*Associate Editor*

*416 4th St*

*Ann Arbor, MI 48103*

*aec (at) ams (dot) org*

**Research Interests**

- Set theory, especially Forcing Axioms, Determinacy, Large Cardinals, Fine structure theory, Infinitary combinatorics.
- Combinatorics, especially its interactions with mathematical logic.

*Some of the entries below are based upon work supported by the National Science Foundation under grant DMS-0801189. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.*

Please let me know of typos or other problems in the papers below.

### Contents: Books, Papers, Reviews, Others, Notes and preprints.

### Books

**Foundations of Mathematics**, Andrés E. Caicedo, James Cummings, Peter Koellner, and Paul B. Larson, eds., Contemporary Mathematics, vol. 690, Amer. Math. Soc., Providence, RI, 2017. DOI: 10.1090/conm/690. MR3656304. Zbl 06733965.**Set theory and its applications**, L. Babinkostova, A. E. Caicedo, S. Geschke, M. Scheepers, eds., Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011. DOI: 10.1090/conm/533. MR2777741 (2012c:03006). Zbl 1205.03004.

### Papers

*Coloring the -smooth numbers with colors*, with Thomas A. C. Chartier and Péter Pál Pach. ArXiv. The Electronic Journal of Combinatorics**28 (1)**(2021), #P1.34, 79 pp. DOI: https://doi.org/10.37236/8492. MR4245267. Zbl 07310935.*The fourteen Victoria Delfino problems and their status in the year 2020*, with Benedikt Löwe. Hamburger Beiträge zur Mathematik preprint server. In**The Cabal Seminar. Volume IV: Large Cardinals, Determinacy and Other Topics**, Lecture Notes in Logic, vol. 49, Cambridge University Press and The Association for Symbolic Logic, 2020, pp. 248–280. DOI: https://doi-org/10.1017/9781316863534.013.*Inner-model reflection principles*, with Neil Barton, Gunter Fuchs, Joel David Hamkins, Jonas Reitz, and Ralf Schindler. ArXiv. PhilArchive. Studia Logica**108 (3)**(2020), 573–595. DOI: https://doi.org/10.1007/s11225-019-09860-7. MR095027. Zbl 07210301.*Of puzzles and partitions. Introducing Partiti*, with Brittany C. Shelton. ArXiv. Mathematics Magazine**91 (1)**(2018), 20–23. DOI: 10.1080/0025570X.2018.1403233. MR3756401. Zbl 07024498.*Downward transference of mice and universality of local core models*, with Martin Zeman. ArXiv. The Journal of Symbolic Logic**82 (2)**(2017), 385–419. DOI: 10.1017/jsl.2016.68. MR3663410. Zbl 06756180.*Topological Ramsey numbers and countable ordinals*, with Jacob Hilton. ArXiv. In**Foundations of Mathematics**, Contemporary Mathematics, vol. 690, Amer. Math. Soc., Providence, RI, 2017, pp. 85–118. DOI: 10.1090/conm/690/13864. MR3656308. Zbl 06767144.*Square principles in extensions*, with Paul Larson, Grigor Sargsyan, Ralf Schindler, John Steel, and Martin Zeman. ArXiv. Israel Journal of Mathematics**217 (1)**(2017), 231–261. DOI: 10.1007/s11856-017-1444-8. MR3625110. Zbl 06709006.*Definability of small puncture sets*, with John Clemens, Clinton Conley, and Benjamin Miller. Fundamenta Mathematicae**215 (1)**(2011), 39–51. DOI: 10.4064/fm215-1-2. MR2851700. Zbl 1254.03085.*and projective well-orderings of the reals*, with Sy Friedman. The Journal of Symbolic Logic**76 (4)**(2011), 1126–1136. DOI: 10.2178/jsl/1318338842. MR2895389 (2012m:03123). Zbl 1252.03107.*A trichotomy theorem in natural models of*, with Richard Ketchersid. In**Set Theory and Its Applications**, Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227–258. DOI: 10.1090/conm/533/10510. MR2777751 (2012i:03146). Zbl 1245.03079.**Erratum:**In p. 234, in the paragraph preceding Theorem 2.5, the assumption that there is a common pre-well-ordering of length in both models is missing. See here.*Cardinal preserving elementary embeddings*. In**Logic Colloquium 2007**, Françoise Delon, Ulrich Kohlenbach, Penelope Maddy, Frank Stephan, eds., Lecture Notes in Logic, Cambridge University Press – ASL (2010), 14–31. DOI: 10.1017/CBO9780511778421.004. MR2668226 (2011j:03122). Zbl 1245.03077.*Regressive functions on pairs*. European Journal of Combinatorics**31 (3)**(2010), 803–812. DOI: 10.1016/j.ejc.2009.07.010. MR2587031 (2011e:05270). Zbl 1247.05255.*Goodstein’s function*. Revista Colombiana de Matemáticas**41 (2)**(2007), 381–391. MR2585906 (2011c:03139). Zbl 1156.03053.*Projective well-orderings of the reals*, with Ralf Schindler. Archive for Mathematical Logic**45 (7)**(2006), 783–793. DOI: 10.1007/s00153-006-0002-6. MR2266903 (2008b:03068). Zbl 1110.03035.*Real-valued measurable cardinals and well-orderings of the reals*. In**Set Theory. Centre de Recerca Matemàtica Barcelona, 2003-2004**, Joan Bagaria, Stevo Todorcevic, eds., Trends in Mathematics, Birkhäuser (2006), 83–120. DOI: 10.1007/3-7643-7692-9_4. MR2267147 (2007g:03064). Zbl 1111.03044.*The bounded proper forcing axiom and well-orderings of the reals*, with Boban Velickovic. Mathematical Research Letters**13 (3)**(2006), 393–408. DOI: 10.4310/MRL.2006.v13.n3.a5. MR2231126 (2007d:03076). Zbl 1113.03039.*Projective well-orderings and bounded forcing axioms*. The Journal of Symbolic Logic**70 (2)**(2005), 557–572. DOI: 10.2178/jsl/1120224728. MR2140046 (2006b:03060). Zbl 1096.03062.*Exponenciación de cardinales infinitos*. Lecturas Matemáticas**19 (2)**(1998), 87–125. MR1856205 (2002f:03093). Zbl 1063.03533.

