View source for Characterization theorems for logics

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First-order logic (cf. also [[Logical calculus|Logical calculus]]) is well-suited for mathematics, e.g.:

1) There is a sound and complete proof calculus (completeness theorem). The decidability of many theories has been proven using the completeness theorem. (Cf. also [[Completeness (in logic)|Completeness (in logic)]]; [[Sound rule|Sound rule]].)

2) There is a system of first-order logical axioms for [[Set theory|set theory]] (e.g., ZFC) that serves as a basis for mathematics.

3) There is a balance between syntax and semantics, e.g., implicitly definable concepts are explicitly definable (Beth's theorem; cf. also [[Beth definability theorem|Beth definability theorem]]).

4) Semantic results such as the compactness theorem and the [[Löwenheim–Skolem theorem|Löwenheim–Skolem theorem]] are valuable model-theoretic tools and lead to an enrichment of mathematical methods. Mainly in the period from 1950 to 1970, much effort was spent in finding languages which strengthen first-order logic but are still simple enough to yield general principles which are useful in investigating and classifying models. In particular, taking into account the situation for first-order logic, many logicians attempted to find logics satisfying analogues of the theorems mentioned above. However, results due to P. Lindström [[#References|[a3]]] limit this search. Or, to state it more positively, Lindström proved the following characterization theorems for first-order logic:

First-order logic is a maximal logic with respect to expressive power satisfying the compactness theorem and the Löwenheim–Skolem theorem.

First-order logic is a maximal logic satisfying the completeness theorem and the Löwenheim–Skolem theorem.

These results were the starting point for investigations trying to order the diversity of extensions of first-order logic, for a systematic study of the relationship between different model-theoretic properties of logics, and for a search for further characterizations theorems for first-order and other logics.

Most of the results obtained can be found in [[#References|[a1]]]. Two characterization theorems obtained for other logics are:

a) $\mathcal{L} _ { \infty  \omega}$ is a maximal bounded logic with the Karp property [[#References|[a2]]];

b) for topological structures, the logic $L _ { t }$ of  "invariant sentences"  is a maximal logic satisfying the compactness theorem and the Löwenheim–Skolem theorem [[#References|[a4]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  J. Barwise,   S. Feferman,   "Model-theoretic logics" , Springer  (1985)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Barwise,   "Axioms for abstract model theory"  ''Ann. Math. Logic'' , '''7'''  (1974)  pp. 221–265</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P. Lindström,   "On extensions of elementary logic"  ''Theoria'' , '''35'''  (1969)  pp. 1–11</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  M. Ziegler,   "A language for topological structures which satisfies a Lindström-theorem"  ''Bull. Amer. Math. Soc.'' , '''82'''  (1976)  pp. 568–570</td></tr></table>