Constructible universe

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The Constructible universe (denoted $L$) was invented by Kurt Gödel as a transitive inner model of GCH + ZFC (assuming the consistency of ZFC) showing that ZFC cannot disprove GCH. It was then shown to be an important model of ZFC for its satisfying of other axioms, thus making them consistent with 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 ZFC, and implies many axioms which are consistent with ZFC to be true. 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 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 $X$).

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 $\beth$-fixed point or $\omega$, $|L_{\omega+\alpha}|<|V_{\omega+\alpha}|$. 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 $0^{\#}$ exists, then every uncountable cardinal $\kappa$ has $L\models\kappa\;\mathrm{is}\;\mathrm{totally}\;\mathrm{ineffible}$ (and therefore the smallest actually totally ineffible cardinal $\lambda$ has many more large cardinal properties in $L$).

However, if $\kappa$ is 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 $0^{\#}$ exists.

Statements True in $L$

Here is a list of statements true in $L$:

  • ZFC (and therefore the Axiom of Choice)
  • GCH
  • $V=L$ (and therefore $V$ $=$ $HOD$)
  • The Diamond Principle
  • The Clubsuit Principle
  • The Falsity of Suslin's Hypothesis

Using Other Logic Systems than First-order Logic

When using second order logic in the definition of $\mathrm{def}$, the new hierarchy is called $L_\alpha^{II}$. Interestingly, $L^{II}=HOD$. When using $\mathcal{L}_{\kappa,\kappa}$, the hierarchy is called $L_\alpha^{\mathcal{L}_{\kappa,\kappa}}$, and $L\subseteq L^{\mathcal{L}_{\kappa,\kappa}}\subseteq L(V_\kappa)$. Therefore, $V\neq L^{\mathcal{L}_{\kappa,\kappa}}$ iff $V\neq L$. Finally, when using $\mathcal{L}_{\infty,\infty}$, it turns out that the result is $V$.

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 ZFC closed under sequences of length $<\kappa$.

References

  • Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003.
  • user46667, Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem), URL (version: 2014-03-17): https://mathoverflow.net/q/156940
  • Chang, C. C. (1971), "Sets Constructible Using $\mathcal{L}_{\kappa,\kappa}$", Axiomatic Set Theory, Proc. Sympos. Pure Math., XIII, Part I, Providence, R.I.: Amer. Math. Soc., pp. 1–8