Abstract
We show that Martin’s Maximum${}^{++}$ implies Woodin’s ${\mathbb P}_{\rm max}$ axiom $(*)$. This answers a question from the 1990s and amalgamates two prominent axioms of set theory which were both known to imply that there are $\aleph _2$ many real numbers.
-
@MISC{stevo,
author = {Todor\v{c}ević,
Stevo},
title = {The power set of $\omega_1$ and the continuum problem},
url = {http://logic.harvard.edu/Todorcevic_Structure4.pdf},
zblnumber = {},
sortyear={2022},
} -
[ARS]
U. Abraham, M. Rubin, and S. Shelah, "On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph_1$-dense real order types," Ann. Pure Appl. Logic, vol. 29, iss. 2, pp. 123-206, 1985.
@ARTICLE{ARS,
author = {Abraham, Uri and Rubin, Matatyahu and Shelah, Saharon},
title = {On the consistency of some partition theorems for continuous colorings, and the structure of {$\aleph_1$}-dense real order types},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {29},
year = {1985},
number = {2},
pages = {123--206},
issn = {0168-0072},
mrclass = {03E35 (03E05 06A05 54A35)},
mrnumber = {0801036},
mrreviewer = {J. M. Henle},
doi = {10.1016/0168-0072(84)90024-1},
url = {https://doi.org/10.1016/0168-0072(84)90024-1},
zblnumber = {0585.03019},
} -
[david-ralf]
D. Asperó and R. Schindler, "Bounded Martin’s Maximum with an asterisk," Notre Dame J. Form. Log., vol. 55, iss. 3, pp. 333-348, 2014.
@ARTICLE{david-ralf,
author = {Asperó,
David and Schindler, Ralf},
title = {Bounded {M}artin's {M}aximum with an asterisk},
journal = {Notre Dame J. Form. Log.},
fjournal = {Notre Dame Journal of Formal Logic},
volume = {55},
year = {2014},
number = {3},
pages = {333--348},
issn = {0029-4527},
mrclass = {03E57 (03E35 03E55 03E60)},
mrnumber = {3263531},
mrreviewer = {Peter Holy},
doi = {10.1215/00294527-2688051},
url = {https://doi.org/10.1215/00294527-2688051},
zblnumber = {1338.03101},
} -
[baumgartner]
J. E. Baumgartner, "All $\aleph _{1}$-dense sets of reals can be isomorphic," Fund. Math., vol. 79, iss. 2, pp. 101-106, 1973.
@ARTICLE{baumgartner,
author = {Baumgartner, James E.},
title = {All {$\aleph \sb{1}$}-dense sets of reals can be isomorphic},
journal = {Fund. Math.},
fjournal = {Polska Akademia Nauk. Fundamenta Mathematicae},
volume = {79},
year = {1973},
number = {2},
pages = {101--106},
issn = {0016-2736},
mrclass = {02K05 (06A05)},
mrnumber = {0317934},
mrreviewer = {John Hickman},
doi = {10.4064/fm-79-2-101-106},
url = {https://doi.org/10.4064/fm-79-2-101-106},
zblnumber = {0274.02037},
} -
[cantor]
H. Cantor, "Über eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen," J. Reine Angew. Math., vol. 77, pp. 258-262, 1874.
@ARTICLE{cantor,
author = {Cantor, Herrn},
title = {Über eine {E}igenschaft des {I}nbegriffs aller reellen algebraischen {Z}ahlen},
journal = {J. Reine Angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
volume = {77},
year = {1874},
pages = {258--262},
issn = {0075-4102},
mrclass = {DML},
mrnumber = {1579605},
doi = {10.1515/crll.1874.77.258},
url = {https://doi.org/10.1515/crll.1874.77.258},
zblnumber = {},
} -
[cs]
B. Claverie and R. Schindler, "Increasing $u_2$ by a stationary set preserving forcing," J. Symbolic Logic, vol. 74, iss. 1, pp. 187-200, 2009.
@ARTICLE{cs,
author = {Claverie, Benjamin and Schindler, Ralf},
title = {Increasing {$u_2$} by a stationary set preserving forcing},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {74},
year = {2009},
number = {1},
pages = {187--200},
issn = {0022-4812},
mrclass = {03E35 (03E40)},
mrnumber = {2499426},
mrreviewer = {Renling Jin},
doi = {10.2178/jsl/1231082308},
url = {https://doi.org/10.2178/jsl/1231082308},
zblnumber = {1163.03026},
} -
[cs2]
B. Claverie and R. Schindler, "Woodin’s axiom $(\ast)$, bounded forcing axioms, and precipitous ideals on $\omega_1$," J. Symbolic Logic, vol. 77, iss. 2, pp. 475-498, 2012.
@ARTICLE{cs2,
author = {Claverie, Benjamin and Schindler, Ralf},
title = {Woodin's axiom {$(\ast)$},
bounded forcing axioms, and precipitous ideals on {$\omega_1$}},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {77},
year = {2012},
number = {2},
pages = {475--498},
issn = {0022-4812},
mrclass = {03E35 (03E45 03E57)},
mrnumber = {2963017},
mrreviewer = {A. Kanamori},
doi = {10.2178/jsl/1333566633},
url = {https://doi.org/10.2178/jsl/1333566633},
zblnumber = {1250.03111},
} -
[cohen] P. J. Cohen, Set Theory and the Continuum Hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966.
