Summary: On the strengths and weaknesses of weak squares
Menachem Magidor and Chris Lambie-Hanson
The term "square" refers not just to one but to an entire family of combinatorial principles.
The strongest is denoted by " " or by "Global ," and there are many interesting weaker
square principles. Before introducing any particular square principle, we provide some moti-
vating applications. In this section, the term "square" will serve as a generic term for "some
particular square principle."
1. Jensen introduced square principles based on his work regarding the fine structure of
L. In his first application, he showed that, in L, there exist -Suslin trees for every
uncountable cardinal that is not weakly compact.
2. Let T be a countable theory with a distinguished predicate R. A model of T is said to
be of type (,µ) if the cardinality of the model is and the cardinality of the model's
interpretation of R is µ. For cardinals ,,, and , (,) (,) is the assertion
that for every countable theory T, if T has a model of type (,), then it has a model
of type (,). Chang showed that under GCH, (1,0) (+
,) holds for all regular
. Jensen later showed that under GCH+square, (1,0) (+
,) holds for singular