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WEAK DIAMOND AND OPEN COLORINGS
WEAK DIAMOND OPEN COLORINGS
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2015/8/17
The purpose of this article is to prove the relative consistency of certain statements about open colorings with 2ℵ0 < 2ℵ1. In particular both OCA and the statement that every 1-1 function...
Parametrized ♦ principles
Parametrized principles
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2015/8/17
We will present a collection of guessing principles which have a similar relationship to as cardinal invariants of the continuum have to CH. The purpose is to provide a means for systematically analyz...
THE PROPER FORCING AXIOM,PRIKRY FORCING,AND THE SINGULAR CARDINALS HYPOTHESIS
FORCING AXIOM PRIKRY FORCING SINGULAR CARDINALS
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2015/8/17
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous refl...
A SOLUTION TO THE L SPACE PROBLEM
SOLUTION L SPACE PROBLEM
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2015/8/17
In [23], Todorcevic gives a survey of basis problems in combinatorial set theory,listing nine theorems and six working conjectures.
COMPACT SPACES WITH HEREDITARILY NORMAL SQUARES
COMPACT SPACES HEREDITARILY NORMAL SQUARES
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2015/8/17
In 1948, Katetov proved the following metrization theorem. Theorem 1.1. [3] If X is a compact space1 and every subspace of X3 is normal, then X is metrizable.
ARONSZAJN LINES AND THE CLUB FILTER
ARONSZAJN LINES CLUB FILTER
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2015/8/17
The purpose of this note is to demonstrate that a weak form of club guessing on ω1 implies the existence of an Aronszajn line with no Countryman suborders. An immediate consequence is that the existen...
A Gδ IDEAL OF COMPACT SETS STRICTLY ABOVE THE NOWHERE DENSE IDEAL IN THE TUKEY ORDER
Gδ IDEAL COMPACT SETS STRICTLY NOWHERE DENSE IDEAL TUKEY ORDER
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2015/8/17
We prove that there is a Gδ σ-ideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a qu...
A UNIVERSAL ARONSZAJN LINE
UNIVERSAL ARONSZAJN LINE
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2015/8/17
The purpose of this note is to define an Aronszajn line ηC and prove that under the assumption of PFA it is universal for the class of Aronszajn lines. Moreover ηC can be easily described in terms of ...
The Proper Forcing Axiom
Forcing axiom Martin’s Axiom OCA Open Coloring Axiom PID P-ideal Dichotomy proper forcing PFA
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2015/8/17
The Proper Forcing Axiom is a powerful extension of the Baire Category Theorem which has proved highly effective in settling mathematical statements which are independent of ZFC. In contrast to the Co...
SET MAPPING REFLECTION
MAPPING REFLECTION
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2015/8/17
The goal of these lectures is to give an exposition of the concept of an open stationary set, an associated reflection principle (for lack of a better word), and a list of examples of how this sort of...
ITERATED FORCING AND THE CONTINUUM HYPOTHESIS
ITERATED FORCING CONTINUUM HYPOTHESIS
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2015/8/17
The notes which follow reflect the content of a two day tutorial which took place at the Fields Institute on 5/29 and 5/30 in 2009. Most of the content has existed in the literature for some time (pri...
A BOOLEAN ACTION OF C(M,U(1)) WITHOUT A SPATIAL MODEL AND A RE-EXAMINATION OF THE CAMERON–MARTIN THEOREM
A BOOLEAN ACTION WITHOUT A SPATIAL MODEL RE-EXAMINATION CAMERON–MARTIN THEOREM
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2015/8/17
We will demonstrate that if M is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from M to U(1) on a separable probability algebra which pr...
FORCING AXIOMS AND THE CONTINUUM HYPOTHESIS,PART II:TRANSCENDING ω1-SEQUENCES OF REAL NUMBERS
FORCING AXIOMS CONTINUUM HYPOTHESIS TRANSCENDING ω1-SEQUENCES OF REAL NUMBERS
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2015/8/17
The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah.
FORCING AXIOMS AND THE CONTINUUM HYPOTHESIS
FORCING AXIOMS CONTINUUM HYPOTHESIS
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2015/8/17
One way to formulate the Baire Category Theorem is that no compact space can be covered by countably many nowhere dense sets. Soon after Cohen’s discovery of forcing, it was realized that it was natur...
THE UTILITY OF THE UNCOUNTABLE
UTILITY UNCOUNTABLE
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2015/8/17
In my lecture at the 2011 Congress on Logic, Methodology, and the Philosophy of Science in Nancy, France, I spoke on an additional axiom of set theory—the Proper Forcing Axiom—which has proved very su...