@BOOK{cohen,
author = {Cohen, Paul J.},
title = {Set Theory and the Continuum Hypothesis},
publisher = {W. A. Benjamin, Inc., New York-Amsterdam},
year = {1966},
pages = {vi+154},
mrclass = {02.65},
mrnumber = {0232676},
mrreviewer = {A. Lévy},
zblnumber = {0182.01301},
} -
[ds]
P. Doebler and R. Schindler, "$\Pi_2$ consequences of BMM plus NS is precipitous and the semiproperness of stationary set preserving forcings," Math. Res. Lett., vol. 16, iss. 5, pp. 797-815, 2009.
@ARTICLE{ds,
author = {Doebler, Philipp and Schindler, Ralf},
title = {{$\Pi_2$} consequences of {B}{M}{M} plus {N}{S} is precipitous and the semiproperness of stationary set preserving forcings},
journal = {Math. Res. Lett.},
fjournal = {Mathematical Research Letters},
volume = {16},
year = {2009},
number = {5},
pages = {797--815},
issn = {1073-2780},
mrclass = {03E57 (03E05 03E35 03E40)},
mrnumber = {2576698},
doi = {10.4310/MRL.2009.v16.n5.a4},
url = {https://doi.org/10.4310/MRL.2009.v16.n5.a4},
zblnumber = {1206.03044},
} -
[ds2] P. Doebler and R. Schindler, "The extender algebra and vagaries of $\Sigma^2_1$ absoluteness," Münster J. Math., vol. 6, iss. 1, pp. 117-166, 2013.
@ARTICLE{ds2,
author = {Doebler, Philipp and Schindler, Ralf},
title = {The extender algebra and vagaries of {$\Sigma^2_1$} absoluteness},
journal = {Münster J. Math.},
fjournal = {Münster Journal of Mathematics},
volume = {6},
year = {2013},
number = {1},
pages = {117--166},
issn = {1867-5778},
mrclass = {03E55 (03E35 03E40 03E45)},
mrnumber = {3148210},
mrreviewer = {Radek Honz\'ık},
zblnumber = {1348.03050},
} -
[Farah]
I. Farah, "All automorphisms of the Calkin algebra are inner," Ann. of Math. (2), vol. 173, iss. 2, pp. 619-661, 2011.
@ARTICLE{Farah,
author = {Farah, Ilijas},
title = {All automorphisms of the {C}alkin algebra are inner},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {173},
year = {2011},
number = {2},
pages = {619--661},
issn = {0003-486X},
mrclass = {03E35 (46L05)},
mrnumber = {2776359},
mrreviewer = {Edward Azoff},
doi = {10.4007/annals.2011.173.2.1},
url = {https://doi.org/10.4007/annals.2011.173.2.1},
zblnumber = {1250.03094},
} -
[new?]
S. Feferman, H. M. Friedman, P. Maddy, and J. R. Steel, "Does mathematics need new axioms?," Bull. Symbolic Logic, vol. 6, iss. 4, pp. 401-446, 2000.
@ARTICLE{new?,
author = {Feferman, Solomon and Friedman, Harvey M. and Maddy, Penelope and Steel, John R.},
title = {Does mathematics need new axioms?},
journal = {Bull. Symbolic Logic},
fjournal = {The Bulletin of Symbolic Logic},
volume = {6},
year = {2000},
number = {4},
pages = {401--446},
issn = {1079-8986},
mrclass = {03A05 (00A30 03E55)},
mrnumber = {1814122},
mrreviewer = {E. Mendelson},
doi = {10.2307/420965},
url = {https://doi.org/10.2307/420965},
zblnumber = {0977.03002},
} -
[FMW]
Q. Feng, M. Magidor, and H. Woodin, "Universally Baire sets of reals," in Set Theory of the Continuum, Springer, New York, 1992, vol. 26, pp. 203-242.
@INCOLLECTION{FMW,
author = {Feng, Qi and Magidor, Menachem and Woodin, Hugh},
title = {Universally {B}aire sets of reals},
booktitle = {Set Theory of the Continuum},
venue = {{B}erkeley, {CA},
1989},
series = {Math. Sci. Res. Inst. Publ.},
volume = {26},
pages = {203--242},
publisher = {Springer, New York},
year = {1992},
mrclass = {03E15 (03E55)},
mrnumber = {1233821},
mrreviewer = {Jakub Jasiński},
doi = {10.1007/978-1-4613-9754-0\_15},
url = {https://doi.org/10.1007/978-1-4613-9754-0_15},
zblnumber = {0781.03034},
} -
[FMS]
M. Foreman, M. Magidor, and S. Shelah, "Martin’s maximum, saturated ideals, and nonregular ultrafilters. I," Ann. of Math. (2), vol. 127, iss. 1, pp. 1-47, 1988.
