Faithful to negation

Negation Trust (ger .: negation complete ) is a property of sequences of predicate logic expressions. This property is also called syntactically complete (in English-language literature also syntactically complete , deductively complete or maximally complete ) - in order to avoid confusion with other terms of completeness . ${\displaystyle \Phi }$

Definition: A set of predicate logic expressions is called faithful to negation if for any expression : ${\displaystyle \Phi }$ ${\displaystyle \phi }$

${\displaystyle \Phi \vdash \phi }$or . ${\displaystyle \Phi \vdash \neg \phi }$

It can also be expressed differently: A formal system, given by the set of axioms , is true to negation or syntactically complete if every further axiom that cannot be derived from leads to a contradiction. ${\displaystyle \Phi }$ ${\displaystyle \Phi }$

meaning

The meaning of the term faithful to negation lies in its role as a proof tool for Henkin's theorem , which in turn is used as a tool for an alternative proof of Godel's completeness theorem of first-order predicate logic .

literature

Hans Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic. Spektrum Akademischer Verlag, Heidelberg 2007, ISBN 3-8274-1691-4 .