Home

# Diagonal Prikry extensions

## 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”.

## References

Hide All
[1]Cummings, J., Collapsing successors of singulars, Proceedings of the American Mathematical Society, vol. 125 (1997), no. 9, pp. 27032709.
[2]Cummings, J., Notes on singular cardinal combinatorics, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 3, pp. 251282.
[3]Cummings, J., Iterated forcing and elementary embeddings, To appear in the Handbook of Set Theory.
[4]Cummings, J., Foreman, M., and Magidor, M., Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 3598.
[5]Cummings, J., Foreman, M., and Magidor, M., The non-compactness of square, this Journal, vol. 68 (2003), no. 2, pp. 637643.
[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. 211243.
[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. 5575.
[8]Foreman, M., Some problems in singular cardinals combinatorics, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 3, pp. 309322.
[9]Foreman, M. and Magidor, M., A very weak square principle, this Journal, vol. 62 (1997), no. 1, pp. 175196.
[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. 138.
[11]Gitik, M., Prikry-Type Forcings, To appear in the Handbook of Set Theory.
[12]Gitik, M., Personal communication.
[13]Gitik, M. and Magidor, M., Extender based forcings, this Journal, vol. 59 (1994), no. 2, pp. 445460.
[14]Gitik, M. and Sharon, A., On SCH and the approachability property, Proceedings of the American Mathematical Society, vol. 136 (2008), no. 1, pp. 311320.
[15]Jech, T., On the cofinality of countable products of cardinal numbers, A tribute to paul erdos. Cambridge Univ. Press, Cambridge, 1990, pp. 289305.
[16]Kanamori, Akihiro, The Higher Infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[17]Kunen, K., Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971), pp. 407413.
[18]Laver, R., Making the supercompactness of n indestructible under K-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385388.
[19]Laver, R., Implications between strong large cardinal axioms, Annals of Pure and Applied Logic, vol. 90 (1997), no. 1-3, pp. 7990.
[20]Laver, R., Personal communication.
[21]Magidor, M., On the singular cardinals problem. I, Israel Journal of Mathematics, vol. 28 (1977), no. 1-2, pp. 131.
[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. 769830.
[23]Neeman, I., Aronszajn trees and failure of the singular cardinal hypothesis, To appear in the Journal of Mathematical Logic.
[24]Sharon, A., Weak squares, scales, stationary reflection and the failure of SCH, Ph.D. thesis, Tel Aviv University, 2005.
[25]Shelah, S., Whitehead groups may be not free, even assuming CH. I, Israel Journal of Mathematics, vol. 28 (1977), no. 3, pp. 193204.
[26]Shelah, S., Whitehead groups may not be free, even assuming CH. II, Israel Journal of Mathematics, vol. 35 (1980), no. 4, pp. 257285.
[27]Sinapova, D., A model for a very good scale and a bad scale, this Journal, vol. 73 (2008), no. 4, pp. 13611372.
[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. 365372.
[29]Woodin, W. H., Personal communication.

# Diagonal Prikry extensions

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *