COMPACTNESS OFAND STRONG AXIOMS OF DETERMINACY - N. Trang, Structure theory ofand its applications. Journal of Symbolic Logic, vol. 80 (2015), no. 1, pp. 29–55. - N. Trang, Supercompactness can be equiconsistent with measurability. Notre Dame Journal of Formal Logic, vol. 62 (2021), no. 4, pp. 593–618. - N. Trang and T. Wilson, Determinacy from strong compactness of . Annals of Pure and Applied Logic, vol. 172 (2021), no. 6, Article no. 102944, 30pp. - D. Ikegami and N. Trang, On supercompactness of $\omega _1$, Advances in Mathematical Logic (T. Arai, M. Kikuchi, S. Kuroda, M. Okada, T. Yorioka, editors), Springer, Proceedings Mathematics & Statistics, Singapore, 369, 2021, pp. 27–45.

The Association for Symbolic Logic publishes analytical reviews of selected books and articles in the field of symbolic logic. The reviews were published in The Journal of Symbolic Logic from the founding of the journal in 1936 until the end of 1999. The Association moved the reviews to this bulletin, beginning in 2000.

The Reviews Section is edited by Graham Leach-Krouse (Managing Editor), Albert Atserias, Mark van Atten, Clinton Conley, Johanna Franklin, Dugald Macpherson, Antonio Montalbán, Valeria de Paiva, Christian Retoré, Marion Scheepers, and Nam Trang. Authors and publishers are requested to send, for review, copies of books to ASL, Department of Mathematics, University of Connecticut, 341 Mansfield Road, U-1009, Storrs, CT 06269-1009, USA.

For an uncountable cardinal $\kappa $ and a set X, we say that $\kappa $ is X-supercompact if there is a fine, normal, $\kappa $ -complete measure on $\wp _{\kappa }(X)$ and say that $\kappa $ is supercompact if it is X-supercompact for any set X. Although this is a rather strong large cardinal property, even $\omega _1$ can be supercompact in the absence of the axiom of choice. Indeed, Takeuti showed that if $\kappa $ is a supercompact cardinal and $g\subseteq \mathrm {Col}(\omega , {<}\kappa )$ is V-generic, then $V(\mathbb {R}^{V[g]})\models \omega _1$ is supercompact (cf. G. Takeuti, A relativization of axioms of strong infinity to $\omega _1$ . Annals of the Japan Association for Philosophy of Science , vol. 3 (1970), pp. 191–204). It turned out that the (partial) supercompactness of $\omega _1$ is particularly interesting because of its connection to the axiom of determinacy ( $\mathsf {AD}$ ) and its strengthenings. Through this review, we give a brief summary of the progress made on this topic over the last 10 years.

Let us start with a classical result by Solovay: Under the axiom of determinacy for games on reals ( $\mathsf {AD}_{\mathbb {R}}$ ), $\omega _1$ is $\mathbb {R}$ -supercompact witnessed by the club filter on $\wp _{\omega _1}(\mathbb {R})$ (cf. R. M. Solovay, The independence of $\mathsf {DC}$ from $\mathsf {AD}$ , Large Cardinals, Determinacy and Other Topics: The Cabal Seminar Volume IV (A. S. Kechris, B. Löwe, J. R. Steel, editors), Cambridge University Press, Cambridge, Lecture Notes in Logic, 49, 2021, pp. 66–95). As a corollary of this, one can easily show that $\mathsf {AD}_{\mathbb {R}}$ implies that $\omega _1$ is ${<}\Theta $ -supercompact, i.e., $\kappa $ -supercompact for any $\kappa <\Theta :=\mathrm{sup} \{\alpha \in \mathrm {Ord}\mid \text {There is a surjection } f\colon \mathbb {R}\to \alpha \}$ . This corollary, however, can be greatly improved using inner model theory. The key result is due to Steel and Woodin, who showed that if $\mathsf {AD}$ holds in $L(\mathbb {R})$ , $\mathsf {HOD}^{L(\mathbb {R})}$ can be characterized as a fine structural model (cf. J. R. Steel and W. H. Woodin, HOD as a core model, Ordinal Definability and Recursion Theory: The Cabal Seminar Volume III (A. S. Kechris, B. Löwe, J. R. Steel, editors), Cambridge University Press, Cambridge, Lecture Notes in Logic, 43, 2016, pp. 257–345). Based on this analysis of $\mathsf {HOD}^{L(\mathbb {R})}$ , Woodin showed that under $\mathsf {AD}\,+\,V=L(\mathbb {R})$ , $\omega _1$ is ${<}\Theta $ -supercompact. Also, using Woodin’s method, Neeman showed the uniqueness of supercompact measures on $\wp _{\omega _1}(\alpha )$ for any $\alpha <\Theta $ under the same assumption (cf. I. Neeman, Inner models and ultrafilters in $L(\mathbb {R})$ . Bulletin of Symbolic Logic , vol. 13 (2007), no. 1, pp. 31–53). Woodin also observed that the hypothesis $\mathsf {AD}\,+\,V=L(\mathbb {R})$ of these results can be replaced with $\mathsf {AD}^+$ , which is a technical strengthening of $\mathsf {AD}$ . Note that it is conjectured that $\mathsf {AD}^+$ is equivalent to $\mathsf {AD}$ . Although the full details of Woodin’s observation have never been written down, one can find discussion about this in J. R. Steel, Ordinal definability in models of determinacy: Introduction to part V, Ordinal Definability and Recursion Theory: The Cabal Seminar Volume III (A. S. Kechris, B. Löwe, J. R. Steel, editors), Cambridge University Press, Cambridge, Lecture Notes in Logic, 43, 2016, pp. 3–48, and G. Sargsyan, $\mathsf {AD}_{\mathbb {R}}$ implies that all sets of reals are $\Theta $ universally Baire. Archive for Mathematical Logic , vol. 60 (2021), pp. 1–15.

