Embedding Orders Into Cardinals With $D{C}_{\kappa }$

Abstract

Jech proved that every partially ordered set can be embedded into the cardinals of some model of $ZF$. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $ZF+D{C}_{<\kappa }$ for any regular $\kappa$. We use this theorem to show that for all $\kappa$, the assumption of $D{C}_{\kappa }$ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model. We use this extension to give a large cardinals-free proof of independence of the weak choice principle known as $WISC$.

11 pages; some improvements as suggested by the referee; minor technical corrections in section 4, journal reference

Cited by in corpus (5)

• The weak choice principle WISC may fail in the category of sets
• Fodor's lemma can fail everywhere
• Broad Infinity and Generation Principles
• Combinatorial Properties and Dependent choice in symmetric extensions based on Lévy Collapse
• On certain 2-categories admitting localisation by bicategories of fractions