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Zuhair Al-Johar
Zuhair Al-Johar
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The proof that $AC$ is independent of $\sf ZF$ axioms is done by forcing and constructibility, and this doesn'tthese don't beg any consistency strength more than that of $\sf ZF$.

Is there a known similar proof of independence of $AC$ from $\sf Z$ that is done at the consistency level of $\sf Z$ itself?

More generally what is the smallest consistency level we need to prove that $AC$ is independent from axioms of $\sf Z$?

The proof that $AC$ is independent of $\sf ZF$ axioms is done by forcing, and this doesn't beg any consistency strength more than that of $\sf ZF$.

Is there a known similar proof of independence of $AC$ from $\sf Z$ that is done at the consistency level of $\sf Z$ itself?

More generally what is the smallest consistency level we need to prove that $AC$ is independent from axioms of $\sf Z$?

The proof that $AC$ is independent of $\sf ZF$ axioms is done by forcing and constructibility, and these don't beg any consistency strength more than that of $\sf ZF$.

Is there a known similar proof of independence of $AC$ from $\sf Z$ that is done at the consistency level of $\sf Z$ itself?

More generally what is the smallest consistency level we need to prove that $AC$ is independent from axioms of $\sf Z$?

Source Link
Zuhair Al-Johar
Zuhair Al-Johar
  • 11.3k
  • 1
  • 13
  • 47

Is there a proof of independence of AC from Z that is done in Z?

The proof that $AC$ is independent of $\sf ZF$ axioms is done by forcing, and this doesn't beg any consistency strength more than that of $\sf ZF$.

Is there a known similar proof of independence of $AC$ from $\sf Z$ that is done at the consistency level of $\sf Z$ itself?

More generally what is the smallest consistency level we need to prove that $AC$ is independent from axioms of $\sf Z$?