Let $\Gamma$ be a consistent set of $L$-sentences with infinite cardinality. If it has an infinite model, then there exists a model for $$ \Gamma' =\Gamma \cup \{\lnot c_a = c_b : a \neq b \}, $$ where $c_i$ is a constant symbol.

How would I go about proving this? I'm imagining that I can extend the model for $\Gamma$ to have a larger domain (to include more constant symbols) but I don't know how to show that it is actually a model for $\Gamma'$. Maybe with compactness?

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    $\begingroup$ Yes, compactness. $\endgroup$ – Asaf Karagila Dec 8 '18 at 20:43
  • $\begingroup$ Yes, with compactness. But it should be assumed that the constant symbols $c_i$ do not appear in $\Gamma$, otherwise the statement is not true. $\endgroup$ – Alex Kruckman Dec 8 '18 at 20:44
  • $\begingroup$ @AlexKruckman Why is that assumption necessary? $\endgroup$ – bof Dec 8 '18 at 20:45
  • $\begingroup$ For example, in the vocabulary $(c_a,c_b)$, $\Gamma$ could be the set $\{c_a = c_b\}$. $\endgroup$ – Alex Kruckman Dec 8 '18 at 20:48
  • $\begingroup$ @AlexKruckman Thanks, that makes sense. Now, compactness requires that every finite subset of $\Gamma'$ has a model but I'm not sure how I would go about proving that for every subset. Any hints? $\endgroup$ – bof Dec 8 '18 at 20:53

A finite subset of $\Gamma'$ is a finite subset of $\Gamma$, together with the assertions that finitely many $c_1,\dots,c_n$ are pairwise distinct.

Take any infinite model of $\Gamma$ (which exists by assumption), interpret $c_1,\dots,c_n$ as distinct elements, and interpret the rest of the constant symbols arbitrarily (we are free to do this, since the constant symbols are not mentioned in $\Gamma$). This is a model of our finite subset of $\Gamma'$. That's all there is to it.

This is the standard proof of the (upwards) Löwenheim-Skolem theorem.

  • $\begingroup$ Thank you. This was very helpful. $\endgroup$ – bof Dec 8 '18 at 21:05
  • $\begingroup$ I'm sorry but I have another question. Why can't we take any infinite model of $\Gamma$ and interpret $c_i$ as distinct elements and use it for the model of $\Gamma'$? What's special about finite subsets? $\endgroup$ – bof Dec 8 '18 at 21:19
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    $\begingroup$ Because there might be more constants then there are elements of the model. $\endgroup$ – Alex Kruckman Dec 8 '18 at 21:39
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    $\begingroup$ The point of the Löwenheim-Skolem theorem os to take a theory that has some infinite model (say a countably infinite one) and prove that it has models of all larger cardinities. $\endgroup$ – Alex Kruckman Dec 8 '18 at 21:41

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