@ARTICLE{FMS,
author = {Foreman, M. and Magidor, M. and Shelah, S.},
title = {Martin's maximum, saturated ideals, and nonregular ultrafilters. {I}},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {127},
year = {1988},
number = {1},
pages = {1--47},
issn = {0003-486X},
mrclass = {03E50 (03E35 03E40 03E55)},
mrnumber = {0924672},
mrreviewer = {F. R. Drake},
doi = {10.2307/1971415},
url = {https://doi.org/10.2307/1971415},
zblnumber = {0645.03028},
} -
[goedel]
K. Gödel, "The consistency of the axiom of choice and the generalized continuum hypothesis," Proc. Nat. Acad. U.S.A., vol. 24, iss. 12, pp. 556-557, 1938.
@ARTICLE{goedel,
author = {Gödel, Kurt},
title = {The consistency of the axiom of choice and the generalized continuum hypothesis},
journal = {Proc. Nat. Acad. U.S.A.},
volume = {24},
year = {1938},
number = {12},
pages = {556--557},
doi = {10.1073/pnas.24.12.556},
url = {https://doi.org/10.1073/pnas.24.12.556},
zblnumber = {0020.29701},
} -
[what_is_ch]
K. Gödel, "What is Cantor’s continuum problem?," Amer. Math. Monthly, vol. 54, pp. 515-525, 1947.
@ARTICLE{what_is_ch,
author = {Gödel, Kurt},
title = {What is {C}antor's continuum problem?},
journal = {Amer. Math. Monthly},
fjournal = {American Mathematical Monthly},
volume = {54},
year = {1947},
pages = {515--525},
issn = {0002-9890},
mrclass = {02.0X},
mrnumber = {0023780},
mrreviewer = {B. Jónsson},
doi = {10.2307/2304666},
url = {https://doi.org/10.2307/2304666},
zblnumber = {0038.03003},
} -
[goedel-feferman] K. Gödel, Collected Works. Vol. II, The Clarendon Press, Oxford Univ. Press, New York, 1990.
@BOOK{goedel-feferman,
author = {Gödel, Kurt},
title = {Collected Works. {V}ol. {II}},
note = {publications 1938--1974; edited and with a preface by Solomon Feferman},
publisher = {The Clarendon Press, Oxford Univ. Press, New York},
year = {1990},
pages = {xviii+407},
isbn = {0-19-503972-6},
mrclass = {01A75 (03-03)},
mrnumber = {1032517},
mrreviewer = {Pierre Kerszberg},
zblnumber = {0698.01023},
} -
[godel_nonCH] K. Gödel, "Some considerations leading to the probable conclusion that the true power of the continuum is $\aleph_2$," in Collected Works. Vol. II, The Clarendon Press, Oxford Univ. Press, New York, 1990, pp. 420-422.
@INCOLLECTION{godel_nonCH,
author = {Gödel, Kurt},
title = {Some considerations leading to the probable conclusion that the true power of the continuum is $\aleph_2$},
booktitle = {Collected Works. {V}ol. {II}},
pages = {420--422},
note = {publications 1938--1974; edited and with a preface by Solomon Feferman et al.},
publisher = {The Clarendon Press, Oxford Univ. Press, New York},
year = {1990},
isbn = {0-19-503972-6},
mrclass = {01A75 (03-03)},
mrnumber = {1032517},
mrreviewer = {Pierre Kerszberg},
zblnumber = {},
} -
[hilbert] D. Hilbert, "Mathematische Probleme," Nachr. K. Ges. Wiss. Göttingen, Math.-Phys. Klasse (Göttinger Nachrichten), vol. 3, pp. 253-297, 1900.
@ARTICLE{hilbert,
author = {Hilbert, D.},
title = {Mathematische Probleme},
journal = {Nachr. K. Ges. Wiss. Göttingen, Math.-Phys. Klasse (Göttinger Nachrichten)},
volume = {3},
year = {1900},
pages = {253--297},
zblnumber = {31.0068.03},
} -
@book {jech,
author = {Jech, Thomas},
TITLE = {Set theory},
SERIES = {Springer Monographs in Mathematics},
NOTE = {The third millennium edition, revised and expanded},
PUBLISHER = {Springer-Verlag, Berlin},
YEAR = {2003},
PAGES = {xiv+769},
ISBN = {3-540-44085-2},
MRCLASS = {03Exx (03-01 03-02)},
MRNUMBER = {1940513},
MRREVIEWER = {Eva Coplakova},
doi = {10.1007/3-540-44761-X},
url = {https://doi.org/10.1007/3-540-44761-X},
zblnumber = {1007.03002},
} -
[ronald_l_forcing] R. Jensen, $\mathcal{L}$-forcing.
@MISC{ronald_l_forcing,
author = {Jensen, Ronald},
title = {{$\mathcal{L}$}-forcing},
note = {handwritten notes; available on author's webpage},
zblnumber = {},
sortyear={2022},
} -
[stacking]
R. Jensen, E. Schimmerling, R. Schindler, and J. Steel, "Stacking mice," J. Symbolic Logic, vol. 74, iss. 1, pp. 315-335, 2009.
