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"
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
On the other hand,
in , 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.
- ZFC + there is a weakly compact cardinal
- ZFC + ω2 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
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.
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+.
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 .)
The main goal of the workshop is to present
the proof from  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
- William Mitchell,
Aronszajn trees and the independence of the transfer property,
Ann. Math. Logic 5 (1972/73), 21-46
- Uri Abraham, Aronszajn trees on aleph2 and
aleph3, Ann. Pure Appl. Logic 24 (1983), no. 3, 213-230
Menachem Magidor and Saharon Shelah,
The tree property at successors of singular cardinals,
Arch. Math. Logic 35 (1996), no. 5-6, 385-404
I. Neeman, Aronszajn trees and failure of the Singular Cardinal Hypothesis
A block of rooms is temporarily set aside at the
To get the following rates, mention CMU Appalachian Set Theory and
Other hotels which offer discounted rates
to those visiting the CMU Mathematical Sciences Department include:
- 2 bedroom suite = $134/night + $18.76 tax
- 1 bedroom suite = $112/night + $15.68 tax
- studio = $89/night + $12.46 tax
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.
- 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