Abstract
An old question is whether there is a countable complete first order theory T such that T has a universal model of cardinality \(\lambda> {\aleph _0} {\underline{iff}}\, \lambda = 2^{< \lambda } > {\aleph _0} \). We solve it here for the class singular cardinals.
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Notes
Here the case \(\delta ' \ne \delta \) is not really needed, but in some other versions, it is helpful.
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The author thanks Alice Leonhardt for the beautiful typing. References like [4, Th0.2=Ly5] means the label of Th.0.2 is y5. The reader should note that the version in my website is usually more updated than the one in the mathematical archive. First typed May 10, 2019. Supported by ISF 1838/19 (The Israel Science Foundation) and by ERC (European Research Council) grant 338821. Paper 1162 on author’s list. The author states that there is no conflict of interest.
