Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-27T22:18:06.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results on higher Suslin trees

Published online by Cambridge University Press:  12 March 2014

R. David
R. David*
Affiliation:
Département de Mathématiques, Université de Savoie, 73011 Chambéry, France
Get access Rights & Permissions [Opens in a new window]

Extract

Higher Suslin trees have become a tool in some forcing constructions in set theory (see, for example, [D1] and [D2]). Most of the constructions using ω1 Suslin trees can be extended to κ+ Suslin trees for any regular cardinal κ. Some of these are given in §1.

In many such constructions, sequences of Suslin trees are used. In §II we show, in various ways, that the generalization to sequences, even ω-sequences, of κ+ Suslin trees cannot be done.

In these constructions the Suslin trees are used as forcing poset (the forcing adds a branch in the tree). There is another way to kill a Suslin tree, namely by adding a big antichain. Some results on this forcing are given in §III.

Our notation is standard. If T is a tree and x ∈ T, then ∣x∣ is the height of x in T. We define Tα (or T(α)) = {x ∈ T: ∣x∣ = α} and T∣α = {x ∈ T: ∣x∣ < α}.

If p and q are forcing conditions, p ≤ q means that p has more information than q.

If (Tα: α ∈ I) is a sequence of trees, Π Tα will always mean the set of (xα: α ∈ I) such that xα ∈ Tα and ∣xα∣ = ∣xβ∣ for α, β ∈ I.

For functions b, T,… we denote by b ∣α, T∣ α,… their restriction to α.

If x is a sequence of ordinals and α is an ordinal, x∧α is the sequence obtained by concatenating α at the end of x.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 2 , June 1990 , pp. 526 - 536
Copyright
Copyright © Association for Symbolic Logic 1990

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

[BJW]Beller, A., Jensen, R. B., and Welch, P. H., Coding the universe, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
[D1]David, R., A very absolute real singleton, Annals of Mathematical Logic, vol. 23 (1982), pp. 101–120.CrossRefGoogle Scholar
[D2]David, R., reals, Annals of Mathematical Logic, vol. 23 (1982), pp. 121–125.CrossRefGoogle Scholar
[DJ]Devlin, K. J. and Johnsbråten, H., The Souslin problem, Lecture Notes in Mathematics, vol. 405, Springer-Verlag, Berlin, 1974.CrossRefGoogle Scholar
[JE]Jech, T. J., Set theory, Academic Press, New York, 1978.Google Scholar
[J]Jensen, R. B., Coding the universe by a real, handwritten notes, 1975.Google Scholar
[JJ]Jensen, R. B. and Johnsbråten, H., A new construction of a non-constructible subset of ω, Fundamenta Mathematicae, vol. 81 (1974), pp. 279–290.CrossRefGoogle Scholar
[LS]Laver, R. and Shelah, S., The ℵ2-Souslin hypothesis, Transactions of the American Mathematical Society, vol. 264 (1981), pp. 411–417.Google Scholar
[SH]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[S-ST]Shelah, S. and Stanley, L., S-forcing. I: A “black box” theorem for morasses, with application to super-Souslin trees, Israel Journal of Mathematics, vol. 43 (1982), pp. 185–224.CrossRefGoogle Scholar
[SO]Solovay, R. M., A nonconstructible set of integers, Transactions of the American Mathematical Society, vol. 127 (1967), pp. 50–75.Google Scholar
[ST]Stanley, L., Another absolute singleton, handwritten notes, 1983.Google Scholar