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I'm working through a proof of the completeness theorem for first order predicate calculus that was given in lectures and I'm stuck on proving a step that wasn't elaborated in class.

Suppose $\phi$ is a formula in first-order predicate calculus and let $\phi '$ be the formula obtained by replacing every occurrence of $c_n$ by $c_{2n}$ where the $c_i$ are the constants in our language $\mathcal{L}$.

Let $\Delta ' := \{ \phi '\ |\ \phi \in \Delta \}$

Then I want to prove:

  1. $\Delta$ consistent $\Rightarrow$ $\Delta '$ consistent

  2. $\Delta '$ has a model $\Rightarrow$ $\Delta $ has a model

Can someone please explain why this is the case.

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Mathmo
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  • For the first one, if $\Delta'$ is inconsistent take a proof of contradiction and replace $c_{2n}$ by $c_n$ whenever it occurs; in the second case pick any model of $\cal L$ and reinterpret the constants (and maybe the relations too if need be), then set that $c_{2n+1}=c_0$ or something. – Asaf Karagila Jun 01 '14 at 18:03
  • What are meant by the constants in the language ? Field theory has two constants $0$ and $1$, DLO has no constants, does he mean variables ? Or are they the Henkin witnesses ? Who is giving these lectures anyways ? – Rene Schipperus Jun 01 '14 at 18:09
  • Likely "Henkin witnesses" indexed by $\mathbb{N}$. – Malice Vidrine Jun 01 '14 at 18:14
  • Oh, OK fine then. – Rene Schipperus Jun 01 '14 at 18:15

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