Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/the-definable-tree-property-for-successors-of-cardinals-0RUBaULR4V
References (30)
-
≤ satisfies the λ + − c -
(1986)
Multiple Forcing -
* ) = dom( f p ) ∪ dom( f q ), 2. For α ∈ dom( f p ), f * (α) = f p (α), 3. For α ∈ dom( f q )\dom -
U Abraham (1983)
Aronszajn trees on $$\aleph _2$$ ℵ 2 and $$\aleph _3$$ ℵ 3Ann. Pure Appl. Logic, 24
-
J. Cummings (2010)
Iterated Forcing and Elementary Embeddings -
Dima Sinapova (2012)
The tree property at ℵω+1The Journal of Symbolic Logic, 77
-
R. Schindler, J. Steel (2009)
The self-iterability of L[E]The Journal of Symbolic Logic, 74
-
Assuming the consistency of a supercompact cardinal and a weakly compact above it, it is consistent to have tree property on both ℵ 2 and ℵ 3 -
Natasha Dobrinen, S. Friedman (2008)
Homogeneous iteration and measure one covering relative to HODArchive for Mathematical Logic, 47
-
M. Foreman, M. Magidor, R. Schindler (2001)
The consistency strength of successive cardinals with the tree propertyJournal of Symbolic Logic, 66
-
Then it is clear that q * ≤ q and that π(q * ) ≤ * p. The lemma follows -
J Cummings (2010)
Iterated Forcing and Elementary Embeddings. Handbook of Set Theory -
Assume H is i(P(E, κ, λ))-generic over V and let G be the filter generated by π -
By Lemma 7 G is P(E, κ, λ)-generic over V , and in V -
M. Magidor, S. Shelah (1995)
The tree property at successors of singular cardinalsArchive for Mathematical Logic, 35
-
G. Leversha (2005)
Set theory: the third millennium edition , by Thomas Jech. Pp. 769. £77. 2003. ISBN 3 540 44085 2 (Springer).The Mathematical Gazette, 89
-
U. Abraham (1983)
Aronszajn trees on aleph2 and aleph3Ann. Pure Appl. Log., 24
-
J. Cummings, M. Foreman (1998)
The Tree PropertyAdvances in Mathematics, 133
-
I. Glicksberg (1971)
When is ∗₁ closed?Transactions of the American Mathematical Society, 160
-
R Schindler, J Steel (2009)
The self-Iterability of $$L[E]$$ L [ E ]J. Symb. Logic, 74
-
Carmi Merimovich (2011)
Supercompact extender based Prikry forcingArchive for Mathematical Logic, 50
-
It follows that V and V[G] have the same bounded subsets of κ and (κ + ) V[G] = λ -
All V-cardinals in the interval (κ, λ) are collapsed, 6. λ is preserved in V -
The consistency of tree property on both ℵ 2 and ℵ 3 implies the consistency of ” 0 ♯ exists” -
Amir Leshem (2000)
On the consistency of the Definable Tree Property on ℵ1Journal of Symbolic Logic, 65
-
A * is an E( f * )-tree -
In the next lemma we show that the above forcing has enough homogeneity properties -
W. Mitchell (1972)
Aronszajn trees and the independence of the transfer propertyAnnals of Mathematical Logic, 5
-
T Jech (2003)
Set Theory: The Third Millennium Edition. Revised and Expanded, Springer Monographs in Mathematics -
I. Neeman (2009)
Aronszajn Trees and Failure of the singular cardinal HypothesisJ. Math. Log., 9
- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by Springer-Verlag Berlin Heidelberg
- Subject
- Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-016-0494-7
- Publisher site
- See Article on Publisher Site
Abstract
Strengthening a result of Leshem (J Symb Logic 65(3):1204–1214, 2000), we prove that the consistency strength of $$\textit{GCH}$$ GCH together with the definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many $$\varPi ^{1}_1$$ Π 1 1 -reflecting cardinals. Moreover it is proved that if $$\kappa $$ κ is a supercompact cardinal and $$\lambda > \kappa $$ λ > κ is measurable, then there is a generic extension of the universe in which $$\kappa $$ κ is a strong limit singular cardinal of cofinality $$\omega , ~ \lambda =\kappa ^+,$$ ω , λ = κ + , and the definable tree property holds at $$\kappa ^+$$ κ + . Additionally we can have $$2^\kappa > \kappa ^+,$$ 2 κ > κ + , so that $$\textit{SCH}$$ SCH fails at $$\kappa $$ κ .
Journal
Archive for Mathematical Logic – Springer Journals
Published: Jun 9, 2016