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sav08:first-order_theories [2008/03/19 15:13] vkuncak created |
sav08:first-order_theories [2008/03/19 22:02] damien |
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| ====== First-Order Theories ====== | ====== First-Order Theories ====== | ||
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| + | (Building on [[First-Order Logic Semantics]].) | ||
| **Definition:** A //first-order theory// is a set $T$ of [[First-Order Logic Syntax|sentences]]. | **Definition:** A //first-order theory// is a set $T$ of [[First-Order Logic Syntax|sentences]]. | ||
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| Let $T = Conseq(\{Ref,Sym,Tra\})$. Let us answer the following: | Let $T = Conseq(\{Ref,Sym,Tra\})$. Let us answer the following: | ||
| * Is $T$ ++consistent?|Yes, take, for example, ordering on integers.++ | * Is $T$ ++consistent?|Yes, take, for example, ordering on integers.++ | ||
| - | * Is $T$ ++complete?|No, take, for example, $\exists x. \forall y. x \le y$.++ | + | * Is $T$ ++complete?|No, take, for example, $\exists x. \forall y. x \le y$. It is true with the ordering on $\mathbb{N}$, but false with the ordering on $\mathbb{Z}$.++ |