Now let us consider $\mathbb {R}$ -supercompactness of $\omega _1$ . This is not very strong assumption in terms of consistency strength because if one starts with a measurable cardinal, then Takeuti’s model would satisfy “ $\mathsf {DC}\,+\,\omega _1$ is $\mathbb {R}$ -supercompact.” The theory “ $\mathsf {AD}\,+\,\mathsf {DC}\,+\,\omega _1$ is $\mathbb {R}$ -supercompact” is also much weaker than $\mathsf {AD}_{\mathbb {R}}$ . Indeed, Woodin showed that the following theories are equiconsistent:

1. (1-1) $\mathsf {ZFC}\,+\,$ there are $\omega ^2$ many Woodin cardinals.

2. (1-2) $L(\mathbb {R}, \mu )\models $ “ $\mathsf {ZF}\,+\,\mathsf {DC}\,+\,\mathsf {AD}\,+\,\omega _1\text { is } \mathbb {R}\text {-supercompact}$ ,” where $\mu $ is the club filter on $\wp _{\omega _1}(\mathbb {R})$ .

The model $L(\mathbb {R}, \mu )$ is called the Solovay model. In the first paper under review, Structure theory of $L(\mathbb {R}, \mu )$ and its applications, Trang starts with a proof of this equiconsistency. The forward direction is a variant of derived model construction in J. R. Steel, The derived model theorem, Logic Colloquium 2006 (S. B. Cooper, H. Geuvers, A. Pillay, J. Väänänen, editors), Cambridge University Press, Cambridge, Lecture Notes in Logic, 32, 2009, p. 280–327. The reverse direction makes use of Prikry forcing associated with the measure $\mu $ on $\wp _{\omega _1}(\mathbb {R})$ as in P. Koellner and W. H. Woodin, Large cardinals from determinacy, Handbook of Set Theory (M. Foreman, A. Kanamori, editors), Springer Dordrecht, Dordrecht, vol. 3 , 2010, pp. 1951–2119. Using these techniques, Trang gave two kinds of applications. One is the $\mathsf {HOD}$ analysis in $L(\mathbb {R}, \mu )$ , which is a natural adaptation of Steel–Woodin’s result for $L(\mathbb {R})$ . The other is the construction of a certain ideal on $\wp _{\omega _1}(\mathbb {R})$ in the $\mathbb {P}_{\max }$ extension of $L(\mathbb {R}, \mu )$ . It is also shown that existence of such an ideal is equiconsistent with (1-2). One can find further results on the Solovay model in N. Trang, Determinacy in $L(\mathbb {R}, \mu )$ . Journal of Mathematical Logic , vol. 14 (2014), no. 1, Article no. 1450006, 23pp and D. Rodríguez and N. Trang, $L(\mathbb {R}, \mu )$ is unique. Advances in Mathematics , vol. 324 (2018), pp. 355–393.

