i am a little bit confused on 1 thing. can a statement that is undecidable in a axiomatic system just be added as a axiom to the original system and never lead to a contridiction?(adsbygoogle = window.adsbygoogle || []).push({});

for example godel and cohen showed that the continuum hypothesis is independent of ZFC. does this mean then that if we add the CH or its negation as an axiom to ZFC we will still have a system that is still consistent (assuming ZFC is already consistent)? Does this mean then that if I add the negation of the CH i can show that there is some cardinal x such that aleph_0<x<c?

this is the wiki article that is confusing me-

http://en.wikipedia.org/wiki/Axiom_of_choice#Independence_of_AC

quote from article: "By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (ZF). This means that neither it nor its negation can be proven to be true in ZF. Consequently, assuming the axiom of choice, or its negation, will never lead to a contradiction that could not be obtained without that assumption."

so if the CH( or ~CH) can be added to ZFC(since it was proven independent just like AC), then why is it still unsolved?

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# Undecidable statement made into axiom?

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