Abstract
A uniform method to define a (restricted iterated) forcing notion to collapse a huge cardinal to a small one to obtain models with various types of highly saturated ideals over small cardinals is presented. The method is discussed in great technical details in the first chapter, while in the second chapter the application of the method is shown on three different models: Model I with an ℵ1-complete ℵ2-saturated ideal over ℵ1 that satisfies Chang’s conjecture, Model II with an ℵ1-complete ℵ3-saturated ideal over ℵ3, and Model III with an ℵ1-complete (ℵ2, ℵ2, ℵ0)-saturated ideal over ℵ1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baumgartner, J.E., Iterated forcing, Surveys in Set Theory, London Math. Soc. Lecture Note Series, 87, Cambridge University Press, 1983.
Baumgartner, J.E., Taylor, A., Saturation properties of ideals in generic extensions I, II, Trans. Amer. Math. Soc. 270 (1982), no.2, 557–574, and 271 (1982), no.2, 587–609.
Franek, F., Some results about saturated ideals and about isomorphisms of κ -trees, Ph.D. thesis, University of Toronto, 1983.
Franek, F., Certain values of completeness and saturatedness of a uniform ideal rule out certain sizes of the underlying index set, Canadian Mathematical Bulletin, 28 (1985), pp. 501–505
Forman, M., Laver, R., There is a model in which every ℵ 2 -chromatic graph of size ℵ 2 has an ℵ 1 -chromatic subgraph of size ℵ 1 , a handwritten note.
Forman, M., Magidor, M., Shelah, S. Marten’s Maximum, Saturated Ideals and non-regular ultrafilters I,II, preprint, to appear.
Jech, T.T., Set theory, Academic Press, 1980.
Kunen, K., Set theory, North-Holland, 1980.
Kunen, K., Saturated ideals, J. Symbolic Logic 43 (1978), no.1, 65–76.
Laver, R., An (ℵ 2 ,ℵ 2 ,ℵ 0 )-saturated ideal on ℵ 1 , Logic Colloquium 10, 173–180, North-Holland, 1982.
Magidor, M., On the existence of non-regular ultrafilters and the cardinality of ultrapowers, Trans. Amer. Math. Soc. 249 (1979), no.1, 97–111.
Magidor, M., Kanamori, A., The evolution of large cardinal axioms in set theory, Higher Set Theory, 99–275, Lecture Notes in Math., 669, Springer, Berlin, 1978.
Mitchell, W.J., Hypermeasurable cardinals, Logic Colloquium ’78 (Mons, 1978), pp.303–316, Stud. Logic Foundations Math., 97, North-Holland, 1979.
Solovay, R.M., Real-valued measurable cardinals, Axiomatic Set Theory, (Proc. Sympos. Pure Math., Vol XIII, Part I, Univ. California, Los Angeles, California, 1967), 397–428.
Solovay, R.M., Reinhardt, W.N., Kanamori, A., Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), no.1, 73–116.
Ulam, S., Zur Masstheorie in der algemainem Mengenlehre, Fund. Math. 16 (1930).
Woodin, H., An ℵ 1 -dense ideal on ℵ 1 , handwritten notes.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this paper
Cite this paper
Franek, F. (1989). Saturated ideals obtained via restricted iterated collapse of huge cardinals. In: Steprāns, J., Watson, S. (eds) Set Theory and its Applications. Lecture Notes in Mathematics, vol 1401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097333
Download citation
DOI: https://doi.org/10.1007/BFb0097333
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51730-6
Online ISBN: 978-3-540-46795-3
eBook Packages: Springer Book Archive