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arXiv · 1909.09333

Weak Vop\v{e}nka's Principle does not imply Vop\v{e}nka's Principle

Trevor M. Wilson
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Abstract

Vop\v{e}nka's Principle says that the category of graphs has no large discrete full subcategory, or equivalently that the category of ordinals cannot be fully embedded into it. Weak Vop\v{e}nka's Principle is the dual statement, which says that the opposite category of ordinals cannot be fully embedded into the category of graphs. It was introduced in 1988 by Ad\'{a}mek, Rosick\'{y}, and Trnkov\'{a}, who showed that it follows from Vop\v{e}nka's Principle and asked whether the two statements are equivalent. We show that they are not. However, we show that Weak Vop\v{e}nka's Principle is equivalent to the generalization of itself known as Semi-Weak Vop\v{e}nka's Principle.

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BibTeXRIS

Trevor M. Wilson. 2019-09-20. Weak Vop\v{e}nka's Principle does not imply Vop\v{e}nka's Principle. https://doi.org/10.1016/j.aim.2020.106986

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