Vaught's Theorem states that a complete first-order theory with only infinite models can never have exactly two countable models up to isomorphism, though it may have exactly one, or three, or more, or infinitely many. Named for Robert Lawson Vaught, it is a classical result of model theory illustrating that the number of countable models a theory admits is constrained in subtle, non-obvious ways, a question the related Vaught conjecture pursues further for uncountably many models.
Facts
StatementThe number of countable models of a complete theory cannot be 2. 1 Connections
Has Statement Form
Source Vaught Conjecture (Wikipedia)
In Branch
Source Vaught Conjecture (Wikipedia)
Proved By
Source Vaught's conjecture (Wikipedia)
Sources
1. Vaught's conjecture (Wikipedia)
Vaught's theorem
Vaught proved that the number of countable models of a complete theory cannot be 2.
Proved By: Robert Lawson Vaught, Lead sentence, attribution
originally proposed by Robert Lawson Vaught in 1961
View the SourceVaught Conjecture (Wikipedia)
In Group: Mathematical Logic and Model Theory, Section: Vaught's theorem
Vaught proved that the number of countable models of a complete theory cannot be 2.
Has Statement Form: Impossibility Theorem, Section: Vaught's theorem
Vaught proved that the number of countable models of a complete theory cannot be 2.
In Branch: Model Theory, Article text
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961.
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