A minimal model for ¬𝐶𝐻: iteration of Jensen’s reals

Abraham, Uri

doi:10.1090/S0002-9947-1984-0722767-0

A minimal model for $\neg \textrm {CH}$: iteration of Jensen’s reals
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by Uri Abraham

Trans. Amer. Math. Soc. 281 (1984), 657-674

DOI: https://doi.org/10.1090/S0002-9947-1984-0722767-0

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Abstract:

A model of ${\text {ZFC}} + {2^{\aleph _0}} = {\aleph _2}$ is constructed which is minimal with respect to being a model of $\neg {\text {CH}}$. Any strictly included submodel of ${\text {ZF}}$ (which contains all the ordinals) satisfies ${\text {CH}}$. In this model the degrees of constructibility have order type ${\omega _2}$. A novel method of using the diamond is applied here to construct a countable-support iteration of Jensen’s reals: In defining the $\alpha {\text {th}}$ stage of the iteration the diamond "guesses" possible $\beta > \alpha$ stages of the iteration.

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