@ARTICLE{stacking,
author = {Jensen, Ronald and Schimmerling, Ernest and Schindler, Ralf and Steel, John},
title = {Stacking mice},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {74},
year = {2009},
number = {1},
pages = {315--335},
issn = {0022-4812},
mrclass = {03E45 (03E55)},
mrnumber = {2499432},
mrreviewer = {Alessandro Andretta},
doi = {10.2178/jsl/1231082314},
url = {https://doi.org/10.2178/jsl/1231082314},
zblnumber = {1161.03031},
} -
[kanamori]
A. Kanamori, The Higher Infinite, Large Cardinals in Set Theory from Their Beginnings, 2nd ed. ed., Springer-Verlag, Berlin, 2003.
@BOOK{kanamori,
author = {Kanamori, Akihiro},
title = {The {H}igher {I}nfinite, {L}arge {C}ardinals in {S}et {T}heory from {T}heir {B}eginnings},
fjournal = {Springer Monographs in Mathematics},
series= {Springer Monogr. Math.},
issn = {1439-7382},
edition = {2nd ed.},
isbn = {3-540-00384-3/hbk},
pages = {xxii + 536},
year = {2003},
Publisher = {Springer-Verlag, Berlin},
zblnumber = {1022.03033},
doi = {10.1007/978-3-540-88867-3},
url = {https://doi.org/10.1007/978-3-540-88867-3},
} -
[kei71] J. H. Keisler, Model theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, North-Holland Publishing Co., Amsterdam-London, 1971, vol. 62.
@BOOK{kei71,
author = {Keisler, H. Jerome},
title = {Model theory for Infinitary Logic. {L}ogic with Countable Conjunctions and Finite Quantifiers},
series = {Stud. Logic Found. Math.},
volume = {62},
publisher = {North-Holland Publishing Co., Amsterdam-London},
year = {1971},
pages = {x+208 pp. (loose erratum)},
mrclass = {02H10 (02B25 02K15)},
mrnumber = {0344115},
mrreviewer = {Andreas Blass},
zblnumber = {0222.02064},
} -
[kei73] J. H. Keisler, "Forcing and the omitting types theorem," in Studies in Model Theory, , 1973, vol. 8, pp. 96-133.
@INCOLLECTION{kei73,
author = {Keisler, H. Jerome},
title = {Forcing and the omitting types theorem},
booktitle = {Studies in Model Theory},
pages = {96--133},
series = {MAA Stud. Math.},
volume = {8},
year = {1973},
mrclass = {02H05},
mrnumber = {0337571},
mrreviewer = {James H. Schmerl},
zblnumber = {},
} -
@misc{peter,
author = {Koellner, P.},
title = {The {C}ontinuum {H}ypothesis, The {S}tanford {E}ncyclopedia of {P}hilosophy},
note = {Spring 2019 Edition, E. N. Zalta (ed.)},
url = {https://plato.stanford.edu/archives/spr2019/entries/continuum-hypothesis/ },
zblnumber = {},
} -
[plarson]
P. Larson, "Martin’s Maximum and the ${\Bbb P}_{ max}$ axiom ($*$)," Ann. Pure Appl. Logic, vol. 106, iss. 1-3, pp. 135-149, 2000.
@ARTICLE{plarson,
author = {Larson, Paul},
title = {Martin's {M}aximum and the {${\Bbb P}_{\rm max}$} axiom ({$*$})},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {106},
year = {2000},
number = {1-3},
pages = {135--149},
issn = {0168-0072},
mrclass = {03E35 (03E05 03E40 03E55 03E65)},
mrnumber = {1785758},
mrreviewer = {A. Kanamori},
doi = {10.1016/S0168-0072(00)00020-8},
url = {https://doi.org/10.1016/S0168-0072(00)00020-8},
zblnumber = {0973.03068},
} -
[paul!] P. Larson, "Showing OCA in ${\Bbb P}_{ max}$-style extensions," Kobe J. Math., vol. 18, iss. 2, pp. 115-126, 2001.
@ARTICLE{paul!,
author = {Larson, Paul},
title = {Showing {OCA} in {${\Bbb P}_{\rm max}$}-style extensions},
journal = {Kobe J. Math.},
fjournal = {Kobe Journal of Mathematics},
volume = {18},
year = {2001},
number = {2},
pages = {115--126},
issn = {0289-9051},
mrclass = {03E35 (03E40 03E45)},
mrnumber = {1907668},
mrreviewer = {Renling Jin},
zblnumber = {1005.03045},
} -
[paul]
P. B. Larson, The Stationary Tower. Notes on a Course by W. Hugh Woodin, Amer. Math. Soc., Providence, RI, 2004, vol. 32.
@BOOK{paul,
author = {Larson, Paul B.},
title = {The Stationary Tower. Notes on a Course by W. Hugh Woodin},
series = {Univ. Lecture Series},
volume = {32},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2004},
pages = {x+132},
isbn = {0-8218-3604-8},
mrclass = {03-02 (03E15 03E35 03E40 03E55 03E60)},
mrnumber = {2069032},
mrreviewer = {Miroslav Repick\'{y}},
doi = {10.1090/ulect/032},
url = {https://doi.org/10.1090/ulect/032},
zblnumber = {1072.03031},
} -
[plarson2]
P. B. Larson, "Martin’s Maximum and definability in $H(\aleph_2)$," Ann. Pure Appl. Logic, vol. 156, iss. 1, pp. 110-122, 2008.
