In the context of asking about the classification of finite simple groups, the question arose: what exactly is meant by a "classification"?

Perhaps unsurprisingly, there is in fact a whole branch of model theory called classification theory, and an epynomous book by Shelah which apparently is unreadable. My impression is that the notion of "classifiable" in this subject is that of a stable theory, but probably there are other notions I'm not aware of. Unfortunately, this notion seems best suited for talking about infinite models of a theory. Moreover, it's not clear to me that this definition gives a complete answer to my question:

What does it mean to say that the models of a theory (with some cardinality limit, particularly the case where the models are required to be finite) admit a classification?

In the context of asking about the classification of finite simple groups, the question arose: what exactly is meant by a "classification"?

Perhaps unsurprisingly, there is in fact a whole branch of model theory called classification theory, and an epynomous book by Shelah which apparently is unreadable. My impression is that the notion of "classifiable" in this subject is that of a stable theory, but probably there are other notions I'm not aware of. Unfortunately, this notion seems best suited for talking about infinite models of a theory. Moreover, it's not clear to me that this definition gives a complete answer to my question:

What does it mean to say that the models of a theory (with some cardinality limit, particularly the case where the models are required to be finite) admit a classification?

In the context of asking about the classification of finite simple groups, the question arose: what exactly is meant by a "classification"?

Perhaps unsurprisingly, there is in fact a whole branch of model theory called classification theory, and an epynomous book by Shelah which apparently is unreadable. My impression is that the notion of "classifiable" in this subject is that of a stable theory, but probably there are other notions I'm not aware of. Unfortunately, this notion seems best suited for talking about infinite models of a theory. Moreover, it's not clear to me that this definition gives a complete answer to my question:

What does it mean to say that the models of a theory (with some cardinality limit) admit a classification?

In the context of asking about the classification of finite simple groups, the question arose: what exactly is meant by a "classification"?

Perhaps unsurprisingly, there is in fact a whole branch of model theory called classification theory, and an epynomous book by Shelah which apparently is unreadable. My impression is that the notion of "classifiable" in this subject is that of a stable theory, but probably there are other notions I'm not aware of. Unfortunately, this notion seems best suited for talking about infinite models of a theory. Moreover, it's not clear to me that this definition gives a complete answer to my question:

What does it mean to say that the models of a theory (with some cardinality limit, particularly the case where the models are required to be finite) admit a classification?