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Johnson, Jesse

doi:10.1016/j.apal.2014.04.003

Cited by 1 publication

(2 citation statements)

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“…There is some work on relative computable categoricity in the context of κcomputability. Johnson [13,12] showed that for κ = ℵ 1 , the Zil ber field of cardinality ℵ 1 is not computably categorical, but it is relatively ∆ 0 2 -categorical. By contrast, the Zil ber cover is relatively computably categorical.…”

Section: 12mentioning

confidence: 99%

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Uniform Procedures in Uncountable Structures

Greenberg¹,

Melnikov²,

Knight³

et al. 2018

J. symb. log.

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This article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of ${\Cal R}^n $ contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.

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Section: 12mentioning

confidence: 99%

“…By contrast, the Zil ber cover is relatively computably categorical. Johnson [12,13] proved a transfer theorem for categoricity of "abstract excellent classes". For these classes, there is a unique model in each uncountable power.…”

Section: Inseparable Structuresmentioning

confidence: 99%