So this is a theorem [1] from A Shorter Model Theory and I'm being unable to prove this when it seems like proving this would be quite intuitive and straightforward. Does anyone know a proof?

Let L be the empty signature and A an L-structure so that A is simply a set. Let X be any subset of A, and Y be a subset of dom(A) which is definable in A by a formula of some logic of signature L, using parameters from X. Then Y is either a subset of X, or the complement in dom(A) of a subset of X.

[1] Theorem 2.1.2, page 27.

  • $\begingroup$ You have written $X$ as a subset of $X$. Could you please edit it? $\endgroup$ – Kyle Dec 29 '14 at 8:45

When $L=\varnothing$ every bijection is an automorphism. If $Y$ is definable over $X$, then every bijection that fixes $X$ pointwise fixes $Y$ setwise. This can only occur if $Y\subseteq X$ or $A\smallsetminus Y\subseteq X$.

  • $\begingroup$ You probably meant to write that $Y$ contains the complement in the latter case. $\endgroup$ – tomasz Dec 30 '14 at 0:56
  • $\begingroup$ @tomasz Corrected.Thank you. $\endgroup$ – Primo Petri Dec 30 '14 at 7:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.