A Jech–Kunen tree is a set-theoretic tree with properties that are incompatible with the

generalized continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent t ...

. It is named after

Thomas Jech Thomas J. Jech ( cs, Tomáš Jech, ; born January 29, 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years. Life He was educated at Charles University (his advisor was Petr Vopěnka) and from ...

and

Kenneth Kunen Herbert Kenneth Kunen (August 2, 1943August 14, 2020) was a professor of mathematics at the University of Wisconsin–Madison who worked in set theory and its applications to various areas of mathematics, such as set-theoretic topology and ...

, both of whom studied the possibility and consequences of its existence.

Definition

A ''ω''₁-tree is a

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

with

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

\aleph_1

and height ''ω''₁, where ''ω''₁ is the

first uncountable ordinal In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. ...

and

\aleph_1

is the associated

cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...

. A Jech–Kunen tree is a ''ω''₁-tree in which the number of branches is greater than

\aleph_1

and less than

2^

Existence

found the first

model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

in which this tree exists, and showed that, assuming the

continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent ...

and

2^ > \aleph_2

, the existence of a Jech–Kunen tree is equivalent to the existence of a compact

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

with

weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar q ...

\aleph_1

and cardinality strictly between

\aleph_1

and

2^

References

* * *{{citation, last=Jin, first=Renling, title=The differences between Kurepa trees and Jech-Kunen trees, journal=

Archive for Mathematical Logic '' Archive for Mathematical Logic'' is a peer-reviewed mathematics journal published by Springer Science+Business Media. It was established in 1950 and publishes articles on mathematical logic. Abstracting and indexing The journal is abstracted an ...

, volume=32, page=369–379, year=1993 Trees (set theory) Independence results