1
SUBTLE CARDINALS AND LINEAR
ORDERINGS
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
[email protected]
http://www.math.ohiostate.edu/~friedman/
April 9, 1998
INTRODUCTION
The subtle, almost ineffable, and ineffable cardinals were
introduced in an unpublished 1971 manuscript of R. Jensen and
K. Kunen, and a number of basic facts were proved there.
These concepts were extended to that of ksubtle, kalmost
ineffable, and kineffable cardinals in [Ba75], where a
highly developed theory is presented.
This important level of the large cardinal hierarchy was
discussed in some detail in the survey article [KM78],
section 20. However, discussion was omitted in the subsequent
[Ka94]. This level is associated with the discrete/finite
combinatorial independence results in [Fr98] and [Fr ].
In section 1 we give a self contained treatment of the basic
facts about this level of the large cardinal hierarchy, which
were established in [Ba75]. In particular, we give a proof
that the ksubtle, kalmost ineffable, and kineffable
cardinals define three properly intertwined hierarchies with
the same limit, lying strictly above “total indescribability”
and strictly below “arrowing
w
.”
The innovation here is presented in section 2, where we take
a distinctly minimalist approach. Here the subtle cardinal
hierarchy is characterized by very elementary properties that
do not mention closed unbounded or stationary sets. This
development culminates in a characterization of the hierarchy
by means of a striking universal second order property of
linear orderings.
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As is usual in set theory, we treat cardinals and ordinals as
von Neumann ordinals. We use
w
for the first limit ordinal,
which is also N. For sets X and integers k
≥ 1
, we let S
k
(X)
be the set of all k element subsets of X.
To orient the reader, we mention two results proved in this
paper.
The first is an important result from [Ba75] which is perhaps
the simplest way of defining the kineffable cardinals from
the point of view of set theoretic Ramsey theory.
We say that
l
is kSRP if and only if
l
is a limit ordinal,
and for every f:S
k
(
l
)
Æ
{0,1}, there exists a stationary E
l
such that f is constant on S
k
(E). Here SRP stands for
"stationary Ramsey property."
In [Ba75] it is proved that for all k ≥ 0 and ordinals
l
,
l
is kineffable if and only if
l
is a regular cardinal with
(k+1)SRP.
The second is a new result that is the most elementary way we
know of defining the least ksubtle cardinal.
We say that a linear ordering (X,<) is kcritical if and only
if it has no endpoints, and:
for all regressive f:X
k
Æ
X, there exists b
1
< .
.. < b
k+1
such that f(b
1
,...,b
k
) = f(b
2
,...,b
k+1
).
We prove that for k ≥ 0, the least ksubtle cardinal is the
least cardinality of a (k+1)critical linear ordering.
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 Math, Set Theory, sK, Lemma, cardinals

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