Acta Mathematica

Forcing axioms and the continuum hypothesis. Part II: transcending ω1-sequences of real numbers

Justin Tatch Moore

Full-text: Open access


The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the continuum hypothesis. This answers a longstanding problem of Shelah.


Dedicated to Fennel, Laurel and Stephanie.

Article information

Acta Math., Volume 210, Number 1 (2013), 173-183.

Received: 27 October 2011
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]
Secondary: 03E57: Generic absoluteness and forcing axioms [See also 03E50]

Completely proper forcing Continuum hypothesis Forcing axiom Iterated forcing

2013 © Institut Mittag-Leffler


Moore, Justin Tatch. Forcing axioms and the continuum hypothesis. Part II: transcending ω 1 -sequences of real numbers. Acta Math. 210 (2013), no. 1, 173--183. doi:10.1007/s11511-013-0092-z.

Export citation


  • Asperό, D., Larson, P. & Moore, J.T., Forcing axioms and the continuum hypothesis. Acta Math., 210 (2013), 1–29.
  • Devlin, K. J. & Shelah, S., A weak version of ◊ which follows from $ {2^{{{\aleph_0}}}}<{2^{{{\aleph_1}}}} $. Israel J. Math., 29 (1978), 239–247.
  • Eisworth, T., Milovich, D. & Moore, J. T., Iterated forcing and the continuum hypothesis, in Appalachian Set Theory 2006–2012, London Math. Society Lecture Notes Series, 406, pp. 207–244. Cambridge Univ. Press, Cambridge, 2013.
  • Eisworth, T. & Nyikos, P., First countable, countably compact spaces and the continuum hypothesis. Trans. Amer. Math. Soc., 357 (2005), 4269–4299.
  • Kunen, K., Set Theory. Studies in Logic and the Foundations of Mathematics, 102. North-Holland, Amsterdam, 1983.
  • Larson, P., The Stationary Tower. University Lecture Series, 32. Amer. Math. Soc., Providence, RI, 2004.
  • Moore, J.T., ω1 and −ω1 may be the only minimal uncountable linear orders. Michigan Math. J., 55 (2007), 437–457.
  • Shelah, S., Proper and Improper Forcing. Perspectives in Mathematical Logic. Springer, Berlin–Heidelberg, 1998.
  • — On what I do not understand (and have something to say). I. Fund. Math., 166 (2000), 1–82.
  • Todorčević, S., Walks on Ordinals and their Characteristics. Progress in Mathematics, 263. Birkhäuser, Basel, 2007.