To determine if there exists a complete, finitely axiomatizable first-order theory ( T ) with exactly 3 countable non-isomorphic models, we can rely on key results in model theory, specifically the Engeler-Sacks-Procopescu theorem.

Key Theorem:

The Engeler-Sacks-Procopescu theorem states that a complete, finitely axiomatizable first-order theory ( T ) is either ω-categorical (i.e., h...

Yes. A very classical way to get exactly three nonisomorphic countable models (the Ehrenfeucht phenomenon) can be carried out in a finite language, and in fact by a single first-order sentence.

Here is one explicit construction.

Language

• One binary relation symbol < (intended as a linear order).
• One unary function symbol f (intended as an order automorphism).

Axioms (a finite list)

• < is a...

Complete, Finitely Axiomatizable Theory with Exactly 3 Countable Models

This is an interesting question at the intersection of model theory, finite axiomatizability, and Vaught's conjecture. Let me provide what's known about this situation:

Short Answer

No such theory exists. A complete, finitely axiomatizable first-order theory cannot have exactly 3 countable models up to isomorphism. ...

This is an excellent question about the intersection of finite axiomatizability and the spectrum problem in model theory.

Answer: No, such a theory cannot exist.

Here's why:

Theorem (Ryll-Nardzewski): A complete theory in a countable language is -categorical (has exactly one countable model up to isomorphism) if and only if for each , there are only finitely many -type...

No such example is known at present, and it is not known whether one can exist.
In fact the problem is open already for the weaker requirement “finite language’’ – nobody has yet produced a complete theory in a finite language that has a finite number > 1 of non-isomorphic countable models, and nobody has been able to prove that such a theory is impossible.

Background

1. Let I(T,ℵ₀) be the n...

No. In fact, all of the “classical’’ constructions of complete theories with exactly three (or, more generally, finitely many >1) countable models use an infinite axiom‐schema (or an infinite language) and one does not know how to collapse them to a single finite axiom in a finite relational language. To this day no one has produced a complete, finitely axiomatizable first‐order T with exactly t...

Excellent question. This delves into a classical and very important part of model theory concerning the number of countable models of a theory.

The answer to your question is no, such a theory does not exist.

The key result that resolves this is a cornerstone of model theory related to Vaught's Conjecture.

The Main Theorem

A complete, finitely axiomatizable theory in a countable first-...

Nobody knows.

More precisely, the following two questions are still open.

1. Does there exist a complete first–order theory in a countable language that has a finite number > 1 of countable, pairwise non-isomorphic models (in particular, 2 or 3)?
2. Does there exist such a theory that can in addition be axiomatized by finitely many sentences (equivalently, by a single sentence)?

Your questi...