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Math Help - tautology

  1. #1
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    tautology

    I try to undestand the generalized tautology theorem (not sure wether it's the correct english expression) and its proof.

     \phi_1,...,\phi_n \models \phi \Rightarrow \phi_1,...,\phi_n \vdash \phi

    Doesn't it simply state the completeness of the axiom system (if it holds forall phi1..n)?

    In the proof there is a curious step. It states that via modus ponens you can transform the first in the second statement:
    \phi_1\vdash\phi_1 \rightarrow ...\rightarrow \phi_n \rightarrow \phi

    \phi_1\vdash\phi_2 \rightarrow ...\rightarrow \phi_n \rightarrow \phi

    I undestand the transformation, but I don't see modus ponens applied here. Anyone knows?

    ps: Btw, where can I look up translations of mathematical expressions (e.g. completeness/Vollständigkeit,...) for English/German? I'm not used doing mathematics in English
    Last edited by kumpel; March 6th 2010 at 06:30 AM. Reason: so many mistakes...
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  2. #2
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    Quote Originally Posted by kumpel View Post
    It states that via modus ponens you can transform the first in the second statement:
    \phi_1\vdash\phi_1 \rightarrow ...\rightarrow \phi_n \rightarrow \phi

    \phi_1\vdash\phi_2 \rightarrow ...\rightarrow \phi_n \rightarrow \phi

    I undestand the transformation, but I don't see modus ponens applied here.
    Solved. The other questions are still open.
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  3. #3
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    Doesn't it simply state the completeness of the axiom system (if it holds forall phi1..n)?
    Yes, this is just completeness. It is "generalized" to distinguish it from the one without open assumptions: \models\phi\Rightarrow{}\vdash\phi.

    Btw, where can I look up translations of mathematical expressions (e.g. completeness/Vollständigkeit,...) for English/German? I'm not used doing mathematics in English
    I don't know German, but you may search for a Wikipedia article in German (e.g., Gödelscher Vollständigkeitssatz) and then look at the same article in English (there is a link on the left).
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