Definition of a Denumerably categorical set of sentences

Question

Boolos, Burgess, and Jeffrey in "Computability and Logic" on page 147 define

A set $\Gamma$ is denumerably categorical if any two denumerable models are isomorphic.

What does this definition mean? (As stated: If you can find any two isomorphic models, then the set $\Gamma$ is denumerably categorical.)

My question is in the other direction: Is it correct to say: if a set of sentences is denumerably categorical, then all denumerable models are isomorphic?

Thanks

Answer 1

It seems like you're reading "any two denumerable models are isomorphic" to mean "there exist denumerable models $M$ and $N$ such that $M$ and $N$ are isomorphic."

But the intended meaning of "any two denumerable models are isomorphic" is "for all $M$ and $N$, if $M$ and $N$ are denumerable models, then $M$ and $N$ are isomorphic".

This is an unfortunate ambiguity in natural language.

Is it correct to say: if a set of sentences is denumerably categorical, then all denumerable models are isomorphic?

Yes.

I'll also point out that the term "countable" is much more common than "denumerable" for a structure with domain of size $\aleph_0$. And correspondingly, the term "countably categorical" (or "$\aleph_0$-categorical", or sometimes "$\omega$-categorical") is much more common than "denumerably categorical".

Answer 2

The fragment you quote is a definition.

My question is in the other direction: Is it correct to say: if a set of sentences is denumerably categorical, then all denumerable models are isomorphic?

Yes, by definition.

What does this definition mean?

It means that if there exists a countable model of $\Gamma$, then any other countable model of $\Gamma$ is isomorphic.

The definition you quote is just a special case of categoricity in power, you might try searching for that if you want to understand it.