Next, we consider $\wp (\mathbb {R})$ -supercompactness of $\omega _1$ . The easiest way to get such supercompactness together with $\mathsf {AD}$ is by assuming much stronger determinacy than $\mathsf {AD}_{\mathbb {R}}$ . Indeed, it is a folklore result that “ $\mathsf {DC}\,+\,\mathsf {AD}_{\mathbb {R}}\,+$ there is a normal $\mathbb {R}$ -complete measure on $\Theta $ ” implies that $\omega _1$ is $\wp (\mathbb {R})$ -supercompact. (cf. N. Trang, Derived models and supercompact measures on $\wp _{\omega _1}(\wp (\mathbb {R}))$ . Mathematical Logic Quarterly , vol. 61 (2015), no. 1–2, pp. 56–65.) In the second paper under review, Supercompactness can be equiconsistent with measurability, it is shown that the following theories are equiconsistent:

1. (2-1) $\mathsf {ZF}\,+\,\mathsf {DC}\,+\,\mathsf {AD}^+\,+\,\mathsf {AD}_{\mathbb {R}}\,+$ there is a normal $\mathbb {R}$ -complete measure on $\Theta $ .

2. (2-2) $\mathsf {ZF}\,+\,\mathsf {DC}\,+\,\mathsf {AD}^+\,+\,\mathsf {AD}_{\mathbb {R}}\,+\Theta $ is regular $\,+\,\omega _1$ is $\wp (\mathbb {R})$ -supercompact.

In the paper, Trang first proved Woodin’s theorem on Vopeňka forcing over $\mathsf {HOD}$ in a determinacy model, which is now considered as a standard tool. Assuming (2-2), a model of (2-1) is obtained as a symmetric extension via Vopeňka forcing of some $\mathsf {ZFC}$ model that has $V_{\Theta }^{\mathsf {HOD}}$ as its rank initial segment and carries a normal measure on $\Theta $ . The argument to find such a $\mathsf {ZFC}$ model is based on the $\mathsf {HOD}$ analysis in G. Sargsyan, Hod mice and the mouse set conjecture. Memoirs of the American Mathematical Society , vol. 236 (2015), no. 1111, viii+172 pp. It is worth noting that a similar argument can be found in R. Atmai and G. Sargsyan, Hod up to $\mathsf {AD}_{\mathbb {R}}\,+\,\Theta $ is measurable. Annals of Pure Applied Logic , vol. 170 (2019), no. 1, pp. 95–108, where they analyze $\mathsf {HOD}$ in a minimal model of (2-1).

Unlike $\mathbb {R}$ -compactness of $\omega _1$ , $\wp (\mathbb {R})$ -supercompactness of $\omega _1$ entails some strong form of determinacy. The third paper under review, Determinacy from strong compactness of $\omega _1$ , Trang and Wilson showed that “ $\mathsf {DC}\,+\,\omega _1$ is $\wp (\mathbb {R})$ -supercompact” implies the existence of a sharp for a transitive model of $\mathsf {AD}_{\mathbb {R}}\,+\,\mathsf {DC}$ including all reals and ordinals. (The exact consistency strength of $\wp (\mathbb {R})$ -supercompactness of $\omega _1$ is still unknown.) This follows from one of their main results, which claims the following theories are equiconsistent:

1. (3-1) $\mathsf {ZF}\,+\,\mathsf {DC}\,+\,\mathsf {AD}_{\mathbb {R}}$ .

2. (3-1) $\mathsf {ZF}\,+\,\mathsf {DC}\,+\,\omega _1\text { is } \wp (\mathbb {R})\text {-strongly compact}$ .

Here, we say that $\omega _1$ is X-strongly compact if there is a fine countably complete measure on $\wp _{\omega _1}(X)$ . The forward direction is easy because by the proof of the aforementioned folklore result, in the minimal model of $\mathsf {ZF}\,+\,\mathsf {DC}\,+\,\mathsf {AD}_{\mathbb {R}}$ , $\omega _1$ is $\wp (\mathbb {R})$ -strongly compact. Most of the paper is devoted to showing the other direction by a technique called the core model induction, which combines inner model theory with descriptive set theory to obtain a model of determinacy. The strong compactness of $\omega _1$ is mainly used for the descriptive set theoretic part: Under $\mathsf {ZF}\,+\,\mathsf {DC}$ , if $\Gamma $ is an inductive-like scaled pointclass and $\omega _1$ is $\mathrm {Env}(\Gamma )$ -strongly compact, then there is a scale on a universal $\check {\Gamma }$ set, each of whose prewellorderings is in $\mathrm {Env}(\Gamma )$ . This fact is essentially proved in Wilson’s PhD thesis, Contributions to descriptive inner model theory (2012).