@ARTICLE{plarson2,
author = {Larson, Paul B.},
title = {Martin's {M}aximum and definability in {$H(\aleph_2)$}},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {156},
year = {2008},
number = {1},
pages = {110--122},
issn = {0168-0072},
mrclass = {03E35 (03E40 03E55 03E65)},
mrnumber = {2474445},
mrreviewer = {John Krueger},
doi = {10.1016/j.apal.2008.06.012},
url = {https://doi.org/10.1016/j.apal.2008.06.012},
zblnumber = {1153.03035},
} -
[larson-handbook]
P. B. Larson, "Forcing over models of determinacy," in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 2121-2177.
@INCOLLECTION{larson-handbook,
author = {Larson, Paul B.},
title = {Forcing over models of determinacy},
booktitle = {Handbook of Set Theory. {V}ols. 1, 2, 3},
pages = {2121--2177},
publisher = {Springer, Dordrecht},
year = {2010},
mrclass = {03E57 (03E35 03E45 03E55 03E60 03E65)},
mrnumber = {2768703},
mrreviewer = {Ralf-Dieter Schindler},
doi = {10.1007/978-1-4020-5764-9\_25},
url = {https://doi.org/10.1007/978-1-4020-5764-9_25},
zblnumber = {1198.03063},
} -
[l-s]
A. Lévy and R. M. Solovay, "Measurable cardinals and the continuum hypothesis," Israel J. Math., vol. 5, pp. 234-248, 1967.
@ARTICLE{l-s,
author = {Lévy, A. and Solovay, R. M.},
title = {Measurable cardinals and the continuum hypothesis},
journal = {Israel J. Math.},
fjournal = {Israel Journal of Mathematics},
volume = {5},
year = {1967},
pages = {234--248},
issn = {0021-2172},
mrclass = {02.68},
mrnumber = {0224458},
mrreviewer = {G. Fodor},
doi = {10.1007/BF02771612},
url = {https://doi.org/10.1007/BF02771612},
zblnumber = {0289.02044},
} -
[luzin] N. Luzin, On some new results in descriptive function theory, 1935.
@MISC{luzin,
author = {Luzin, N.},
title = {On some new results in descriptive function theory},
note = {(in Russian), Moscow-Leningrad},
year = {1935},
zblnumber = {},
} -
@MISC{menachem,
author = {Magidor, M.},
title = {Some set theories are more equal},
url = {logic.harvard.edu/EFI_Magidor.pdf},
zblnumber = {},
} -
[martin-solovay]
D. A. Martin and R. M. Solovay, "Internal Cohen extensions," Ann. Math. Logic, vol. 2, iss. 2, pp. 143-178, 1970.
@ARTICLE{martin-solovay,
author = {Martin, D. A. and Solovay, R. M.},
title = {Internal {C}ohen extensions},
journal = {Ann. Math. Logic},
fjournal = {Annals of Mathematical Logic},
volume = {2},
year = {1970},
number = {2},
pages = {143--178},
issn = {0003-4843},
mrclass = {02.60},
mrnumber = {0270904},
mrreviewer = {L. Bukovsk\'{y}},
doi = {10.1016/0003-4843(70)90009-4},
url = {https://doi.org/10.1016/0003-4843(70)90009-4},
zblnumber = {0222.02075},
} -
[proj-determinacy]
D. A. Martin and J. R. Steel, "A proof of projective determinacy," J. Amer. Math. Soc., vol. 2, iss. 1, pp. 71-125, 1989.
@ARTICLE{proj-determinacy,
author = {Martin, Donald A. and Steel, John R.},
title = {A proof of projective determinacy},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {2},
year = {1989},
number = {1},
pages = {71--125},
issn = {0894-0347},
mrclass = {03E15 (03E55 03E60)},
mrnumber = {0955605},
mrreviewer = {Thomas J. Jech},
doi = {10.2307/1990913},
url = {https://doi.org/10.2307/1990913},
zblnumber = {0668.03021},
} -
@ARTICLE{justin,
author = {Moore, Justin Tatch},
title = {Set mapping reflection},
journal = {J. Math. Log.},
fjournal = {Journal of Mathematical Logic},
volume = {5},
year = {2005},
number = {1},
pages = {87--97},
issn = {0219-0613},
mrclass = {03E65 (03E05 03E50)},
mrnumber = {2151584},
mrreviewer = {Andrzej Ros\l anowski},
doi = {10.1142/S0219061305000407},
url = {https://doi.org/10.1142/S0219061305000407},
zblnumber = {1082.03042},
} -
[Moore5]
J. T. Moore, "A five element basis for the uncountable linear orders," Ann. of Math. (2), vol. 163, iss. 2, pp. 669-688, 2006.
