“Löwenheim-Skolem theorem”的概念、定义、翻译、参考文献-科学参考

Löwenheim-Skolem theorem

Logic

A theorem established by Leopold Löwenheim (1878–1957) and Thoralf Skolem (1887–1963). The theorem applies to classical first-order logic, and has upward and downward parts. A model of a set of sentences is an interpretation which makes all of its members true. The cardinality of an interpretation is the cardinality of its domain. The cardinality of a language is the cardinality of its set of non-logical symbols. Upward part: if a set of sentences has a model of any infinite cardinality $κ$ , it has a model of any cardinality greater than $κ$ . Downward part: if a set of sentences in language $ℒ$ has a model of any infinite cardinality, $κ$ , it has a model of any cardinality less than $κ$ and greater than or equal to the cardinality of $ℒ$ . In particular, if, as usual, the language is countable, the set of sentences has a countable model.

Philosophy

Theorem stating that any class of well-formed formulae of the predicate calculus that has a model, has a model with a denumerable domain. See also Skolem paradox.

单词	Löwenheim-Skolem theorem
释义	Löwenheim-Skolem theorem Logic A theorem established by Leopold Löwenheim (1878–1957) and Thoralf Skolem (1887–1963). The theorem applies to classical first-order logic, and has upward and downward parts. A model of a set of sentences is an interpretation which makes all of its members true. The cardinality of an interpretation is the cardinality of its domain. The cardinality of a language is the cardinality of its set of non-logical symbols. Upward part: if a set of sentences has a model of any infinite cardinality $κ$ , it has a model of any cardinality greater than $κ$ . Downward part: if a set of sentences in language $ℒ$ has a model of any infinite cardinality, $κ$ , it has a model of any cardinality less than $κ$ and greater than or equal to the cardinality of $ℒ$ . In particular, if, as usual, the language is countable, the set of sentences has a countable model. Philosophy Theorem stating that any class of well-formed formulae of the predicate calculus that has a model, has a model with a denumerable domain. See also Skolem paradox.
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