Notes and papers

Associate Editor

Mathematical Reviews

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.

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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.

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• 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.)

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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.

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Others

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

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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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