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Contrasting constructible universe and von Neumann universe

In constructible universe, GCH (Generalized Continuum Hypothesis) is true. If so, what is the point of distinguishing between constructible universe and von Neumann universe? Each stage will be virtually equal, as $2^{\aleph_0} = \aleph_1$.

What am I getting wrong?

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    I gave (what I think is) a pretty comprehensive answer in here: http://math.stackexchange.com/a/114642/622 If you read it and you think I'm wrong in thinking that this answer also answered your question here, please let me know and I will write you something here.2012-05-30
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    But in L, GCH is true, meaning that each stage would be exactly how Von Neumann universe's stages form (power set)2012-05-30
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    No. This is **very** wrong. Read my answer, please. The definition of $L$ is not "take power sets" but rather take "definable sets". There are other definitions too, but they do not "automatically generate $L$". The von Neumann construction generates **every** universe of ZFC, even those not satisfying $V=L$.2012-05-30
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    user1894, what is your mathematical background? You are asking a lot of questions which show that you are interested in the topics, but you lack the structured frame of a course or a book. Have you studied *any* [academic] mathematics before, any set theory related courses?2012-06-01

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