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On page 307 of Vaught's paper "Denumerable models of complete theories", theorem 2.1.2 states that there is a denumerable model $\frak{A}$ of $T$ such that if, for each $j\in \omega, \:P_j$ is a non-principal prime ideal (now called ultrafilter) of $F_{p_j+1}(T)$, then there is a denumerable model $\frak{A}$ of $T$ such that $P_0(\mathfrak{A}), P_1(\mathfrak{A}), \cdots$ are all empty. Then Vaught said that 2.1.2 was proved by Ehrenfeucht.

I wonder which of Ehrenfeucht's papers contains theorem 2.1.2.

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    $\begingroup$ I don’t know what would be the original source, but if I guess correctly what all the notation means, then this is just the omitting types theorem (stated for complete types). $\endgroup$ Apr 6, 2021 at 7:19
  • $\begingroup$ I know it is (original of) omitting types theorem. It looks like having better form. So omitting types theorem should give some credit to Ehrenfeucht. $\endgroup$ Apr 6, 2021 at 16:34

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The Omitting Types Theorem has its roots in the independent work of Leon Henkin and Steven Orey in the early 1950’s on $\omega$-logic. This line of work was advanced further by Andrzej Grzegorczyk, Andrzej Mostowski, and Czeslaw Ryll-Nardzewski, where a more explicit version of the theorem can be found. The form of the theorem we have used is apparently due to Andrzej Ehrenfeucht around 1955 —- Ehrenfeucht never published his version.
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