Definition:Let M and N be V-structures. If M and N models the same V-sentences, then M and N are said to be elementarily equivalent, denoted $M \equiv N.$

Example: the $V_{<}$-structures $\mathbb{Q}_{<}$ and $\mathbb{R}_{<}$ are elementairly equivalent.

My question:Is not this sentence: $\forall x((x = \pi) \implies$ (x =x))? and the second structure modeles it? while the first one doesnt as $\pi$ not in $\mathbb{Q}$!


Is $\pi$ even a symbol in $V$? If not this is not a sentence to begin with.

Note that if $\pi$ is in the language then it has to have some interpretation in $\Bbb Q$, but that won't be the same number you think about, but rather some rational number.

  • $\begingroup$ that was cool. Well, how can I prove that they are elementairly equivalent.? $\endgroup$ – Lo52 Sep 12 '13 at 0:59
  • 3
    $\begingroup$ You can use something like the Tarski-Vaught criterion to show the rationals are an elementary submodel of the reals. Or you can show that the theory of dense ordered sets without endpoints is complete, and therefore every two models are elementarily equivalent. $\endgroup$ – Asaf Karagila Sep 12 '13 at 1:00

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