@ARTICLE{Moore5,
author = {Moore, Justin Tatch},
title = {A five element basis for the uncountable linear orders},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {163},
year = {2006},
number = {2},
pages = {669--688},
issn = {0003-486X},
mrclass = {03E35 (03E05 03E40 03E55)},
mrnumber = {2199228},
mrreviewer = {Maxim R. Burke},
doi = {10.4007/annals.2006.163.669},
url = {https://doi.org/10.4007/annals.2006.163.669},
zblnumber = {1143.03026},
} -
[moore]
J. T. Moore, "What makes the continuum $\aleph_2$," in Foundations of Mathematics, Amer. Math. Soc., Providence, RI, 2017, vol. 690, pp. 259-287.
@INCOLLECTION{moore,
author = {Moore, Justin Tatch},
title = {What makes the continuum {$\aleph_2$}},
booktitle = {Foundations of Mathematics},
series = {Contemp. Math.},
volume = {690},
pages = {259--287},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2017},
mrclass = {03E50 (03E35 03E57 03E65)},
mrnumber = {3656315},
mrreviewer = {Matteo Viale},
doi = {10.1090/conm/690},
url = {https://doi.org/10.1090/conm/690},
zblnumber = {1423.03194},
} -
[trang-sargsyan] G. Sargsyan and N. Trang, The largest Suslin Axiom.
@MISC{trang-sargsyan,
author = {Sargsyan, G. and Trang, N.},
title = {The largest {S}uslin {A}xiom},
note = {available on {N. T}rang's website},
zblnumber = {},
} -
[mlq0002d]
R. Schindler, "Semi-proper forcing, remarkable cardinals, and bounded Martin’s maximum," MLQ Math. Log. Q., vol. 50, iss. 6, pp. 527-532, 2004.
@ARTICLE{mlq0002d,
author = {Schindler, Ralf},
title = {Semi-proper forcing, remarkable cardinals, and bounded {M}artin's maximum},
journal = {MLQ Math. Log. Q.},
fjournal = {MLQ. Mathematical Logic Quarterly},
volume = {50},
year = {2004},
number = {6},
pages = {527--532},
issn = {0942-5616},
mrclass = {03E35 (03E55)},
mrnumber = {2096166},
mrreviewer = {Tetsuya Ishiu},
doi = {10.1002/malq.200410002},
url = {https://doi.org/10.1002/malq.200410002},
zblnumber = {1058.03052},
} -
@BOOK{book,
author = {Schindler, Ralf},
title = {Set Theory. Exploring Independence and Truth},
series = {Universitext},
publisher = {Springer, Cham},
year = {2014},
pages = {x+332},
isbn = {978-3-319-06724-7; 978-3-319-06725-4},
mrclass = {03-02 (03E02 03E35 03E55 03E60)},
mrnumber = {3243739},
mrreviewer = {A. Kanamori},
doi = {10.1007/978-3-319-06725-4},
url = {https://doi.org/10.1007/978-3-319-06725-4},
zblnumber = {1296.03002},
} -
[both]
R. Schindler, "Woodin’s axiom ($*$), or Martin’s Maximum, or both?," in Foundations of Mathematics, Amer. Math. Soc., Providence, RI, 2017, vol. 690, pp. 177-204.
@INCOLLECTION{both,
author = {Schindler, Ralf},
title = {Woodin's axiom ({$*$}), or {M}artin's {M}aximum, or both?},
booktitle = {Foundations of Mathematics},
series = {Contemp. Math.},
volume = {690},
pages = {177--204},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2017},
mrclass = {03E57 (03E45 03E55 03E60 03E65)},
mrnumber = {3656312},
mrreviewer = {Xianghui Shi},
doi = {10.1090/conm/690},
url = {https://doi.org/10.1090/conm/690/13868},
zblnumber = {1423.03198},
} -
[scott] D. Scott, "Measurable cardinals and constructible sets," Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., vol. 9, pp. 521-524, 1961.
@ARTICLE{scott,
author = {Scott, Dana},
title = {Measurable cardinals and constructible sets},
journal = {Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.},
fjournal = {Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques},
volume = {9},
year = {1961},
pages = {521--524},
issn = {0001-4117},
mrclass = {02.68},
mrnumber = {0143710},
mrreviewer = {Andr\'{a}s Hajnal},
zblnumber = {0154.00702},
} -
[Shelah-Whitehead]
S. Shelah, "Infinite abelian groups, Whitehead problem and some constructions," Israel J. Math., vol. 18, pp. 243-256, 1974.
@ARTICLE{Shelah-Whitehead,
author = {Shelah, Saharon},
title = {Infinite abelian groups, {W}hitehead problem and some constructions},
journal = {Israel J. Math.},
fjournal = {Israel Journal of Mathematics},
volume = {18},
year = {1974},
pages = {243--256},
issn = {0021-2172},
mrclass = {02K05 (02K35 04A20 20K35)},
mrnumber = {0357114},
mrreviewer = {Paul C. Eklof},
doi = {10.1007/BF02757281},
url = {https://doi.org/10.1007/BF02757281},
zblnumber = {0318.02053},
} -
[solovay] R. M. Solovay, "$2^{\aleph_0}$ can be anything it ought to be," in The Theory of Models. Proceedings of the 1963 International Symposium at Berkeley, North-Holland, Amsterdam, 1965, p. 435.
