Set-theoretic arguments often make use of the fact that a particular property $\phi$ is local, in the sense that instances of the property can be verified by checking certain facts in only a bounded part of the set-theoretic universe, such as inside some rank-initial segment $V_{\theta }$ or inside the collection $H_{\kappa }$ of all sets of hereditary size less than $\kappa$. It turns out that this concept is exactly equivalent to the property being ${\mathrm{\Sigma }}_2$ expressible in the language of set theory.

Theorem. For any assertion $\phi$ in the language of set theory, the following are equivalent:

1. $\phi$ is ZFC-provably equivalent to a ${\mathrm{\Sigma }}_2$ assertion.
2. $\phi$ is ZFC-provably equivalent to an assertion of the form “$\mathrm{\exists }\theta V_{\theta }\models \psi$,” where $\psi$ is a statement of any complexity.
3. $\phi$ is ZFC-provably equivalent to an assertion of the form “$\mathrm{\exists }\kappa H_{\kappa }\models \psi$,” where $\psi$ is a statement of any complexity.

Just to clarify, the ${\mathrm{\Sigma }}_2$ assertions in set theory are those of the form $\mathrm{\exists }x\mathrm{\forall }y{\phi }_0(x,y)$, where ${\phi }_0$ has only bounded quantifiers. The set $V_{\theta }$ refers to the rank-initial segment of the set-theoretic universe, consisting of all sets of von Neumann rank less than $\theta$. The set $H_{\kappa }$ consists of all sets of hereditary size less than $\kappa$, that is, whose transitive closure has size less than $\kappa$.

Proof. ($3\to 2$) Since $H_{\kappa }$ is correctly computed inside $V_{\theta }$ for any $\theta >\kappa$, it follows that to assert that some $H_{\kappa }$ satisfies $\psi$ is the same as to assert that some $V_{\theta }$ thinks that there is some cardinal $\kappa$ such that $H_{\kappa }$ satisfies $\psi$.

($2\to 1$) The statement $\mathrm{\exists }\theta V_{\theta }\models \psi$ is equivalent to the assertion $\mathrm{\exists }\theta \mathrm{\exists }x(x=V_{\theta }\wedge x\models \psi )$. The claim that $x\models \psi$ involves only bounded quantifiers, since the quantifiers of $\psi$ become bounded by $x$. The claim that $x=V_{\theta }$ is ${\mathrm{\Pi }}_1$ in $x$ and $\theta$, since it is equivalent to saying that $x$ is transitive and the ordinals of $x$ are precisely $\theta$ and $x$ thinks every $V_{\alpha }$ exists, plus a certain minimal set theory (so far this is just ${\mathrm{\Delta }}_0$, since all quantifiers are bounded), plus, finally, the assertion that $x$ contains every subset of each of its elements. So altogether, the assertion that some $V_{\theta }$ satisfies $\psi$ has complexity ${\mathrm{\Sigma }}_2$ in the language of set theory.

($1\to 3$) This implication is a consequence of the following absoluteness lemma.

Lemma. (Levy) If $\kappa$ is any uncountable cardinal, then $H_{\kappa }{\prec }_{{\mathrm{\Sigma }}_1}V$.

Proof. Suppose that $x\in H_{\kappa }$ and $V\models \mathrm{\exists }y\psi (x,y)$, where $\psi$ has only bounded quantifiers. Fix some such witness $y$, which exists inside some $H_{\gamma }$ for perhaps much larger $\gamma$. By the Löwenheim-Skolem theorem, there is $X\prec H_{\gamma }$ with $\text{TC}(\{x\})\subset X$, $y\in X$ and $X$ of size less than $\kappa$. Let $\pi :X\cong M$ be the Mostowski collapse of $X$, so that $M$ is transitive, and since it has size less than $\kappa$, it follows that $M\subset H_{\kappa }$. Since the transitive closure of $\{x\}$ was contained in $X$, it follows that $\pi (x)=x$. Thus, since $X\models \psi (x,y)$ we conclude that $M\models \psi (x,\pi (y))$ and so hence $\pi (y)$ is a witness to $\psi (x,\cdot )$ inside $H_{\kappa }$, as desired. QED

Using the lemma, we now prove the remaining part of the theorem. Consider any ${\mathrm{\Sigma }}_2$ assertion $\mathrm{\exists }x\mathrm{\forall }y{\phi }_0(x,y)$, where ${\phi }_0$ has only bounded quantifiers. This assertion is equivalent to $\mathrm{\exists }\kappa H_{\kappa }\models \mathrm{\exists }x\mathrm{\forall }y{\phi }_0(x,y)$, simply because if there is such a $\kappa$ with $H_{\kappa }$ having such an $x$, then by the lemma this $x$ works for all $y\in V$ since $H_{\kappa }{\prec }_{{\mathrm{\Sigma }}_1}V$; and conversely, if there is an $x$ such that $\mathrm{\forall }y{\phi }_0(x,y)$, then this will remain true inside any $H_{\kappa }$ with $x\in H_{\kappa }$. QED

In light of the theorem, it makes sense to refer to the ${\mathrm{\Sigma }}_2$ properties as the locally verifiable properties, or perhaps as semi-local properties, since positive instances of ${\mathrm{\Sigma }}_2$ assertions can be verified in some sufficiently large $V_{\theta }$, without need for unbounded search. A truly local property, therefore, would be one such that positive and negative instances can be verified this way, and these would be precisely the ${\mathrm{\Delta }}_2$ properties, whose positive and negative instances are locally verifiable.

Tighter concepts of locality are obtained by insisting that the property is not merely verified in some $V_{\theta }$, perhaps very large, but rather is verified in a $V_{\theta }$ where $\theta$ has a certain closeness to the parameters or instance of the property. For example, a cardinal $\kappa$ is measurable just in case there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and this is verified inside $V_{\kappa +2}$. Thus, the assertion “$\kappa$ is measurable,” has complexity ${\mathrm{\Sigma }}_1^2$ over $V_{\kappa }$. One may similarly speak of ${\mathrm{\Sigma }}_m^n$ or ${\mathrm{\Sigma }}_m^{\alpha }$ properties, to refer to properties that can be verified with ${\mathrm{\Sigma }}_m$ assertions in $V_{\kappa +\alpha }$. Alternatively, for any class function $f$ on the ordinals, one may speak of $f$-local properties, meaning a property that can be checked of $x\in V_{\theta }$ by checking a property inside $V_{f(\theta )}$.

This post was made in response to a question on MathOverflow.