*Simply definable well-orderings of the reals*, Dissertation, UC Berkeley, 2003. Signature page. (Advisors: John Steel, Hugh Woodin). MR2705176.**Errata:**The complexity of the well-ordering mentioned in p. 36 should be , see here (the optimal complexity of is achieved by more elaborate arguments, see here). Wisest to ignore Chapter 4.*El problema de los cardinales singulares*, Apuntes Matemáticos**37**, Departamento de Matemáticas, Universidad de los Andes, Bogotá, 1996. (This is a reprint of my undergraduate dissertation, written under the supervision of Xavier Caicedo.)

### Reviews

*Reflections—the magic, music and mathematics of Raymond Smullyan*, by Raymond Merrill Smullyan. Mathematical Reviews MR3379889.*Foundational implications of the inner model hypothesis*, by Tatiana Arrigoni and Sy-David Friedman. Mathematical Reviews MR2944490.*Graphs on Euclidean spaces defined using transcendental distances*, by Péter Komjáth and James Schmerl. Mathematical Reviews MR2891155.*Some applications of Sargsyan’s equiconsistency method*, by Arthur Apter. Mathematical Reviews MR2890542.*Regular cardinals without the weak partition property*, by Steve Jackson. Mathematical Reviews MR2907002.*Constructing -morasses for*, by Bernhard Irrgang. Mathematical Reviews MR2850762 (2012k:03113).*Proposing -morasses for*, by Bernhard Irrgang. Mathematical Reviews MR2827553 (2012k:03114).*The realm of the infinite*, by W. Hugh Woodin. Mathematical Reviews MR2767235 (2012k:03125).*On guessing generalized clubs at the successors of regulars*, by Assaf Rinot. Mathematical Reviews MR2781096 (2012f:03105).*A Dedekind finite Borel set*, by Arnold W. Miller. Mathematical Reviews MR2765632 (2012f:03101).*The two-cardinal problem for languages of arbitrary cardinality*, by Luis Miguel Villegas Silva. Mathematical Reviews MR2723767 (2011m:03064).*Generic embeddings associated to an indestructibly weakly compact cardinal,*by Gunter Fuchs. Mathematical Reviews MR2720661 (2011m:03094).*On coding uncountable sets of reals*, by Joan Bagaria and Vladimir Kanovei. Mathematical Reviews MR2681346 (2011h:03109).*More on regular and decomposable ultrafilters in ZFC*, by Paolo Lipparini. Mathematical Reviews MR2681340 (2012f:03099).*The consistency strength of the tree property at the double successor of a measurable cardinal*, by Natasha Dobrinen and Sy Friedman. Mathematical Reviews MR2640069 (2011h:03106).*On changing coﬁnality of partially ordered sets*, by Moti Gitik. Mathematical Reviews MR2648158 (2011g:03112).*Strong logics of first and second order*, by Peter Koellner. Mathematical Reviews MR2656116 (2011e:03016).*Cardinal characteristics and projective wellorders*, by Vera Fischer and Sy Friedman. Mathematical Reviews MR2601019 (2011h:03100).*The determinacy of long games*, by Itay Neeman. The Bulletin of Symbolic Logic**12 (2)**(2006), 299–302. MR2962446.

### Others

*Introducing Tribus*, with Brittany C. Shelton. Mathematics Magazine**92 (1)**(2019), 18. DOI: 10.1080/0025570X.2019.1542215.

### Notes and preprints

- Barna’s inequality. December 30, 2014.
- Determinacy and Jónsson cardinals. April 9, 2012.
- Woodin’s proof of the second incompleteness theorem for set theory. November 4, 2010.
**Note:**Rather than requiring that is a hereditary property of models, we must require that proves this. Thanks to Robert Solovay for pointing this out. - A linear algebra proof of the fundamental theorem of algebra, May 18, 2010.
**Note:**Lorenzo Mattioli has identified a small gap in the proof of Lemma 3.2; a corrected version should be posted shortly. - Defining non-empty small sets from families of finite sets, with John Clemens, Clinton Conley, and Benjamin Miller. December 4, 2009.
- A review of sharps, version of November 2004.
- Is Turing equivalence universal?, version of August 2004.

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