The Vaught conjecture, proposed by Robert Vaught in 1961 in model theory, states that the number of countable models of a complete first-order theory in a countable language is either finite, the countable infinite cardinal aleph-null, or the cardinality of the continuum. Michael Morley narrowed the possibilities, leaving the conjecture open only for the case of aleph-one models when the continuum hypothesis fails; a claimed counterexample announced in 2002 had not been independently verified as of 2021.
Facts
Partially Attested
Progress Toward ResolutionMorley resolved all cases except aleph-one models when the continuum hypothesis fails; Knight's announced counterexample is unverified as of 2021. 1 A counterexample announced by Robin Knight is unverified, so the conjecture is treated as open. StatementThe number of countable models of a complete first-order theory in a countable language is finite, aleph-null, or the cardinality of the continuum. 1 Proposed Year Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Vaught Conjecture (Wikipedia)
Posed By
Source Vaught Conjecture (Wikipedia)
Sources
1. Vaught Conjecture (Wikipedia)
Lead section
originally proposed by Robert Lawson Vaught in 1961
Progress
As of 2021, the counterexample has not been verified.
Lead section [statement] [2]
the number of countable models of a first-order complete theory in a countable language is finite or
In Branch: Model Theory, Lead sentence
ture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961.
Posed By: Robert Lawson Vaught, Lead paragraph
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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