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I wish to check my understanding regarding the definitions of contradiction, consistency and completeness, as well as their relation to syntax and semantics.

Let us make the following definitions: $\def\bS{\textbf{S}} \def\f{\varphi}$

Definition: a sentence $\chi$ is a contradiction iff $\vdash \neg\chi$.

Definition: a set of sentences is consistent iff there is no contradiction $\chi$ for which $\bS\vdash\chi$. Otherwise it is called inconsistent.

Definition: a set of sentences $\bS$ is complete iff given any sentence $\f$, then either $\bS\vdash \f$ or $\bS\vdash \neg\f$. Otherwise it is called incomplete.

It seems to me that all the previous definitions could be made in terms of semantics by replacing '$\vdash$' by '$\models$'. Because of the Completeness Theorem, it is irrelevant which to use, yet previous to proving said theorem we could distuinguish between the syntactic and semantic version of these definitions.

Am I wrong?

Sam
  • 5,208
  • The $\vdash$ is about derivability in some proof system. That makes it a syntactic notion, and that makes $\vdash$ only meaningful relative to some proof system. Indeed, in one proof system one might be able to derive something that in a different system one cannot. $\vDash$ has nothing to do with proof systems; it considers the conditions under which some statement would evaluate to True or False. Now, in practice, we use proof systems that are both sound and complete ... and that is what makes the $\vDash$ and $\vdash$ collapse in practice as well. But they are really different concepts. – Bram28 Feb 07 '25 at 14:36

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