23 Übersetzungen
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Anwendungsbeispiele Englisch
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- The completeness theorem and the compactness theorem are two cornerstones of first-order logic.
- Each "condition" is a finite piece of information – the idea is that only finite pieces are relevant for consistency, since, by the compactness theorem, a theory is satisfiable if and only if every finite subset of its axioms is satisfiable.
- First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.
- The compactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable.
- Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic.
- The ultraproduct construction in model theory uses ultrafilters to produce a new model starting from a sequence of [...] -indexed models; for example, the compactness theorem can be proved this way.
© dict.cc English-German dictionary 2024
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