This is a homework question so please don't give the full answer, just an approach will do.

Question: Let $T=D_u\cup\{c_i<c_{i+1}\mid i\in\mathbb N\}$ where $D_u$ is the theory of dense linear orders without end points.

(a) Show that $T$ has three non-isomorphic models of size $\aleph_0$, (you may use that $D_u$ is $\omega$-categorical).

(b) Show that $T$ is complete.

(a) was easy enough but I'm stuck on (b).

Now my first attempt was to simply say "$T$ is consistent because it is satisfiable, $D_u$ is complete by Vaught's test, so $T$ is complete." That's missing a couple of steps but you get the idea.

Then I realised that $D_u$ isn't actually complete here, because the language is $\{<\}\cup\{c_i\mid i\in\mathbb N\}$, which means we have sentences like $c_1<c_2$.

So because $D_u$ is complete with respect to $\{<\}$, the only problem sentences can be those that involve constants, but I don't see where to go from there.

  • $\begingroup$ Care to explain how the EF games would help, I'm in the same class and I'm stuck with the same problem and still don't see the answer, even when looking at EF games. A bit more help would be appreciated! $\endgroup$ – matti0006 Feb 22 '14 at 19:47
  • $\begingroup$ Hint: if any two models of $T$ are elementarily equivalent, then $T$ must be complete. EF games are a way of showing that structures are elementarily equivalent. But please don't post comments as answers. $\endgroup$ – Alex Kruckman Feb 22 '14 at 19:52
  • $\begingroup$ sorry, I don't have enough reputation to comment:(. I didn't know what you said, but if that's the case I see what I should do now, thanks a bundle! $\endgroup$ – matti0006 Feb 22 '14 at 19:52

I think for part a) you also want that "non-isomorphic models of size $\aleph_{0}$". For b) a back and forth won't work (since the models are not isomorphic). What about using EF games?

  • $\begingroup$ Ah thank you, it turned out to be pretty easy once I read up on EF games. $\endgroup$ – user99412 Feb 22 '14 at 19:39
  • $\begingroup$ you are welcome! $\endgroup$ – UserB1234 Feb 23 '14 at 19:13

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