Hostname: page-component-84b7d79bbc-g5fl4 Total loading time: 0 Render date: 2024-07-30T23:51:44.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Morasses and the Lévy-collapse

Published online by Cambridge University Press:  12 March 2014

P. Komjáth
Get access Rights & Permissions [Opens in a new window]

Extract

For several old problems in combinatorial set theory A. Hajnal and the present author [2] showed that on collapsing a sufficiently Mahlo cardinal to ω1 by the Lévy-collapse one gets a model where these problems are solved in the “counter-example” direction. The authors of [2] have speculated that the theorems of that paper should hold in L, and this, in fact, was shown for some of the results by Todorčević and Velleman [7,8]. The observation that collapsing a large cardinal to ω1 may give rise to L-like constructions is not new. As it was shown long ago by Silver and Rowbottom, there is a Kurepa-tree if a strongly inaccessible cardinal is Lévy-collapsed to ω1. In [5] it is proved that even Silver's W holds in that model. Here we show that even a quagmire exists there, but not necessarily a morass. To be more exact, we show that if κ < λ are the first two strongly inaccessible cardinals, first λ is Lévy-collapsed to κ+, and then κ is Lévy-collapsed to then there is no ω1-morass with built-in diamond in the resulting model (GCH is assumed). If λ is Mahlo, there is not even a morass.

Our notations are standard. For excellent survey papers on morass-like principles and their uses in combinatorial set theory see [4,5,6].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 1 , March 1987 , pp. 111 - 115
Copyright
Copyright © Association for Symbolic Logic 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Baumgartner, J. E., A new class of order types, Annals of Mathematical Logic, vol. 9 (1976), pp. 187–222.CrossRefGoogle Scholar
[2] Hajnal, A. and Komjáth, P., Some higher-gap examples in combinatorial set theory, Annals of Pure and Applied Logic (to appear).Google Scholar
[3] Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
[4] Kanamori, A., Morass-level combinatorial principles, Patras Logic Symposion (Metakides, G., editor), North-Holland, Amsterdam, 1982, pp. 339–358.CrossRefGoogle Scholar
[5] Kanamori, A., On Silver's and related principles, Logic Colloquium '80 (van Dalen, D. et al., editors), North-Holland, Amsterdam, 1982, pp. 153–172.Google Scholar
[6] Kanamori, A., Morasses in combinatorial set theory, Surveys in set theory (Mathias, A., editor), London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 167–196.CrossRefGoogle Scholar
[7] Todorčević, S., Aronszajn treees and partitions, Israel Journal of Mathematics, vol. 52 (1985), pp. 53–58.CrossRefGoogle Scholar
[8] Velleman, D., On a combinatorial principle of Hajnal and Komjáth, this Journal, vol. 51 (1986), pp. 1056–1060.Google Scholar
[9] Velleman, D., Morasses, diamond, and forcing, Annals of Mathematical Logic, vol. 23 (1982), pp. 199–281.CrossRefGoogle Scholar
[10] Velleman, D., Simplified morasses with linear limits, this Journal, vol. 49 (1984), pp. 1001–1021.Google Scholar