# Countable models of first order theory of rationals

There is exactly one countable model (upto isomorphism) of the first order theory of $$(\mathbb Q,<)$$.

I am reading about spectrum of complete theories where the spectrum of various theories are stated (but not proved). I am looking for a reference where the above statement is proved, so that I can prove the other statements similarly, and so I can understand how can one prove such statements.

## 2 Answers

But that initial claim [before it was edited] isn't true. You know from the upward Lowenheim-Skolem theorem that any theory with a countable model also has models of all larger cardinalities (which therefore won't be isomorphic with the countable model).

What is true is that all countable models of the theory of dense linear orders without endpoints look the same. It is still false that there is exactly one model: there are still many models. But at least these, the countable models, are all isomorphic.

That special result is proved in any model theory text! Or googling we find e.g. https://www.math.ucsd.edu/~sbuss/CourseWeb/Math260_2012WS/Mar05Ricketts.pdf

• You're right, I was sloppy. I have edited my question. I couldn't find the proof of the result in most model theory texts I looked (eg: Tent and Ziegler, Hodges) Commented Nov 15, 2021 at 13:48
• @modeltheory In Tent-Ziegler this is the first example after Theorem 2.3.4. I don't have a copy of Hodges available right now but I'd be very surprised if it's not in it Commented Nov 15, 2021 at 14:09
• @AlessandroCodenotti You're right. The statements in the texts were actually stated in terms different to what I'm familiar to, so I was not able to find the proof until Peter's posted the link. Commented Nov 15, 2021 at 14:16
• @modeltheory Indeed it's also in Hodges: Example 3 in Section 3.2 (p. 79 in A Shorter Model Theory). Peter Smith is not exaggerating when he writes that the result is proved in any model theory text. It is the original and canonical example of a back-and-forth argument, and the back-and-forth technique is central to model theory. I would be very surprised to find any introductory model theory book which did not contain a proof! Commented Nov 15, 2021 at 14:22

The statement is proved in Theorem 2.5 here.

• Thank you. The proof here was easier to understand than others I was finally able to find. Commented Nov 15, 2021 at 14:17