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 &omega_{1} does not.

The situation at ω_{2} 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.

- ZFC + there is a weakly compact cardinal
- ZFC + ω
_{2}has the tree property

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
κ_{n} 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 + aleph_{ω} is a strong limit cardinal
and the tree property holds at aleph_{ω+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 2^{λ} = 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
2^{λ} = λ^{+}. (See [2].)
The main goal of the workshop is to present
the proof from [6] that the answer is no.

- 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 - Matthew Foreman, Some Problems in Singular Cardinals Combinatorics, Notre Dame Journal of Formal Logic, Volume 46, Number 3 (2005), 309-322 [Project Euclid]
- William Mitchell, Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972/73), 21-46 [ScienceDirect]
- Uri Abraham, Aronszajn trees on aleph
_{2}and aleph_{3}, Ann. Pure Appl. Logic 24 (1983), no. 3, 213-230 [ScienceDirect] - Menachem Magidor and Saharon Shelah, The tree property at successors of singular cardinals, Arch. Math. Logic 35 (1996), no. 5-6, 385-404 [SpringerLink]
- I. Neeman, Aronszajn trees and failure of the Singular Cardinal Hypothesis [Link]

To get the following rates, mention CMU Appalachian Set Theory and code 3096NW.

- 2 bedroom suite = $134/night + $18.76 tax
- 1 bedroom suite = $112/night + $15.68 tax
- studio = $89/night + $12.46 tax

- Your name, university affiliation, mailing address, phone number and email address
- Your professional status and
- undergraduate students: please describe your background in set theory
- graduate students: please tell us your year and the name of your thesis advisor if you have one
- faculty: please tell us whether you hold a federal research grant

- A brief statement about your interest in the workshop