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Diagonal Prikry extensions

Published online by Cambridge University Press:  12 March 2014

James Cummings  and
Matthew Foreman
James Cummings
Affiliation:
Mathematical Sciences Department, Carnegie Mellon University, Pittsburgh PA 15215, USA. E-mail: [email protected]
Matthew Foreman
Affiliation:
Department of Mathematics, University of California, Irvine, Irvine CA 92697, USA. E-mail: [email protected]
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Extract

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then

1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.

2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.

Strong covering implies weak covering.

In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include

1. X is “0# exists”, Kx is L, Y is “strongly”.

2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.

3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 4 , December 2010 , pp. 1383 - 1402
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Cummings, J., Collapsing successors of singulars, Proceedings of the American Mathematical Society, vol. 125 (1997), no. 9, pp. 2703–2709.CrossRefGoogle Scholar
[2]Cummings, J., Notes on singular cardinal combinatorics, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 3, pp. 251–282.CrossRefGoogle Scholar
[3]Cummings, J., Iterated forcing and elementary embeddings, To appear in the Handbook of Set Theory.Google Scholar
[4]Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.CrossRefGoogle Scholar
[5]Cummings, J., Foreman, M., and Magidor, M., The non-compactness of square, this Journal, vol. 68 (2003), no. 2, pp. 637–643.Google Scholar
[6]Cummings, J., Foreman, M., and Magidor, M., Canonical structure in the universe of set theory. I, Annals of Pure and Applied Logic, vol. 129 (2004), no. 1–3, pp. 211–243.CrossRefGoogle Scholar
[7]Cummings, J., Foreman, M., and Magidor, M., Canonical structure in the universe of set theory. II, Annals of Pure and Applied Logic, vol. 142 (2006), no. 1–3, pp. 55–75.CrossRefGoogle Scholar
[8]Foreman, M., Some problems in singular cardinals combinatorics, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 3, pp. 309–322.Google Scholar
[9]Foreman, M. and Magidor, M., A very weak square principle, this Journal, vol. 62 (1997), no. 1, pp. 175–196.Google Scholar
[10]Gitik, M., Blowing up power of a singular cardinal—wider gaps, Annals of Pure and Applied Logic, vol. 116 (2002), no. 1-3, pp. 1–38.CrossRefGoogle Scholar
[11]Gitik, M., Prikry-Type Forcings, To appear in the Handbook of Set Theory.Google Scholar
[12]Gitik, M., Personal communication.Google Scholar
[13]Gitik, M. and Magidor, M., Extender based forcings, this Journal, vol. 59 (1994), no. 2, pp. 445–460.Google Scholar
[14]Gitik, M. and Sharon, A., On SCH and the approachability property, Proceedings of the American Mathematical Society, vol. 136 (2008), no. 1, pp. 311–320.CrossRefGoogle Scholar
[15]Jech, T., On the cofinality of countable products of cardinal numbers, A tribute to paul erdos. Cambridge Univ. Press, Cambridge, 1990, pp. 289–305.CrossRefGoogle Scholar
[16]Kanamori, Akihiro, The Higher Infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[17]Kunen, K., Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971), pp. 407–413.Google Scholar
[18]Laver, R., Making the supercompactness of n indestructible under K-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385–388.CrossRefGoogle Scholar
[19]Laver, R., Implications between strong large cardinal axioms, Annals of Pure and Applied Logic, vol. 90 (1997), no. 1-3, pp. 79–90.CrossRefGoogle Scholar
[20]Laver, R., Personal communication.Google Scholar
[21]Magidor, M., On the singular cardinals problem. I, Israel Journal of Mathematics, vol. 28 (1977), no. 1-2, pp. 1–31.CrossRefGoogle Scholar
[22]Magidor, M. and Shelah, S., When does almost free imply free! (For groups, transversals, etc.), Journal of the American Mathematical Society, vol. 7 (1994), no. 4, pp. 769–830.CrossRefGoogle Scholar
[23]Neeman, I., Aronszajn trees and failure of the singular cardinal hypothesis, To appear in the Journal of Mathematical Logic.Google Scholar
[24]Sharon, A., Weak squares, scales, stationary reflection and the failure of SCH, Ph.D. thesis, Tel Aviv University, 2005.Google Scholar
[25]Shelah, S., Whitehead groups may be not free, even assuming CH. I, Israel Journal of Mathematics, vol. 28 (1977), no. 3, pp. 193–204.CrossRefGoogle Scholar
[26]Shelah, S., Whitehead groups may not be free, even assuming CH. II, Israel Journal of Mathematics, vol. 35 (1980), no. 4, pp. 257–285.CrossRefGoogle Scholar
[27]Sinapova, D., A model for a very good scale and a bad scale, this Journal, vol. 73 (2008), no. 4, pp. 1361–1372.Google Scholar
[28]Solovay, R. M., Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif, 1971) (Providence, R.I.), American Mathematical Society, 1974, pp. 365–372.Google Scholar
[29]Woodin, W. H., Personal communication.Google Scholar