Łoś–Vaught test

In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic, this means that for every sentence, the theory contains either the sentence or its negation but not both.

Statement

A theory $T$ with signature σ is $\kappa$ -categorical for an infinite cardinal $\kappa$ if $T$ has exactly one model (up to isomorphism) of cardinality $\kappa .$

The Łoś–Vaught test states that if a satisfiable theory is $\kappa$ -categorical for some $\kappa \geq |\sigma |$ and has no finite model, then it is complete.

This theorem was proved independently by Jerzy Łoś (1954) and Robert L. Vaught (1954), after whom it is named.

References

Enderton, Herbert B. (1972), A mathematical introduction to logic, Academic Press, New York-London, p. 147, MR 0337470.
Łoś, Jerzy (1954), "On the categoricity in power of elementary deductive systems and some related problems", Colloquium Mathematicum, 3: 58–62, MR 0061561.
Vaught, Robert L. (1954), "Applications to the Löwenheim-Skolem-Tarski theorem to problems of completeness and decidability", Indagationes Mathematicae, 16: 467–472, MR 0063993.

Mathematical logic

General

Theorems (list)
and paradoxes

Gödel's completeness and incompleteness theorems
Tarski's undefinability
Banach–Tarski paradox
Cantor's theorem, paradox and diagonal argument
Compactness
Halting problem
Lindström's
Löwenheim–Skolem
Russell's paradox

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types of sets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive

Formal systems (list),
language and syntax

Alphabet
Arity
Automata
Axiom schema
Expression
- ground
Extension
- by definition
- conservative
Relation
Formation rule
Grammar
Formula
- atomic
- closed
- ground
- open
Free/bound variable
Language
Metalanguage
Logical connective
- ¬
- ∨
- ∧
- →
- ↔
- =
Predicate
- functional
- variable
- propositional variable
Proof
Quantifier
- ∃
- !
- ∀
- rank
Sentence
- atomic
- spectrum
Signature
String
Substitution
Symbol
- function
- logical/constant
- non-logical
- variable
Term
Theory
- list

Example axiomatic
systems (list)

of arithmetic:
of the real numbers
- Tarski's axiomatization
of Boolean algebras
- canonical
- minimal axioms
of geometry:
- Euclidean:
- non-Euclidean

Principia Mathematica

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Model theory

Computability theory

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