Weak Vopenka's Principle does not imply Vopenka's Principle
Abstract
Vopenka'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 Vopenka'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\'amek, Rosick\'y, and Trnkov\'a, who showed that it follows from Vopenka's Principle and asked whether the two statements are equivalent. We show that they are not. However, we show that Weak Vopenka's Principle is equivalent to the generalization of itself known as Semi-Weak Vopenka's Principle.
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