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

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Karagila A.

Fundamenta Mathematicae

2014

tom 226

nr 2

143-156

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 $DC_{κ}$.

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1 Fundamenta Mathematicae

1 Karagila A.

1 2014

Wyniki wyszukiwania

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