Appalachian set theory

Appalachian set theory

Saturday, February 28, 2009

9:30 a.m. - 6 p.m. with coffee and lunch breaks

Carnegie Mellon University

Lectures in Dougherty Hall 2302

Refreshments in Wean Hall 6220

Itay Neeman : "Aronszajn trees and the Singular Cardinals Hypothesis"

List of participants

Lecture notes by Itay Neeman and Spencer Unger (PDF; Final version 11/10)


König proved that if T is a tree with infinite height but finite levels, then T has an infinite branch. Aronszajn proved that König's lemma becomes false if "finite" and "infinite" are replaced by "countable" and "uncountable".

Let λ be a regular cardinal. Then T is a λ-tree iff T is a tree of height λ and every level of T has cardinality strictly less than λ. An Aronszajn tree on λ is a λ-tree with no λ-branch. If there are no Aronszajn trees on λ, then we say λ has the tree property. Thus ω has the tree property but &omega1 does not.

The situation at ω2 and above is much more complicated. Under GCH, for every regular cardinal κ, there is an Aronszajn tree on κ+. On the other hand, in [3], Mitchell proved the following two theories are equiconsistent.

Recall that μ is a weakly compact cardinal iff μ is an inaccessible cardinal and the tree property holds at μ.

The tree property at successors of singular cardinals is tied up with large cardinals much more powerful than weakly compacts. Suppose that λ is the supremum of supercompact cardinals κn for n = 0, 1, 2, ... Shelah showed that the tree property holds at λ+. Building on this theorem, Magidor and Shelah proved the consistency of ZFC + alephω is a strong limit cardinal and the tree property holds at alephω+1. (See [5].)

Now recall Solovay's fundamental theorem that if κ is a supercompact cardinal, and λ > κ is a singular strong limit cardinal, then 2λ = λ+. (See [1].)

In light of the Solovay and Magidor-Shelah results, Woodin and others asked in 1989 whether the tree property at the successor of a strong limit cardinal λ of countable cofinality implies 2λ = λ+. (See [2].) The main goal of the workshop is to present the proof from [6] that the answer is no.

Suggested reading

  1. Aki Kanamori and Menachem Magidor, The evolution of large cardinal axioms in set theory, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), pp. 99-275, Lecture Notes in Math., 669, Springer, Berlin, 1978
  2. Matthew Foreman, Some Problems in Singular Cardinals Combinatorics, Notre Dame Journal of Formal Logic, Volume 46, Number 3 (2005), 309-322 [Project Euclid]
  3. William Mitchell, Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972/73), 21-46 [ScienceDirect]
  4. Uri Abraham, Aronszajn trees on aleph2 and aleph3, Ann. Pure Appl. Logic 24 (1983), no. 3, 213-230 [ScienceDirect]
  5. Menachem Magidor and Saharon Shelah, The tree property at successors of singular cardinals, Arch. Math. Logic 35 (1996), no. 5-6, 385-404 [SpringerLink]
  6. I. Neeman, Aronszajn trees and failure of the Singular Cardinal Hypothesis [Link]

Local information

A block of rooms is temporarily set aside at the Shadyside Inn.
To get the following rates, mention CMU Appalachian Set Theory and code 3096NW. Other hotels which offer discounted rates to those visiting the CMU Mathematical Sciences Department include:

Participant travel support

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to James Cummings and Ernest Schimmerling by email.