A framework for forcing constructions at successors of singular cardinals

Authors:
James Cummings, Mirna Džamonja, Menachem Magidor, Charles Morgan and Saharon Shelah

Journal:
Trans. Amer. Math. Soc. **369** (2017), 7405-7441

MSC (2010):
Primary 03E35, 03E55, 03E75

DOI:
https://doi.org/10.1090/tran/6974

Published electronically:
May 31, 2017

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while enjoys various combinatorial properties.

As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal of uncountable cofinality where SCH fails and such that there is a collection of size less than of graphs on such that any graph on embeds into one of the graphs in the collection.

**[1]**James E. Baumgartner,*Iterated forcing*, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1-59. MR**823775****[2]**James Cummings,*Iterated forcing and elementary embeddings*, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 775-883. MR**2768691****[3]**James Cummings, Mirna Dzamonja, and Charles Morgan,*Small universal families of graphs on*, J. Symb. Log.**81**(2016), no. 2, 541-569. MR**3519445****[4]**James Cummings, Matthew Foreman, and Menachem Magidor,*Squares, scales and stationary reflection*, J. Math. Log.**1**(2001), no. 1, 35-98. MR**1838355**- [5]
James Cummings and W. Hugh Woodin,
*Generalised Prikry forcing*, book manuscript in preparation, version 2012. **[6]**Mirna Damonja and Saharon Shelah,*Universal graphs at the successor of a singular cardinal*, J. Symbolic Logic**68**(2003), no. 2, 366-388. MR**1976583****[7]**Mirna Dzamonja and Saharon Shelah,*On the existence of universal models*, Arch. Math. Logic**43**(2004), no. 7, 901-936. MR**2096141****[8]**Erik Ellentuck,*A new proof that analytic sets are Ramsey*, J. Symbolic Logic**39**(1974), 163-165. MR**0349393****[9]**Moti Gitik,*The negation of the singular cardinal hypothesis from*, Ann. Pure Appl. Logic**43**(1989), no. 3, 209-234. MR**1007865****[10]**Moti Gitik,*The strength of the failure of the singular cardinal hypothesis*, Ann. Pure Appl. Logic**51**(1991), no. 3, 215-240. MR**1098782****[11]**Moti Gitik,*Prikry-type forcings*, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1351-1447. MR**2768695****[12]**J. M. Henle,*Magidor-like and Radin-like forcing*, Ann. Pure Appl. Logic**25**(1983), no. 1, 59-72. MR**722169****[13]**Thomas Jech,*Set theory*:*The third millennium edition*, revised and expanded ,Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR**1940513**- [14]
Shizuo Kamo,
*Splitting numbers on uncountable regular cardinals*, preprint, 1992. **[15]**Kenneth Kunen,*Set theory*:*An introduction to independence proofs*, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. MR**597342****[16]**Richard Laver,*Making the supercompactness of indestructible under -directed closed forcing*, Israel J. Math.**29**(1978), no. 4, 385-388. MR**0472529****[17]**Richard Laver and Saharon Shelah,*The -Souslin hypothesis*, Trans. Amer. Math. Soc.**264**(1981), no. 2, 411-417. MR**603771****[18]**William Mitchell,*How weak is a closed unbounded ultrafilter?*, Logic Colloquium '80 (Prague, 1980) Stud. Logic Foundations Math., vol. 108, North-Holland, Amsterdam-New York, 1982, pp. 209-230. MR**673794****[19]**William J. Mitchell,*The core model for sequences of measures. I*, Math. Proc. Cambridge Philos. Soc.**95**(1984), no. 2, 229-260. MR**735366****[20]**Lon Berk Radin,*Adding closed cofinal sequences to large cardinals*, Ann. Math. Logic**22**(1982), no. 3, 243-261. MR**670992****[21]**Assaf Rinot,*Jensen's diamond principle and its relative*, Set theory and its applications, Contemp. Math., vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 125-156. MR**2777747****[22]**S. Shelah,*A weak generalization of MA to higher cardinals*, Israel J. Math.**30**(1978), no. 4, 297-306. MR**0505492****[23]**Saharon Shelah,*Cardinal arithmetic*, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR**1318912****[24]**Saharon Shelah,*Proper and improper forcing*, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR**1623206****[25]**Saharon Shelah,*Diamonds*, Proc. Amer. Math. Soc.**138**(2010), no. 6, 2151-2161. MR**2596054**- [26]
D. H. Stewart,
*The consistency of the Kurepa hypothesis with the axioms of set theory*, Master's thesis, University of Bristol, 1966. **[27]**Toshio Suzuki,*About splitting numbers*, Proc. Japan Acad. Ser. A Math. Sci.**74**(1998), no. 2, 33-35. MR**1618475****[28]**Jindaich Zapletal,*Splitting number at uncountable cardinals*, J. Symbolic Logic**62**(1997), no. 1, 35-42. MR**1450512**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
03E35,
03E55,
03E75

Retrieve articles in all journals with MSC (2010): 03E35, 03E55, 03E75

Additional Information

**James Cummings**

Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Email:
jcumming@andrew.cmu.edu

**Mirna Džamonja**

Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom

Email:
M.Dzamonja@uea.ac.uk

**Menachem Magidor**

Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, 91904 Givat Ram, Israel

Email:
mensara@savion.huji.ac.il

**Charles Morgan**

Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom — and — School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom

Email:
charles.morgan@ucl.ac.uk

**Saharon Shelah**

Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, 91904 Givat Ram, Israel — and — Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854

Email:
shelah@math.huji.ac.il

DOI:
https://doi.org/10.1090/tran/6974

Keywords:
Successor of singular cardinal,
iterated forcing,
strong chain condition,
Radin forcing,
forcing axiom,
universal graph,
indestructible supercompact cardinal

Received by editor(s):
March 25, 2014

Received by editor(s) in revised form:
October 9, 2015, April 1, 2016, and April 30, 2016

Published electronically:
May 31, 2017

Additional Notes:
The first author thanks the National Science Foundation for support through grant DMS-1101156. The first, second and fourth authors thank the Institut Henri Poincaré for support through the “Research in Paris” program during the period June 24–29, 2013. The second author thanks EPSRC for support through grants EP/G068720 and EP/I00498. The second, third and fifth authors thank the Mittag-Leffler Institute for support during the month of September 2009. The fourth author thanks EPSRC for support through grant EP/I00498. This publication is denoted [Sh963] in Saharon Shelah’s list of publications. Shelah thanks the United States-Israel Binational Science Foundation (grant no. 2006108), which partially supported this research.

Article copyright:
© Copyright 2017
American Mathematical Society