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METAMATHEMATICS OF COMPARABILITY
Harvey M. Friedman
Department of Mathematics
Ohio State University
[email protected]
http://www.math.ohiostate.edu/~friedman/
September 25, 2000
October 7, 2001
Abstract. A number of comparability theorems have been
investigated from the viewpoint of reverse mathematics. Among
these are various comparability theorems between countable
well orderings ([2],[8]), and between closed sets in metric
spaces ([3],[5]). Here we investigate the reverse mathematics
of a comparability theorem for countable metric spaces,
countable linear orderings, and sets of rationals. The
previous work on closed sets used a strengthened notion of
continuous embedding. The usual weaker notion of continuous
embedding is used here. As a byproduct, we sharpen previous
results of [3],[5].
1. COMPARABILITY OF COUNTABLE WELL ORDERINGS.
In this paper, we assume that the field of all linear
orderings is a subset of
w
, and all linear orderings are
reflexive. (This is an official convention that we break at
the slightest provocation). We normally write
£¢
, and use the
usual notation <
¢
for the irreflexive part.
We say that a linear ordering
£¢
is a well ordering if and
only if every nonempty subset of fld(
£¢
) has a <
¢
least
element. A well ordering is a well founded linear ordering.
THEOREM 1.1. The following are provably equivalent in RCA
0
.
i) ATR
0
;
ii) For any two countable well orderings, there is a
comparison map from one to the other;
iii) For any two countable well orderings, there is an order
preserving map from one into the other.
Proof: For i)
´
ii), see [8], p. 198. By a comparison map we
mean an isomorphism from one onto the other or from one onto
an initial segment determined by a point in the other. For i)
´
iii), see [2]. QED
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2. COMPARABILITY OF COUNTABLE METRIC SPACES.
We begin by proving what we call the comparability of
countable metric spaces without worrying about what axioms
are needed.
Let
C
be any metric space. We allow
C
to be empty.
Let x
C
and
U
C
. An
C
neighborhood of x is an open
subset of
C
that contains x. An
C
clopen neighborhood of x is
a simultaneously open and closed subset of
C
that contains x.
An
C
limit point of
U
is a point in
C
all of whose
C
neighborhoods contain at least two elements from
U
. Note that
C
limit points of
U
do not have to lie in
U
.
Note that
U
C
is closed if and only if every
C
limit point
of
U
lies in
U
.
For each ordinal
a
, we define
C
[
a
] as follows.
C
[0]
=
C
.
C
[
a
+
1
] is the set of all
C
limit points of
C
[
a
].
C
[
l
] is the
intersection of all
C
[
b
],
b
<
l
.
LEMMA 2.1. Let
C
be a metric space. For all ordinals
a
,
C
[
a+1
]
C
[
a
]
and
C
[
a
] is a closed subset of
C
. There is an
ordinal
a
such that
C
[
a
] =
C
[
a+1
]. If
C
[
a
] =
C
[
a+1
] then for
all
b
≥
a
,
C
[
a
] =
C
[
b
]. For
U
C
, every element of
U
is an
C
limit point of
U
if and only if every element of
U
is a
U
limit point of
U
.
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 Math, Topology, Wn, Wj, Compact space, limit point, wn+1

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