Uniquely undefinable elements.(English)Zbl 1184.03036

Summary: There exists a model in a countable language having a unique element which is not definable in $$\mathcal L_{\omega _{1}, \omega }$$. This answers Problem 10.12 from the updated version of A. Miller’s problem list [http://www.math.wisc.edu/~miller/res/problem.pdf].

MSC:

 03C75 Other infinitary logic

Zbl 0828.03017
Full Text:

References:

 [1] Set Theory of the Reals 6 pp 645– (1993) [2] Admissible Sets and Structures. An Approach to Definability Theory (1975) · Zbl 0316.02047 [3] DOI: 10.1016/0003-4843(70)90006-9 · Zbl 0222.02014 [4] Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers 62 (1971)
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