Embedding orders into the cardinals with $DC_{κ}$

• Autorzy: Asaf Karagila
• Języki publikacji: EN

Abstrakty

EN

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 $ZF + DC_{<κ}$ for any regular κ. We use this theorem to show that for all κ, the assumption of $DC_{κ}$ 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 from $DC_{κ}$.

Kategorie tematyczne

• 03E25: Axiom of choice and related propositions
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 226
• Numer: 2
• Strony: 143-156

Daty

wydano

2014

Twórcy

autor

Asaf Karagila

• Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel

Identyfikatory

DOI

10.4064/fm226-2-4