@incollection{solovay,
author={Solovay, R. M.},
title={$2^{\aleph_0}$ can be anything it ought to be},
booktitle={The {T}heory of Models. Proceedings of the 1963 International Symposium at Berkeley},
note={J. Addison et al.\, eds.)},
series={Stud. Logic Found. Math.},
publisher={North-Holland, Amsterdam},
year={1965},
pages={435},
zblnumber = {0202.30701},
} -
[solovay-tennenbaum]
R. M. Solovay and S. Tennenbaum, "Iterated Cohen extensions and Souslin’s problem," Ann. of Math. (2), vol. 94, pp. 201-245, 1971.
@ARTICLE{solovay-tennenbaum,
author = {Solovay, R. M. and Tennenbaum, S.},
title = {Iterated {C}ohen extensions and {S}ouslin's problem},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {94},
year = {1971},
pages = {201--245},
issn = {0003-486X},
mrclass = {04A15},
mrnumber = {0294139},
mrreviewer = {Thomas J. Jech},
doi = {10.2307/1970860},
url = {https://doi.org/10.2307/1970860},
zblnumber = {0244.02023},
} -
[john]
J. R. Steel, "PFA implies ${ AD}^{L(\Bbb R)}$," J. Symbolic Logic, vol. 70, iss. 4, pp. 1255-1296, 2005.
@ARTICLE{john,
author = {Steel, John R.},
title = {P{FA} implies {${\rm AD}^{L(\Bbb R)}$}},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {70},
year = {2005},
number = {4},
pages = {1255--1296},
issn = {0022-4812},
mrclass = {03E45 (03E55)},
mrnumber = {2194247},
mrreviewer = {A. Kanamori},
doi = {10.2178/jsl/1129642125},
url = {https://doi.org/10.2178/jsl/1129642125},
zblnumber = {1103.03047},
} -
[forcing-free]
J. R. Steel, "A stationary-tower-free proof of the derived model theorem," in Advances in Logic, Amer. Math. Soc., Providence, RI, 2007, vol. 425, pp. 1-8.
@INCOLLECTION{forcing-free,
author = {Steel, John R.},
title = {A stationary-tower-free proof of the derived model theorem},
booktitle = {Advances in Logic},
series = {Contemp. Math.},
volume = {425},
pages = {1--8},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2007},
mrclass = {03E45 (03E35 03E60)},
mrnumber = {2322359},
doi = {10.1090/conm/425/08113},
url = {https://doi.org/10.1090/conm/425/08113},
zblnumber = {1124.03024},
} -
[StVW]
J. R. Steel and R. Van Wesep, "Two consequences of determinacy consistent with choice," Trans. Amer. Math. Soc., vol. 272, iss. 1, pp. 67-85, 1982.
@ARTICLE{StVW,
author = {Steel, John R. and Van Wesep, Robert},
title = {Two consequences of determinacy consistent with choice},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the Amer. Math. Soc.},
volume = {272},
year = {1982},
number = {1},
pages = {67--85},
issn = {0002-9947},
mrclass = {03E35 (03E60)},
mrnumber = {0656481},
mrreviewer = {J. M. Henle},
doi = {10.2307/1998951},
url = {https://doi.org/10.2307/1998951},
zblnumber = {0528.03033},
} -
[PWIM]
J. R. Steel, "Projectively well-ordered inner models," Ann. Pure Appl. Logic, vol. 74, iss. 1, pp. 77-104, 1995.
@ARTICLE{PWIM,
author = {Steel, John R.},
title = {Projectively well-ordered inner models},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {74},
year = {1995},
number = {1},
pages = {77--104},
issn = {0168-0072},
mrclass = {03E45 (03E35 03E55)},
mrnumber = {1336414},
mrreviewer = {A. Kanamori},
doi = {10.1016/0168-0072(94)00021-T},
url = {https://doi.org/10.1016/0168-0072(94)00021-T},
zblnumber = {0821.03023},
} -
[todo]
S. Todorvcević, Partition Problems in Topology, Amer. Math. Soc., Providence, RI, 1989, vol. 84.
@BOOK{todo,
author = {Todor\v{c}ević,
Stevo},
title = {Partition Problems in Topology},
series = {Contemp. Math.},
volume = {84},
publisher = {Amer. Math. Soc., Providence, RI},
year = {1989},
pages = {xii+116},
isbn = {0-8218-5091-1},
mrclass = {04-02 (03E05 03E50 54A25)},
mrnumber = {0980949},
mrreviewer = {Eva Coplakova},
doi = {10.1090/conm/084},
url = {https://doi.org/10.1090/conm/084},
zblnumber = {},
} -
[viale]
M. Viale, "Category forcings, $\mathrm{MM}^{+++}$, and generic absoluteness for the theory of strong forcing axioms," J. Amer. Math. Soc., vol. 29, iss. 3, pp. 675-728, 2016.
