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ON SEQUENCES GENERIC IN THE SENSE OF MAGIDOR

Published online by Cambridge University Press:  12 December 2014

GUNTER FUCHS
GUNTER FUCHS*
Affiliation:
THE COLLEGE OF STATEN ISLAND OF THE CITY UNIVERSITY OF NEW YORK/THE CUNY GRADUATE CENTER STATEN ISLAND, NY 10314, USAE-mail: [email protected]
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Abstract

The main result of this paper is a combinatorial characterization of Magidor-generic sequences. Using this characterization, I show that the critical sequences of certain iterations are Magidor-generic over the target model. I then employ these results in order to analyze which other Magidor sequences exist in a Magidor extension. One result in this direction is that if we temporarily identify Magidor sequences with their ranges, then Magidor sequences are maximal, in the sense that they contain any other Magidor sequence that is present in their forcing extension, even if the other sequence is generic for a different Magidor forcing. A stronger result holds if both sequences come from the same forcing: I show that a Magidor sequence is almost unique in its forcing extension, in the sense that any other sequence generic for the same forcing which is present in the same forcing extension coincides with the original sequence at all but finitely many coordinates, and at all limit coordinates. Further, I ask the question: If d ε V[c], where c and d are Magidor-generic over V, then which Magidor forcing can d be generic for? It turns out that it must essentially be a collapsed version of the Magidor forcing for which c was generic. I treat several related questions as well. Finally, I introduce a special case of Magidor forcing which I call minimal Magidor forcing. This approach simplifies the forcing, and I prove that it doesn’t restrict the class of possible Magidor sequences. That is, if c is generic for a Magidor forcing over V, then it is generic for a minimal Magidor forcing over V.

Keywords

03E5503E4003E3503E45Forcinglarge cardinalsiterated ultrapowers
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 4 , December 2014 , pp. 1286 - 1314
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Dehornoy, Patrick, An application of iterated ultrapowers to changing cofinalities. Journal of Symbolic Logic, vol. 48 (1983), no. 2, pp. 225–235.Google Scholar
Fuchs, Gunter, A characterization of generalized Příkrý sequences. Archive for Mathematical Logic, vol. 44 (2005), no. 8, pp. 935–971.CrossRefGoogle Scholar
Gitik, Moti, Prikry-type forcings, Handbook of Set Theory (Foreman, Matthew and Magidor, Menachem, editors), vol. 2, chapter. 16, Springer, Berlin, 2010, pp. 1351–1447.CrossRefGoogle Scholar
Jensen, Ronald B., Dee-subproper forcing, Handwritten notes, 2008, http://www.mathematik.hu-berlin.de/∼raesch/org/jensen.html.Google Scholar
Magidor, Menachem, Changing cofinality of cardinals. Fundamenta Mathematicae, vol. 99 (1978), no. 1, pp. 61–71.Google Scholar
Mathias, A. R. D., On sequences generic in the sense of Prikry. Journal of the Australian Mathematical Society, vol. 15 (1973), pp. 409–414.Google Scholar
Příkrý, Karel, Changing measurable into accessible cardinals. Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 68 (1970), pp. 5–52.Google Scholar