Concept
Löwenheim–Skolem theorem
Overview
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem.
The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism.
As a consequence, first-order theories are unable to control the cardinality of their infinite models.
The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic.
In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Löwenheim–Skolem theorem is.
Hence, in the conclusion of a syllogism we restrict a predicate to a certain object, after having thought it in the major in its whole extent under a certain condition. This complete quantity of the extent in relation to such a condition is called universality (universalitas). To this corresponds totality (universitas) of conditions in the synthesis of intuitions.
Immanuel Kant, The Critique of Pure Reason
States what Löwenheim–Skolem theorem is.
Therefore spaces and times, if real at all, must not be regarded as composed of points and instants.
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Löwenheim–Skolem theorem is.
Philosophers who discussed Löwenheim–Skolem theorem
Related ideas
- First-order logic
- Journal of Symbolic Logic
- Gödel's completeness theorem
- Skolem's paradox
- Springer Science+Business Media
- Interpretation (logic)
- Compactness theorem
- Mathematical logic
- Model theory
- Power set
- Peano axioms
- Berlin
- Second-order logic
- Cantor's theorem
- Stanford Encyclopedia of Philosophy
- Cambridge
- Association for Symbolic Logic
- Structure (mathematical logic)
- Arity
- Heidelberg
- Oxford University Press
- Alonzo Church
- Gödel's incompleteness theorems
- Alfred Tarski
Place in the tradition
Part of First-order logic.