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

TL;DR

This paper proves that Weak Vopěnka's Principle is strictly weaker than Vopěnka's Principle and establishes its equivalence to a generalized form called Semi-Weak Vopěnka's Principle.

Contribution

The paper demonstrates that Weak Vopěnka's Principle does not imply Vopěnka's Principle and introduces its equivalence to Semi-Weak Vopěnka's Principle.

Findings

Weak Vopěnka's Principle does not imply Vopěnka's Principle.

Weak Vopěnka's Principle is equivalent to Semi-Weak Vopěnka's Principle.

The two principles are logically independent.

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.