An ordinal-connection axiom as a weak form of global choice under the GCH

Archive for Mathematical Logic 62 (3):321-332 (2022)
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Abstract

The minimal ordinal-connection axiom $$MOC$$ was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that $$MOC$$ is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, $$MOC$$ is in fact equivalent to the $${{\,\mathrm{GCH}\,}}$$. Our main results then show that $$MOC$$ corresponds to a weak version of global choice in models of the $${{\,\mathrm{GCH}\,}}$$ : it can fail in models of the $${{\,\mathrm{GCH}\,}}$$ without global choice, but also global choice can fail in models of $$MOC$$.

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Powers of regular cardinals.William B. Easton - 1970 - Annals of Mathematical Logic 1 (2):139.

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