages that can be captured by Finite Model Theory is limited, which may prevent it from being useful in certain contexts.

These objections are not unique to Finite Model Theory, and similar criticisms can be raised against other areas of mathematical logic or computer science.

"For every finite model there is one greater"

The notion that "for every finite model there is one greater" does not necessarily invalidate the theory but is a valid criticism in that the notion of a finite model is an arbitrary limitation on the size of a model.

While one of the main ideas behind Finite Model Theory is to study the complexity of logical languages over finite structures, which are finite models with a finite number of objects and relations, it is true that finite models impose a limit on the size of the structures that can be studied. This can be seen as an arbitrary limitation, especially if we consider problems or systems that require very large or infinite structures to be properly modeled.

The study of infinite structures falls under a different area of research, such as Set Theory or Model Theory with infinite structures.

One way to extend the interpretation of a theory under Finite Model Theory to include the study of Set Theory or Model Theory is to consider a hierarchy of models, where each level of the hierarchy consists of models of increasing size. At the lowest level, we have the finite models that are the primary focus of Finite Model Theory. As we move up the hierarchy, we consider larger and larger models, eventually reaching infinite models such as the models used in Set Theory or Model Theory.

In this way, the techniques and ideas developed in Finite Model Theory can be used to study the complexity of logical languages over models of varying sizes. For example, by studying the complexity of logical languages over finite models, we can develop techniques and insights that can be applied to the study of infinite models. Conversely, by studying the complexity of logical languages over infinite models, we can gain a better understanding of the limits and possibilities of the techniques developed in Finite Model Theory.

The study of Set Theory and Model Theory involves many techniques and ideas that are distinct from those used in Finite Model Theory, but there are also many connections and overlaps between the two fields. By extending the interpretation of a theory under Finite Model Theory to include the study of Set Theory or Model Theory, we can explore these connections and gain a deeper understanding of the complexity of logical languages over models of various sizes.

FactEngine works with the view that Ehrenfeucht-Fraïssé games extended to a higher order may be used to extend the interpretation of a theory under Finite Model Theory to include the study of higher-order logics, if we assume no arbitrary restriction on the size of the model covered by the theory, and to gain insights into the complexity of logical languages over models of varying sizes, including infinite models.