Spectrum of a Theory
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High Quality Content by WIKIPEDIA articles! In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, ) for the number of models of T (up to isomorphism) of cardinality . The spectrum problem is to describe the possible behaviors of I(T, ) as a function of . It has been almost completely solved for the case of a countable theory T. Robert Vaught showed that I(T, ) cannot be 2. It is easy to find examples where it is any given no...
High Quality Content by WIKIPEDIA articles! In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, ) for the number of models of T (up to isomorphism) of cardinality . The spectrum problem is to describe the possible behaviors of I(T, ) as a function of . It has been almost completely solved for the case of a countable theory T. Robert Vaught showed that I(T, ) cannot be 2. It is easy to find examples where it is any given non-negative integer other than 2. Morley proved that if I(T, ) is infinite then it must be 0 or 1 or the cardinality of the continuum. It is not known if it can be 1 if the continuum hypothesis is false: this is called the Vaught conjecture and is the main remaining open problem (in 2005) in the theory of the spectrum.
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