Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeMay 25th 2012
   • (edited May 25th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI have expanded [[theory]] adding more basics in classical syntactic approach. I added a new subsection > #### Set-theoretic models for a first-order theory in syntactic approach > The basic concept is of a structure for a first-order language $L$: a set $M$ together with an interpretation of $L$ in $M$. A theory is specified by a language and a set of sentences in $L$. An $L$-structure $M$ is a __model__ of $T$ if for every sentence $\phi$ in $T$, its interpretation in $M$, $\phi^M$ is true ("$\phi$ holds in $M$"). We say that $T$ is __consistent__ or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for $T$ (in our universe). Two theories, $T_1$, $T_2$ are said to be __equivalent__ if they have the same models. > Given a class $K$ of structures for $L$, there is a theory $Th(K)$ consisting of all sentences in $L$ which hold in every structure from $K$. Two structures $M$ and $N$ are __elementary equivalent__ (sometimes written by equality $M=N$, sometimes said "elementarily equivalent") if $Th(M)=Th(N)$, i.e. if they satisfy the same sentences in $L$. Any set of sentences which is equivalent to $Th(K)$ is called __a set of axioms__ of $K$. A theory is said to be __finitely axiomatizable__ if there exist a finite set of axioms for $K$. > A theory is said to be __complete__ if it is equivalent to $Th(M)$ for some structure $M$.

   I have expanded theory adding more basics in classical syntactic approach. I added a new subsection

   Set-theoretic models for a first-order theory in syntactic approach

   The basic concept is of a structure for a first-order language : a set together with an interpretation of in . A theory is specified by a language and a set of sentences in . An -structure is a model of if for every sentence in , its interpretation in , is true (“ holds in ”). We say that is consistent or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for (in our universe). Two theories, , are said to be equivalent if they have the same models.

   Given a class of structures for , there is a theory consisting of all sentences in which hold in every structure from . Two structures and are elementary equivalent (sometimes written by equality , sometimes said “elementarily equivalent”) if , i.e. if they satisfy the same sentences in . Any set of sentences which is equivalent to is called a set of axioms of . A theory is said to be finitely axiomatizable if there exist a finite set of axioms for .

   A theory is said to be complete if it is equivalent to for some structure .