For every $n \in \mathbb{N}$, there is a countable complete theory with $n$ exactly models (up to isomorphism) of cardinality $\aleph_0$, if and only if $n \neq 2$.

This is known as "Vaught's never 2 theorem".

Related reading: spectrum of a theory, Vaught conjecture.

For every $n \in \mathbb{N}$, there is a countable complete theory with exactly $n$ models (up to isomorphism) of cardinality $\aleph_0$, if and only if $n \neq 2$.

This is known as "Vaught's never 2 theorem".

Related reading: spectrum of a theory, Vaught conjecture.

A countable complete theory can have $n$ non-isomorphic models of cardinality $\aleph_0$ iff $n \neq 2$.

This is known as "Vaught's never 2 theorem".

Related reading: spectrum of a theory, Vaught conjecture.

For every $n \in \mathbb{N}$, there is a countable complete theory with $n$ exactly models (up to isomorphism) of cardinality $\aleph_0$, if and only if $n \neq 2$.

This is known as "Vaught's never 2 theorem".

Related reading: spectrum of a theory, Vaught conjecture.