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sav08:complete_recursive_axiomatizations [2008/04/06 16:59]
vkuncak
sav08:complete_recursive_axiomatizations [2009/05/16 10:47] (current)
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**Proof.**++++| **Proof.**++++|
Suppose $Ax$ is a complete recursive axiomatization. ​ There are two cases, depending on whether $Ax$ is consistent. Suppose $Ax$ is a complete recursive axiomatization. ​ There are two cases, depending on whether $Ax$ is consistent.
-  * **Case 1):** The set $Ax$ is inconsistent,​ that is, there are not models for $Ax$.  Then $Conseq(Ax)$ is the set of all first-order sentences and there is a trivial algorithm for checking whether $F \in Conseq(Ax)$:​ always return true.+  * **Case 1):** The set $Ax$ is inconsistent,​ that is, there are no models for $Ax$.  Then $Conseq(Ax)$ is the set of all first-order sentences and there is a trivial algorithm for checking whether $F \in Conseq(Ax)$:​ always return true.
* **Case 2):** The set $Ax$ is consistent. ​ Given $F$, to check whether $F \in Conseq(Ax)$ we run in parallel two complete theorem provers (which exist by [[lecture10|Herbrand theorem]]), the first one trying to prove the formula $F$, the second one trying to prove the formula $\lnot F$; the procedure terminates if any of these theorem provers succeed (the theorem provers simultaneously searches for longer and longer proofs from a larger and larger finite subsets of $Ax$). We show that this is an algorithm that decides $F \in Conseq(Ax)$.   * **Case 2):** The set $Ax$ is consistent. ​ Given $F$, to check whether $F \in Conseq(Ax)$ we run in parallel two complete theorem provers (which exist by [[lecture10|Herbrand theorem]]), the first one trying to prove the formula $F$, the second one trying to prove the formula $\lnot F$; the procedure terminates if any of these theorem provers succeed (the theorem provers simultaneously searches for longer and longer proofs from a larger and larger finite subsets of $Ax$). We show that this is an algorithm that decides $F \in Conseq(Ax)$.
* Because either $F \in Conseq(Ax)$ or $(\lnot F) \in Conseq(Ax)$,​ and theorem provers are complete, one of these theorem provers will eventually halt.  The procedure is therefore an algorithm.     * Because either $F \in Conseq(Ax)$ or $(\lnot F) \in Conseq(Ax)$,​ and theorem provers are complete, one of these theorem provers will eventually halt.  The procedure is therefore an algorithm.