Introduction to remarkable cardinals

This is a talk at the CUNY Set Theory Seminar, March 14, 2014.

The lecture will examine basic properties of remarkable cardinals, which were introduced by Ralf Schindler in [1] to pinpoint the consistency strength of the assumption that the theory of $L(\mathbb R)$ cannot be altered by proper forcing. More on this in a bit.

The model $L(\mathbb R)$ of ${\rm ZF}+{\rm DC}$ is obtained by starting with the real numbers $\mathbb R$ instead of $\emptyset$ in Gödel's construction of $L$. By construction, $L(\mathbb R)$ has all the reals of $V$. A modification of Solovay's original proof shows that if $\kappa$ is inaccessible, then in $L(\mathbb R)$ of the ${\rm Coll}(\omega,\lt\kappa)$ forcing extension all sets of reals have the classical regularity properties such as Lebesgue measurability, the Baire property, and the perfect set property. These regularity properties follow also from assuming the Axiom of Determinacy (${\rm AD}$), and under certain large cardinal hypothesis ($\omega$-many Woodin cardinals with a measurable above them), $L(\mathbb R)$ is a model of ${\rm AD}$. Surprisingly the consistency strength of $L(\mathbb R)\models{\rm AD}$ is deeply connected with the assumption that the theory of $L(\mathbb R)$ cannot be altered by (set) forcing. If $\Gamma$ is a class of forcing notions, let us say that $L(\mathbb R)$ is absolute for $\Gamma$ if for every formula $\varphi(x)$, every real $r\in V$, and every $\mathbb P\in \Gamma$, we have $$L(\mathbb R)\models \varphi(r)\text{ if and only if} \Vdash_{\mathbb P} L(\dot{\mathbb R})\models\varphi(r).$$

Kunen had showed that $L(\mathbb R)$ absoluteness for $ccc$ forcing is precisely equiconsistent with a weakly compact cardinal and a natural question arose about the consistency strength of $L(\mathbb R)$ absoluteness for proper forcing. Preliminary results seemed to indicate that absoluteness for proper forcing might carry consistency strength in the neighborhood of full absoluteness. Schindler had showed that assuming $L(\mathbb R)$ absoluteness for the class of reasonable forcing implies that there is an inner model with a strong cardinal [bibcite key=schindler:reasonableforcing]. Reasonable forcing preserve the stationarity of $[\omega_1]^\omega$ of $V$ to the forcing extension and so in particular generalize proper forcing. Only it turned out that $L(\mathbb R)$ absoluteness by proper forcing is much weaker. Schindler introduced the notion of remarkable cardinals because these are precisely equiconsistent with $L(\mathbb R)$ absoluteness for proper forcing [1].

To motivate the definition of remarkable cardinals, it is good to be familiar with a result of Magidor showing that a cardinal $\kappa$ is supercompact if and only if for every regular cardinal $\lambda>\kappa$, there are $\gamma<\overline\lambda<\kappa$ and an elementary embedding $j:H_{\overline\lambda}\to H_\lambda$ with critical point $\gamma$ and $j(\gamma)=\kappa$ [2]. A cardinal $\kappa$ is remarkable if in the ${\rm Coll}(\omega,\lt\kappa)$ forcing extension, for every regular cardinal $\lambda>\kappa$, there are $\gamma<\overline\lambda<\kappa$ and an elementary embedding $j:H_{\overline\lambda}^V\to H_\lambda^V$ with critical point $\gamma$ and $j(\gamma)=\kappa$. Strong cardinals are remarkable, but remarkable cardinals are much weaker and they are downward absolute to $L$. A remarkable cardinal is totally indescribable meaning that every $n$-th order sentence that holds true in $V_\kappa$ for a remarkable $\kappa$ already holds true over some $V_\alpha$ with $\alpha<\kappa$. It follows that the least measurable $\kappa$, which is describable by the existence of a $\kappa$-complete measure, cannot be remarkable.

Schindler had showed that remarkable cardinals are weaker than $\omega$-Erdős cardinals [3]. Jointly with Welch, we showed that remarkable cardinals nest between 1-iterable and 2-iterable cardinals [4]. More precisely, remarkable cardinals are 1-iterable limits of 1-iterable cardinals and if $\kappa$ is 2-iterable, then $V_\kappa$ is a model of proper class many remarkable cardinals. We introduced the $\alpha$-iterable hierarchy of cardinals (for $1\leq\alpha\leq\omega_1$) by defining that a cardinal $\kappa$ is $\alpha$-iterable if for every $A\subseteq\kappa$, there is a transitive model $M\models{\rm ZFC}^-$ (${\rm ZFC}$ without powerset) of size $\kappa$ with $\kappa\in M$ for which there is a weakly amenable $M$-ultrafilter $U$ on $\kappa$ producing at least $\alpha$-many well-founded iterated ultrapowers. An $M$-ultrafilter measures all subsets of $\kappa$ of $M$ and is $\kappa$-complete for sequences in $M$. Weak amenability is required to define the iterated ultrapower construction because it mandates that the potentially external $M$-ultrafilter is partially internal to $M$. An $M$-ultrafilter is weakly amenable if for every $X\in M$ such that $|X|^M=\kappa$, we have $X\cap U\in M$. 1-iterable cardinals sit much higher in the hierarchy than weakly compact cardinals and the entire $\alpha$-iterable hierarchy is bounded by a Ramsey cardinal. We showed that the $\alpha$-iterable hierarchy truly deserves its name because whenever $\beta<\alpha$ and $\kappa$ is $\alpha$-iterable, then it is a limit of $\beta$-iterable cardinals [4].

References

  1. R.-D. Schindler, “Proper forcing and remarkable cardinals,” Bull. Symbolic Logic, vol. 6, no. 2, pp. 176–184, 2000.
  2. M. Magidor, “On the role of supercompact and extendible cardinals in logic,” Israel J. Math., vol. 10, pp. 147–157, 1971.
  3. R.-D. Schindler, “Proper forcing and remarkable cardinals. II,” J. Symbolic Logic, vol. 66, no. 3, pp. 1481–1492, 2001.
  4. V. Gitman and P. D. Welch, “Ramsey-like cardinals II,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 541–560, 2011.