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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Undefinable classes and definable elements in models of set theory and arithmetic


Author: Ali Enayat
Journal: Proc. Amer. Math. Soc. 103 (1988), 1216-1220
MSC: Primary 03C62; Secondary 03C25, 03C40
DOI: https://doi.org/10.1090/S0002-9939-1988-0955013-2
MathSciNet review: 955013
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Abstract: Every countable model $ {\mathbf{M}}$ of PA or ZFC, by a theorem of S. Simpson, has a "class" $ X$ which has the curious property: Every element of the expanded structure $ ({\mathbf{M}},X)$ is definable. Here we prove:

Theorem A. Every completion $ T$ of PA has a countable model $ {\mathbf{M}}$ (indeed there are $ {{\mathbf{2}}^\omega }$ many such $ {\mathbf{M}}$'s for each $ T$) which is not pointwise definable and yet becomes pointwise definable upon adjoining any undefinable class $ X$ to $ {\mathbf{M}}$.

Theorem B. Let $ {\mathbf{M}} \vDash {\text{ZF}} + ''V = {\text{HOD''}}$ be a well-founded model of any cardinality. There exists an undefinable class $ X$ such that the definable points of $ {\mathbf{M}}$ and $ ({\mathbf{M}},X)$ coincide.

Theorem C. Let $ {\mathbf{M}}\, \vDash {\text{PA}}$ or $ {\text{ZF}} + ''V = {\text{HOD''}}$. There exists an undefinable class $ X$ such that the definable points of $ {\mathbf{M}}$ and $ ({\mathbf{M}},X)$ coincide if one of the conditions below is satisfied.

(A) The definable elements of $ {\mathbf{M}}$ are cofinal in $ {\mathbf{M}}$.

(B) $ {\mathbf{M}}$ is recursively saturated and $ \operatorname{cf}(\mathbf{M}) = \omega $.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0955013-2
Article copyright: © Copyright 1988 American Mathematical Society