1. tautology

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

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

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:
$\displaystyle \phi_1\vdash\phi_1 \rightarrow ...\rightarrow \phi_n \rightarrow \phi$

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

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

2. Originally Posted by kumpel
It states that via modus ponens you can transform the first in the second statement:
$\displaystyle \phi_1\vdash\phi_1 \rightarrow ...\rightarrow \phi_n \rightarrow \phi$

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

I undestand the transformation, but I don't see modus ponens applied here.
Solved. The other questions are still open.

3. 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: $\displaystyle \models\phi\Rightarrow{}\vdash\phi$.

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).