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edited Aug 12, 2018 at 16:21

Asaf Karagila ♦

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set-theory axiom-of-choice

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edited Aug 11, 2018 at 9:29

Zuhair Al-Johar

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Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice?

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asked Aug 10, 2018 at 19:05

Zuhair Al-Johar

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Are there known examples of sets whose power is equal to power of larger sets only in absence of choice?

The question of existence of sets $x,y$ such that

$$|x|<|y| \wedge |P(x)|=|P(y)|$$

is known to be independent of $\text{ZFC}$!

But are there known examples of sets fulfilling the above condition that necessitates violation of choice?

set-theory