Independence of CH

Set Theory · Axiom Academy

The continuum hypothesis is neither provable nor refutable from ZFC 1. Gödel's Result (1938): Con(ZF) → Con(ZFC + CH) Kurt Gödel showed that if ZF is consistent, then so is ZFC together with the continuum hypothesis. He did this by constructing the constructible universe L . In L, every set is built in a very controlled, step-by-step way. Gödel proved that L is a model of ZFC + CH + GCH (the generalized continuum hypothesis). 2. Cohen's Result (1963): Con(ZF) → Con(ZFC + ¬CH) Paul Cohen completed the picture by showing that if ZF is consistent, then so is ZFC together with the negation of CH. He used his newly invented technique of forcing . Cohen started with a model M of ZFC (we can assume M = L, where CH holds). He then forced to add many new real numbers without adding new countable ordinals, creating a model M[G] where CH fails. Combining Gödel's and Cohen's results gives us a complete picture: ZFC + CH is consistent (if ZFC is) ZFC + ¬CH is consistent (if ZFC is) Therefore, ZFC alone cannot settle the question of CH This was a profound discovery. It means that the standard axioms of set theory do not determine the size of the continuum. Both possibilities—CH true and CH false—are compatible with ZFC. 4. What Independence Means Philosophically The independence of CH raises deep philosophical questions: Is there one "true" set theory? Or are there multiple equally valid set-theoretic universes?

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