Jech–Kunen tree explained

A Jech–Kunen tree is a set-theoretic tree with properties that are incompatible with the generalized continuum hypothesis. It is named after Thomas Jech and Kenneth Kunen, both of whom studied the possibility and consequences of its existence.

Definition

\aleph1

and height ω1, where ω1 is the first uncountable ordinal and

\aleph1

is the associated cardinal number. A Jech–Kunen tree is a ω1-tree in which the number of branches is greater than

\aleph1

and less than
\aleph1
2
.

Existence

found the first model in which this tree exists, and showed that, assuming the continuum hypothesis and

\aleph1
2

>\aleph2

, the existence of a Jech–Kunen tree is equivalent to the existence of a compact Hausdorff space with weight

\aleph1

and cardinality strictly between

\aleph1

and
\aleph1
2