The Constructible universe (denoted $L$) was invented by Kurt Gödel as a transitive inner model of [[ZFC]]+[[GCH]] (assuming the consistency of $\text{ZF}$) showing that $\text{ZFC}$ cannot disprove $\text{GCH}$. It was then shown to be an important model of $\text{ZFC}$ for its satisfying of other axioms, thus making them consistent with $\text{ZFC}$. The idea is that $L$ is built up by ranks like $V$. $L_0$ is the empty set, and $L_{\alpha+1}$ is the set of all easily definable subsets of $L_\alpha$. The assumption that $V=L$ (also known as the **Axiom of constructibility**) is undecidable from $\text{ZFC}$, and implies many axioms which are consistent with $\text{ZFC}$. A set $X$ is **constructible** iff $X\in L$. $V=L$ iff every set is constructible.
## Definition
$\mathrm{def}(X)$ is the set of all "easily definable" subsets of $X$ (specifically the $\Delta_0$ definable subsets). More specifically, a subset $x$ of $X$ is in $\mathrm{def}(X)$ iff there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,v_0,v_1...v_n]\}$. Then, $L_\alpha$ and $L$ are defined as follows:
- $L_0=\emptyset$
- $L_{\alpha+1}=\mathrm{def}(L_\alpha)$
- $L_\beta=\bigcup_{\alpha<\beta} L_\alpha$ if $\beta$ is
a limit ordinal
- $L=\bigcup_{\alpha\in\mathrm{Ord}} L_\alpha$
### The Relativized constructible universes $L_\alpha(W)$ and $L_\alpha[W]$
$L_\alpha(W)$ for a class $W$ is defined the same way except $L_0(W)=\text{TC}(\{W\})$ (the transitive closure of $\{W\}$).
$L_\alpha[W]$ for a class $W$ is defined in the same way as $L$ except using $\mathrm{def}_W(X)$, where $\mathrm{def}_W(X)$ is the set of all $x\subseteq X$ such that there is a first-order formula $\varphi$ and $v_0,v_1...v_n\in X$ such that $x=\{y\in X:\varphi^X[y,W,v_0,v_1...v_n]\}$ (because the relativization of $\varphi$ to $X$ is used and $\langle X,\in\rangle$ is not used, this definition makes sense even when $W$ is not in $X$).
$L[W]=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha[W]$ is always a model of $\text{ZFC}$, and always satisfies $\text{GCH}$ past a certain cardinality.
$L(W)=\bigcup_{\alpha\in\mathrm{Ord}}L_\alpha(W)$ is always a model of $\text{ZF}$ but need not satisfy $\text{AC}$ (the axiom of choice).
In particular, $L(\mathbb{R})$ is, under large cardinal assumptions, a model of the [[Axiom of determinacy]]. However, Shelah proved that if $\lambda$ is a strong limit cardinal of uncountable cofinality then $L(\mathcal{P}(\lambda))$ is a model of $\text{AC}$.
## Alternate definition
$L$ and $L[A]$ can also be defined by using [[Jensen's hierarchy]]:
![[Jensen's hierarchy#^264f5d]]
## The difference between $L_\alpha$ and $V_\alpha$
For $\alpha\leq\omega$, $L_\alpha=V_\alpha$.
However, $\|L_{\omega+\alpha}\|=\aleph_0 + \|\alpha\|$ whilst $\|V_{\omega+\alpha}\|=\beth_\alpha$. Unless $\alpha$ is a [[Beth]], $\|L_{\omega+\alpha}\|<\|V_{\omega+\alpha}\|$.
Although $L_\alpha$ is quite small compared to $V_\alpha$, $L$ is a tall model, meaning $L$ contains every ordinal. In fact, $V_\alpha\cap\mathrm{Ord}=L_\alpha\cap\mathrm{Ord}=\alpha$, so the ordinals in $V_\alpha$ are precisely those in $L_\alpha$.
If $0^{\sharp}$ exists (see below), then every uncountable cardinal $\kappa$ has $L\models
quot;$\kappa$ is [[Ineffable]] (and therefore the smallest actually totally ineffable cardinal $\lambda$ has many more large cardinal properties in $L$).
However, if $\kappa$ is [inaccessible](Inaccessible.md "Inaccessible") and $V=L$, then $V_\kappa=L_\kappa$. Furthermore, $V_\kappa\models (V=L)$. In the case where $V\neq L$, it is still true that $V_\kappa^L=L_\kappa$, although $V_\kappa^L$ will not be $V_\kappa$. In fact, $\mathcal{P}(\omega)\not\in V_\kappa^L$ if $0^{\sharp}$ exists.
## Statements True in $L$
Here is a list of statements true in $L$ of any model of $\text{ZF}$:
- $\text{ZFC}$ (and therefore the Axiom of Choice)
- $\text{GCH}$
- $V=L$ (and therefore $V$ $=$ [$\text{HOD}$](HOD.md "HOD"))
- The [[Diamond_principle]]
- The <a href="index.php?title=Clubsuit_principle&action=edit&redlink=1" class="new" title="Clubsuit principle (page does not exist)">clubsuit principle</a>
- The falsity of <a href="index.php?title=Suslin%27s_hypothesis&action=edit&redlink=1" class="new" title="Suslin's hypothesis (page does not exist)">Suslin's hypothesis</a>
## Determinacy of $L(\mathbb{R})$
*Main article: [[Axiom of determinacy#Determinacy of $L( mathbb{R})$]]
## Using other logic systems than first-order logic
*Main article:* [[Inner models from extended logics]]
When using second order logic in the definition of $\mathrm{def}$, the new hierarchy is called $L_\alpha^{II}$. Interestingly, $L^{II}=\text{HOD}$.
When using $\mathcal{L}_{\kappa,\kappa}$, the hierarchy is called $L_\alpha^{\mathcal{L}_{\kappa,\kappa}}$, and $L\subseteq L^{\mathcal{L}_{\kappa,\kappa}}\subseteq L(V_\kappa)$.
Chang's Model is $L^{\mathcal{L}_{\omega_1,\omega_1}}$. Chang proved that $L^{\mathcal{L}_{\kappa,\kappa}}$ is the smallest inner model of $\text{ZFC}$ closed under sequences of length lt;\kappa$.
Finally, when using $\mathcal{L}_{\infty,\infty}$, it turns out that the result is $V$.
## Silver indiscernibles
*To be expanded.*
## Silver cardinals
A cardinal $κ$ is **Silver** if in a set-forcing extension there is a club in $κ$ of generating indiscernibles for $V_κ$ of order-type $κ$.
This is a very strong property downwards absolute to $L$, e.g.:{% cite Gitmana %}
- Every element of a club $C$ witnessing that $κ$ is a Silver cardinal
is
<a href="Rank-into-rank" class="mw-redirect" title="Rank-into-rank">virtually rank-into-rank</a>.
- If $C ∈ V[H]$, a forcing extension by $\mathrm{Coll}(ω, V_κ)$,
is a club in $κ$ of generating indiscernibles for $V_κ$ of
order-type $κ$, then each $ξ ∈ C$ is lt;
ω_1$-<a href="Iterable" class="mw-redirect" title="Iterable">iterable</a>.
## $0^\sharp$
See [[Zero sharp]]