Summary
We prove here the consistency of u>i where: u=Min{|X|:X⫅P(ω) generates a non-principle ultrafilter}, i=Min{|A|:A is a maximal independent family of subsets of ω}
In this we continue Goldstern and Shelah [G1Sh388] where Con(r>u) was proved using a similar but different forcing. We were motivated by Vaughan [V] (which consists of a survey and a list of open problems). For more information on the subject see [V] and [G1Sh388].
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References
[G1Sh388] Goldstern, M., Shelah, S.: Ramsey ultrafilters and the reaping number Con(r<u). Ann. Pure Appl. Logic49, 121–142 (1990)
[V] Vaughan, J.E.: Small uncountable cardinals and topology. In: van Mill, J., Reed, G.M. (eds.) Open problems in topology, pp. 195–218. Elsevier: North-Holland 1990
[Sh-b] Shelah, S.: Proper forcing. Lect. Notes940, 195–209 (1982)
[Sh-f] Shelah, S.: Proper and improper forcing (in preprints)
[Sh326] Shelah, S.: Vive le Difference I, Proceedings of the Conference in Set Theory. MSRI 10/89, to appear
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I thank Alice Leonhardt for the beautiful typing of the manuscript, as well as the referee for meticulous work. Partially supported by the Basic Research Fund, Israeli Academy of Science. Done 11/89-Publ. 407
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Shelah, S. Con(u>i). Arch Math Logic 31, 433–443 (1992). https://doi.org/10.1007/BF01277485
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DOI: https://doi.org/10.1007/BF01277485