Löwenheim

Löwenheim was a German logician and mathematician active in the early 20th century. He is primarily known for his work in the foundations of logic, where his results helped shape the development of model theory and the study of formal languages.

His most enduring contribution is what is now called the Löwenheim–Skolem theorems. In 1915 he showed that if a first-order theory has a model and its language is countable, then the theory has a model whose domain is countable. In subsequent work, Thoralf Skolem extended the ideas to show that if a theory has an infinite model, it has models of every larger infinite cardinality. Together, these results are often referred to as the Löwenheim–Skolem theorems.

The theorems have significant implications for the philosophy and semantics of logic. They demonstrate that first-order logic cannot, in general, determine the cardinality of its models; a theory can have models that are countable as well as models that are uncountable and not elementarily distinguishable by the language. This insight contributed to discussions about categoricity, completeness, and the expressive limits of formal systems, and it prompted a deeper examination of the nature of mathematical structures as described by logical theories.

Löwenheim’s work, together with Skolem’s refinements, remains a foundational element in model theory. The Löwenheim–Skolem theorems are cited as early and important results illustrating how model-theoretic methods interact with formal languages and the size of structures.