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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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    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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Despite pointing out that the answer was essentially given in another thread, I really feel that I need to say something additional here.

What Goedel proved in his famous paper about the consistency of GCH with ZF is that if there is a universe of sets, $V$, satisfying the axioms of ZF then inside of this universe there exists a definable collection of sets in which ZFC+GCH holds. We call this universe $L$, and we can formulate an axiom which tells us that $V=L$, namely the universe is exactly this $L$ and not larger. In fact Goedel proved some more, but right now it is less important.

In contrast, von Neumann proved (in his dissertation, I believe) that if we have a universe of sets in which all the axioms of ZF hold except the axiom of regularity, then we can generate a collection of sets in which ZF holds in full.

This means that if our universe happens to be $L$, we can construct it by two different ways from the empty set:

  1. The constructible hierarchy: we begin with the empty set as $L_0$. If $L_\alpha$ was defined then $L_{\alpha+1}$ is everything which is definable from $L_\alpha$; in limit stages we simply gather all the previously defined sets.

    This is a rather "slow" way of generating the universe, since there are only countably many formulas, we have that for infinite levels, $|\alpha|=|L_\alpha|$. So for example $L_{\omega+1}$ is countable.

  2. The von Neumann hierarchy: we begin with the empty set and simply reiterate the power set operation (and again we take unions at limit stages). This is quite a rapid way to generate sets, since we take everything that the universe has to offer. In this case, $V_{\omega+1}$ is already of size continuum, where as $L_{\omega+1}$ is still countable.

In fact, in the constructible hierarchy we add new subsets of natural numbers at every countable step, where as in the von Neumann hierarchy we added them all in the $\omega+2$-th step.

Some years later, Cohen proved that if $V=L$ was consistent then $V\neq L$ was also consistent, this means that if we have a model of $V=L$ then we can create a model in which there is a set which is not constructible, therefore we will have a model that the constructible hierarchy "misses" some sets. In contrast, again, the von Neumann hierarchy misses nothing, as it takes everything the universe has to offer.

There are other ways to generate $L$, for example The Fine Structure hierarchy which I have to admit I am not too familiar with, but it works differently than both the above constructions.

So to conclude, the constructible hierarchy is very different than the von Neumann hierarchy - even when assuming $V=L$ is true (and much more when assuming $V\neq L$).