@ARTICLE{viale,
author = {Viale, Matteo},
title = {Category forcings, {$\mathrm{MM}^{+++}$},
and generic absoluteness for the theory of strong forcing axioms},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {29},
year = {2016},
number = {3},
pages = {675--728},
issn = {0894-0347},
mrclass = {03E35 (03E40 03E57)},
mrnumber = {3486170},
mrreviewer = {Chris Lambie-Hanson},
doi = {10.1090/jams/844},
url = {https://doi.org/10.1090/jams/844},
zblnumber = {1403.03108},
} -
[hugh83]
H. W. Woodin, "Some consistency results in ${ ZFC}$ using ${ AD}$," in Cabal Seminar 79–81, Springer, Berlin, 1983, vol. 1019, pp. 172-198.
@INCOLLECTION{hugh83,
author = {Woodin, W. Hugh},
title = {Some consistency results in {${\rm ZFC}$} using {${\rm AD}$}},
booktitle = {Cabal Seminar 79--81},
series = {Lecture Notes in Math.},
volume = {1019},
pages = {172--198},
publisher = {Springer, Berlin},
year = {1983},
mrclass = {03E35 (03E60)},
mrnumber = {0730594},
doi = {10.1007/BFb0071701},
url = {https://doi.org/10.1007/BFb0071701},
zblnumber = {0549.03048},
} -
[hugh]
H. W. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co., Berlin, 1999, vol. 1.
@BOOK{hugh,
author = {Woodin, W. Hugh},
title = {The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal},
series = {De Gruyter Ser.Logic Appl.},
volume = {1},
publisher = {Walter de Gruyter \& Co., Berlin},
year = {1999},
pages = {vi+934},
isbn = {3-11-015708-X},
mrclass = {03-02 (03E05 03E15 03E35 03E45 03E55 03E60)},
mrnumber = {1713438},
mrreviewer = {A. Kanamori},
doi = {10.1515/9783110804737},
url = {https://doi.org/10.1515/9783110804737},
zblnumber = {0954.03046},
} -
[hugh-ch1] H. W. Woodin, "The Continuum Hypothesis: Part I," Notices Amer. Math. Soc., vol. 48, iss. 6, pp. 567-576, 2001.
@ARTICLE{hugh-ch1,
author = {Woodin, W. Hugh},
title = {The {C}ontinuum {H}ypothesis: {P}art {I}},
journal = {Notices Amer. Math. Soc.},
fjournal = {Notices of the Amer. Math. Soc.},
volume = {48},
year = {2001},
number = {6},
pages = {567--576},
issn = {0002-9920},
mrclass = {03E50 (03E05 03E15 03E35)},
mrnumber = {1834351},
mrreviewer = {Yehuda Rav},
zblnumber = {0992.03063},
} -
[hugh-ch2] H. W. Woodin, "The Continuum Hypothesis: Part II," Notices Amer. Math. Soc., vol. 48, iss. 7, pp. 681-690, 2001.
@ARTICLE{hugh-ch2,
author = {Woodin, W. Hugh},
title = {The {C}ontinuum {H}ypothesis: {P}art {II}},
journal = {Notices Amer. Math. Soc.},
fjournal = {Notices of the Amer. Math. Soc.},
volume = {48},
year = {2001},
number = {7},
pages = {681--690},
issn = {0002-9920},
mrclass = {03E50 (03A05 03E05 03E15 03E35)},
mrnumber = {1842471},
mrreviewer = {Yehuda Rav},
zblnumber = {1047.03041},
} -
[ICM]
H. W. Woodin, "Strong axioms of infinity and the search for $V$," in Proceedings of the International Congress of Mathematicians. Volume I, 2010, pp. 504-528.
@INPROCEEDINGS{ICM,
author = {Woodin, W. Hugh},
title = {Strong axioms of infinity and the search for {$V$}},
booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians. {V}olume {I}},
pages = {504--528},
publisher = {Hindustan Book Agency, New Delhi},
year = {2010},
mrclass = {03-02 (03E05 03E15 03E25 03E35 03E40 03E60)},
mrnumber = {2827903},
doi = {10.1142/9789814324359_0023 },
url = {https://doi.org/10.1142/9789814324359_0023},
zblnumber = {1252.03001},
} -
[midrasha]
H. W. Woodin, "In search of Ultimate-$L$: the 19th Midrasha Mathematicae Lectures," Bull. Symb. Log., vol. 23, iss. 1, pp. 1-109, 2017.
@ARTICLE{midrasha,
author = {Woodin, W. Hugh},
title = {In search of {U}ltimate-{$L$}: the 19th {M}idrasha {M}athematicae {L}ectures},
journal = {Bull. Symb. Log.},
fjournal = {The Bulletin of Symbolic Logic},
volume = {23},
year = {2017},
number = {1},
pages = {1--109},
issn = {1079-8986},
mrclass = {03E45 (03E55 03E60)},
mrnumber = {3632568},
mrreviewer = {A. Kanamori},
doi = {10.1017/bsl.2016.34},
url = {https://doi.org/10.1017/bsl.2016.34},
zblnumber = {},
} -
[hugh-preprint] H. W. Woodin, The equivalence of Axiom $(*)^+$ and Axiom $(*)^{++}$, 2020.
@MISC{hugh-preprint,
author = {Woodin, W. Hugh},
title = {The equivalence of {A}xiom $(*)^+$ and {A}xiom $(*)^{++}$},
note = {preprint},
year = {2020},
zblnumber = {},
}