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I am working through some exercises from Marker's Model Theory in self-study and I am stuck at Exercise 3.4.1 as I do not know how to formally prove that a theory has quantifier elimination. I am aware of the definitions and possible checks but I cannot come up with a formal proof or an intuitive way to check if they do not have quantifier elimination.

Can someone help me by guiding me through the exercise? I would appreciate the effort as I am really trying to learn. Thank you very much.

Let $L = \{E\}$ where $E$ is a binary relation symbol. For each of the following theories either prove that they have quantifier elimination or give an example showing that they do not have quantifier elimination and a natural $L'\supset L$ in which they do have quantifier elimination.

a) E has infinitely many equivalence classes all of size $2$.

b) $E$ has infinitely many equivalence classes classes all of which are infinite.

c) $E$ has infinitely many equivalence classes of size $2$, infinitely many classes of size $3$, and every class has size $2$ or $3$.

d) $E$ has exactly one equivalence class of size $n$ for each $n < \omega$.

Alex Kruckman
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bluci
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    @William Elliot: I agree that the [relations] tag isn't appropriate here. But why did you remove [logic]? – Alex Kruckman May 23 '20 at 16:55
  • Welcome to Math Stackexchange. Please use MathJax (similar to LaTeX) formatting on this site. I've edited to add this formatting to your post. Here is a tutorial: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Alex Kruckman May 23 '20 at 16:59
  • Do you already know that, to prove quantifier elimination for a theory, it suffices to eliminate the quantifier from formulas of the simple form $\exists x,\phi(x,\vec y)$ where $\phi$ is a conjunction of atomic formulas and negations of atomic formulas? Since your $L$ has only one relation symbol (along with equality), there aren't very many atomic formulas. Try to eliminate the quantifier from $\exists x,\phi$ for the various possible $\phi$'s. If it works, you win. If it doesn't work, that will suggest what to add to $L$ to produce a suitable $L'$. – Andreas Blass May 24 '20 at 02:05
  • @AlexKruckman ok thank you, I will consider that next time! – bluci May 24 '20 at 09:05
  • @AndreasBlass, yes I do know that, however the proofs I have looked at construct some kind of back-and-forth argument (for example for dense linear orders) but I am struggling to come up with a formal proof construction like this myself. – bluci May 24 '20 at 09:07
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    I think you might be conflating two different proofs of the completeness of the theory of dense linear orders without endpoints. One proof goes via $\aleph_0$-categoricity and Vaught's test for completeness; the categoricity is established by a back-and-forth argument. The other proof goes via quantifier-elimination; I don't see any back-and-forth in this second proof. And the second is what you should imitate here, since you're explicitly asked to prove quantifier-elimination. – Andreas Blass May 24 '20 at 12:06
  • For parts (c) and (d) you can take some inspiration from here. – Rick Jun 19 '20 at 15:26

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Hint: For the negative results, note that the set of elements whose classes have $n$ elements for a fixed finite $n$ is definable, but often not quantifier-free definable. On the other hand, after you make these sets quantifier-free definable, you can just prove q.e. directly, like in this example.

tomasz
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