Forum:Can anyone help me about constructible hierarchy?

Constructible hierarchy is defined as: It can also be defined using Gödel operations as: But I'm not clear about how it works. Here are some questions I have: &#123;hyp/^,cos&#125; (talk) 07:13, February 12, 2018 (UTC)
 * \(L_0=\{\}\)
 * \(L_{\alpha+1}=\{\{x\in L_\alpha|(L_\alpha,\in)\models\Phi(x,y_1,\cdots,y_n)\}|\Phi\text{ is first-order formula and }y_1,\cdots,y_n\in L_\alpha\}\)
 * For limit \(\alpha\), \(L_\alpha=\bigcup_{\beta<\alpha}L_\beta\)
 * \(L_0=\{\}\)
 * \(C_0(A)=A\)
 * \(C_{n+1}(A)=\{\{X,Y\}|X,Y\in C_n(A)\}\\ \cup\{X\times Y|X,Y\in C_n(A)\}\\ \cup\{\{(x,y)|x\in X\wedge y\in Y\wedge x\in y\}|X,Y\in C_n(A)\}\\ \cup\{X-Y|X,Y\in C_n(A)\}\\ \cup\{X\cap Y|X,Y\in C_n(A)\}\\ \cup\{\cup X|X\in C_n(A)\}\\ \cup\{\{x|\exists y(x,y)\in X\}|X\in C_n(A)\}\\ \cup\{\{(x,y)|(y,x)\in X\}|X\in C_n(A)\}\\ \cup\{\{((x,y),z)|((x,z),y)\in X\}|X\in C_n(A)\}\\ \cup\{\{((x,y),z)|((y,z),x)\in X\}|X\in C_n(A)\}\)
 * \(L_{\alpha+1}=P(L_\alpha)\cap\bigcup_{n<\omega}C_n(L_\alpha\cup\{L_\alpha\})\)
 * For limit \(\alpha\), \(L_\alpha=\bigcup_{\beta<\alpha}L_\beta\)
 * 1) There is a theorem saying "\(L_\omega=V_\omega\)". Then, how to determine \(L_{\omega+1}\)?
 * 2) What kind of things belong to \(V_{\omega+1}\) but don't belong to \(L_{\omega+1}\)? Can we take an example of them?
 * 3) What does \(L_{\omega+1}\) look like?
 * \(|V_{\omega+1}|=\beth_1\) is the least uncountable term in \(V_\alpha\). For comparison, what's the least ordinal \(\alpha\) such that \(|L_\alpha|>\aleph_0\)?