In ZF¬C, if a set can't be well ordered, what does it say about its cardinality?
The well-ordering theorem follows only if the AoC is true, which means that otherwise, there are sets with no well-ordering.
Supposedly, the Real numbers is one of such sets. Does that mean the real numbers can't be shown to have a direct bijection to any aleph number? And if that's the case, does that mean 2^Aleph_0 is bigger than any aleph number?
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I don't believe so. I think that it means (or at least could mean) ZF without C is incomplete with regards to this question, the way that ZFC is incomplete regarding the continuum hypothesis.