Approximation and cover properties propagate upward

I should like to record here the proof of the following fact, which Jonas Reitz and I first observed years ago, when he was my graduate student, and I recall him making the critical observation.

It concerns the upward propagation of the approximation and cover properties, some technical concepts that lie at the center of my paper, Extensions with he approximation and cover properties have no new large cardinals, and which are also used in my proof of Laver’s theorem on the definability of the ground model, and which figure in Jonas’s work on the ground axiom.

The fact has a curious and rather embarrassing history, in that Jonas and I have seen an unfortunate cycle, in which we first proved the theorem, and then subsequently lost and forgot our own proof, and then lost confidence in the fact, until we rediscovered the proof again. This cycle has now repeated several times, in absurd mathematical comedy, and each time the proof was lost, various people with whom we discussed the issue sincerely doubted that it could be true.  But we are on the upswing now, for in response to some recently expressed doubts about the fact, although I too was beginning to doubt it again, I spent some time thinking about it and rediscovered our old proof! Hurrah!  In order to break this absurd cycle, however, I am now recording the proof here in order that we may have a place to point in the future, to give the theorem a home.

Although the fact has not yet been used in any application to my knowledge, it strikes me as inevitable that this fundamental fact about the approximation and cover properties will eventually find an important use.

Definition. Assume $\delta$ is a cardinal in $V$ and $W\subset V$ is a transitive inner model of set theory.

  • The extension $W\subset V$ satisfies the $\delta$-approximation property if whenever $A\subset W$ is a set in $V$ and $A\cap a\in W$ for any $a\in W$ of size less than $\delta$ in $W$, then $A\in W$.
  • The extension $W\subset V$ satisfies the $\delta$-cover property if whenever $A\subset W$ is a set of size less than $\delta$ in $V$, then there is a covering set $B\in W$ with $A\subset B$ and $|B|^W\lt\delta$.

Theorem. If $W\subset V$ has the $\delta$-approximation and $\delta$-cover properties and $\delta\lt\gamma$ are both infinite cardinals in $V$, then it also has the $\gamma$-approximation and $\gamma$-cover properties.

Proof. First, notice that the $\delta$-approximation property trivially implies the $\gamma$-approximation property for any larger cardinal $\gamma$. So we need only verify the $\gamma$-cover property, and this we do by induction. Note that the limit case is trivial, since if the cover property holds at every cardinal below a limit cardinal, then it trivially holds at that limit cardinal, since there are no additional instances of covering to be treated. Thus, we reduce to the case $\gamma=\delta^+$, meaning $(\delta^+)^V$, but we must allow that $\delta$ may be singular here.

If $\delta$ is singular, then we claim that the $\delta$-cover property alone implies the $\delta^+$-cover property: if $A\subset W$ has size $\delta$ in $V$, then by the singularity of $\delta$ we may write it as $A=\bigcup _{\alpha\in I}A_\alpha$, where each $A_\alpha$ and $I$ have size less than $\delta$. By the $\delta$-cover property, there are covers $A_\alpha\subset B_\alpha\in W$ with $B_\alpha$ of size less than $\delta$ in $W$.  Furthermore, the set $\{B_\alpha\mid\alpha\in I\}$ itself is covered by some set $\mathcal{B}\in W$ of size less than $\delta$ in $W$. That is, we cover the small set of small covers. We may assume that every set in $\mathcal{B}$ has size less than $\delta$, by discarding those that aren’t, and so $B=\bigcup\mathcal{B}$ is a set in $W$ that covers $A$ and has size at most $\delta$ there, since it is small union of small sets, thereby verifying this instance of the $\gamma$-cover property.

If $\delta$ is regular, consider a set $A\subset W$ with $A\in V$ of size $\delta$ in $V$, so that $A=\{a_\xi\mid\xi\lt\delta\}$. For each $\alpha\lt\delta$, the initial segment $\{a_\xi\mid\xi\lt\alpha\}$ has size less than $\delta$ and is therefore covered by some $B_\alpha\in W$ of size less than $\delta$ in $W$.  By adding each $B_\alpha$ to what we are covering at later stages, we may assume that they form an increasing tower: $\alpha\lt\beta\to B_\alpha\subset B_\beta$. The choices $\alpha\mapsto B_\alpha$ are made in $V$.  Let $B=\bigcup_\alpha B_\alpha$, which certainly covers $A$. Observe that for any set $a\in W$ of size less than $\delta$, it follows by the regularity of $\delta$ that $B\cap a=B_\alpha\cap a$ for all sufficiently large $\alpha$.  Thus, all $\delta$-approximations to $B$ are in $W$ and so $B$ itself is in $W$ by the $\delta$-approximation property, as desired. Note that $B$ has size less than $\gamma$ in $W$, because it has size $\delta$ in $V$, and so we have verified this instance of the $\gamma$-cover property for $W\subset V$.

Thus, in either case we’ve established the $\gamma$-cover property for $W\subset V$, and the proof is complete. QED

(Thanks to Thomas Johnstone for some comments and for pointing out a simplification in the proof:  previously, I had reduced without loss of generality to the case where $A$ is a set of ordinals of order type $\delta$; but Tom pointed out that the general case is not actually any harder.   And indeed, Jonas dug up some old notes to find the 2008 version of the argument, which is essentially the same as what now appears here.)

Note that without the $\delta$-approximation property, it is not true that the $\delta$-cover property transfers upward. For example, every extension has the $\aleph_0$-cover property.

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