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As far as I know, difference between constructible universe and von Neumann universe is that the elements of constructible universe, $L$, must be defined by a first-order formula. What I do not get is how this may be a proper subset of von Neumann universe. I know that it is an open question whether $V=L$ is true or not.

More specifically, what does it mean by saying definable by first-order formula? If I am mistaken, can anyone provide me correction and brief review of constructible universe?


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I honestly suggest that you pick a book in set theory and read through it; alternatively take a course which is foundational. There are many fine points which are confusing (especially with independence and different models of set theory), and a course or a good book can be helpful to learn and avoid those confusions later on. –  Asaf Karagila Feb 29 '12 at 1:11
@AsafKaragila can you recommend any good book about set theory? –  user24796 Feb 29 '12 at 3:43
Probably the first part of Jech's Set Theory a good place to start. I heard good things about Enderton's book and Devlin's Joy of Sets. –  Asaf Karagila Mar 2 '12 at 0:15

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up vote 7 down vote accepted

The statement $V=L$ is an additional axiom to the usual axioms of $ZFC$. It is not an open problem whether or not it is true, this is the basic results by Cohen when he first developed the technique of forcing:

Suppose that $ZFC+V=L$ is consistent and has a countable transitive model, we can then produce a countable transitive model of $ZFC+V\neq L$. This shows that one cannot prove nor disprove $V=L$ from the axioms of $ZFC$, but rather needs to either assume that we work in $L$ or that we don't work in $L$.

Just as well there are other definable inner models, $L[A]$ in which we add a predicate $A$ to the language; or $L(A)$ in which we take a model defined when starting from the set $A$ (or from its transitive closure). There are models which may be somewhat larger like $OD$ (sets which are defined by ordinals) and similar extensions. To say that those collections are "definable inner models" means that the collection $\{x\mid\varphi(x,y)\}$ for some formula $x$ with a parameter $y$ is a model of $ZFC$ which is a subclass of the universe.

The question whether or not the universe of set theory in which you work in a given moment satisfies such axiom depends on the model itself, much like working in an algebraically closed field need not mean that $\pi$ is in that field.

The constructible universe starts from the idea of definability. We can write formulas in the language of set theory, these formulas can be satisfied sometimes, for example $\varphi(x):=\forall y(y\notin x)$ will be only true if $x=\varnothing$.

We can define a lot of things, and we can define things with parameters, that is $\varphi(x,y)$ to be some formula that after fixing the variable $y$ to be a certain set only a certain collection of elements placed into $x$ will give a true value.

The construction of $L$ starts with the empty set, we then begin collecting everything which is definable over the empty set, in this case it is not much. We then take things which are definable over that collection, and so on. We continue this over all the ordinals, where in limit stages we simply collect everything we have so far.

The result, as it turns out, is a model of $ZFC$ (granted that we started with a model of $ZF$) and the surprising part is that there is a formula $\varphi(x)$ in the language of set theory which is true exactly when $x$ is in that collection. So the axiom $\forall x.\varphi(x)$ would ensure that the model is exactly $L$.

In contrast, the von Neumann universe simply starts with an empty set and reiterates the power set operation, instead of limiting ourselves to definable things, we take everything that the universe has to offer. This gives us a model of $ZF$ (not necessarily $ZFC$) and this model can be very wild in comparison to $L$.

The construction of $L$ is so rigid that we cannot change it by adding new sets to the universe (e.g. forcing) or by considering inner models (a subclass of the universe which is also a model of $ZF$). The construction of von Neumann holds for every model of $ZF$, so if $M$ was a model of $ZF$ and I start to reiterate the power set (in $M$) operation from $\varnothing$ I will end up with $M$ again. This tells us that every universe of $ZFC$ can be constructed as a von Neumann universe, $L$ as well.

If, however, I started with $M$ and I redid the $L$ construction I will end up with $L$, so if $M\models V\neq L$ the result will be a different model.

To finish with a minor remark, the universe of set theory is not a set but a class. So $L$ is not a subset of the universe but a subclass of the universe, the distinction is important because sets are elements of the universe and (proper) classes are not.

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