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'first order theories'
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| Title of object: |
first order theories |
| Canonical Name: |
FirstOrderTheories |
| Type: |
Definition |
| Created on: |
2002-06-03 00:24:43 |
| Modified on: |
2002-06-03 18:51:14 |
| Classification: |
msc:03B10, msc:03C07 |
| Defines: |
theory, complete theory, axiomatizable theory |
Preamble:
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Content:
Let $L$ be a first-order language. A {\bf theory} in $L$ is a set of sentences of $L$, i.e. a set of formulas of $L$ that have no free variables.
{\bf Definition.} A theory $T$ is said to be {\bf consistent} if and only if $T\not\proves\perp$, where $\perp$ stands for ``false''. In other words, $T$ is consistent if one cannot derive a contradiction from it. If $\varphi$ is a sentence of $L$, then we say $\varphi$ is {\bf consistent with $T$} if and only if the theory $T\cup\{\varphi\}$ is consistent.
{\bf Definition.} A theory $T\subseteq L$ is said to be complete if and only if for every formula $\varphi\in L$, either $T\proves\varphi$ or $T\proves\neg\varphi$.
{\bf Lemma.} A theory $T$ in $L$ is complete if and only if it is maximal consistent. In other words, $T$ is complete if and only if for every $\varphi\not\in T$, $T\cup\{\varphi\}$ is inconsistent.
{\bf Theorem. (Tarski)} Every consistent theory $T$ in $L$ can be extended to a complete theory.
{\bf Proof :} Use Zorn's lemma on the collection of consistent theory extending $T$.\hfill$\diamondsuit$ |
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