In the fourth paper under review, On supercompactness of $\omega _1$ , Ikegami and Trang obtained several structural consequences of full supercompactness of $\omega _1$ . They first show that supercompactness of $\omega _1$ implies $\mathsf {DC}$ . They also show that supercompactness of $\omega _1$ implies determinacy-like consequences such as non-existence of an $\omega _1$ -sequence of distinct reals, $\infty $ -Borelness of sets of reals in the Chang model $\mathsf {CM}:=\bigcup _{\alpha \in \mathrm {Ord}}L({}^{\omega }\alpha )$ , and weak homogeneity of any tree on $\omega \times \mathrm {Ord}$ . One interesting corollary of $\mathsf {DC}$ and the weak homogeneity of the Martin–Solovay trees is the equivalence of $\mathsf {AD}^+$ and $\mathsf {AD}_{\mathbb {R}}$ . Lastly, they show that under the inner model theoretic assumption called Hod Pair Capturing, supercompactness of $\omega _1$ implies determinacy for all Suslin sets of reals. Note that one cannot expect full determinacy here because Takeuti’s model does not satisfy $\mathsf {AD}$ . However, the consistency of “ $\mathsf {AD}_{\mathbb {R}} \,+\, \omega _1$ is supercompact” is proved by Woodin in his unpublished work. He showed that assuming a proper class of Woodin limits of Woodin cardinals, the generalized Chang model $\mathsf {CM}^+ := \bigcup _{\alpha \in \mathrm {Ord}}L({}^{\omega }\alpha )[\mu _{\alpha }]$ , where $\mu _{\alpha }$ is the club filter on $\wp _{\omega _1}({}^{\omega }\alpha )$ , satisfies “ $\mathsf {AD}_{\mathbb {R}}\,+\,\omega _1$ is supercompact.” Ikegami and Trang conjectured that:

1. (4-1) $\mathsf {ZFC}\,+\,$ there is a proper class of Woodin limits of Woodin cardinals,

2. (4-2) $\mathsf {ZF}\,+\,\omega _1$ is supercompact,

3. (4-3) $\mathsf {ZF}\,+\,\mathsf {AD}_{\mathbb {R}}\,+\,\omega _1$ is supercompact,

are equiconsistent.

One thing that the articles under review do not discuss is the relation between supercompactness for $\omega _1$ and long game determinacy. For example, in Trang’s PhD thesis, Generalized Solovay measures, the HOD analysis, and the core model induction (2013), it is shown that the following theories are equivalent over $\mathsf {ZFC}$ :

1. (5-1) There is a sharp for an inner model with $\omega ^2$ many Woodin cardinals.

2. (5-2) All ${<}\omega ^2\text {-}\mathbf {\Pi }^1_1$ games on natural numbers of length $\omega ^3$ are determined.

3. (5-3) There is a sharp for $L(\mathbb {R}, \mu )$ and $L(\mathbb {R}, \mu )\models $ “ $\mathsf {AD}\,+\,\omega _1\text { is } \mathbb {R}\text {-supercompact}$ ,” where $\mu $ is the club filter on $\wp _{\omega _1}(\mathbb {R})$ .

Trang also obtained such equivalence for a sharp for an inner model with $\omega ^{\alpha }$ many Woodin cardinals for any $\alpha <\omega _1$ by introducing generalized Solovay models. Supercompactness of $\omega _1$ seems important beyond determinacy of fixed countable length games too. Steel proved several results on the relation between the theory (2-2) and games ending at the first $\Sigma _n$ -admissible relative to the play (cf. J. R. Steel, Long games, Games, Scales, and Suslin Cardinals: The Cabal Seminar Volume I (A. S. Kechris, B. Löwe, J. R. Steel, editors), Cambridge University Press, Cambridge, Lecture Notes in Logic, 31, 2008, pp. 223–259). Also, based on Neeman’s consistency proof of long game determinacy (cf. I. Neeman, The Determinacy of Long Games De Gruyter, Berlin, De Gruyter Series in Logic and its Applications, 7, 2004, xii+317 pp.), Woodin showed that, assuming a sharp for an inner model with a Woodin limit of Woodin cardinals, it is consistent that $\mathsf {ZFC}\,+$ all games on natural numbers of length $\omega _1$ with payoff sets that are definable from real and ordinal parameters are determined. He then used such determinacy to prove the aforementioned theorem on $\mathsf {CM}^+$ .

Many questions about the supercompactness or strong compactness of $\omega _1$ are still open and seem crucial for a proper understanding of the connection between inner models and determinacy axioms. The four papers under review could be good starting points for anyone interested in tackling such questions.