1
1.4. Thin Set Theorems.
Recall the Thin Set Theorem from section 1.1.
THIN SET THEOREM. For all f
∈
MF there exists A
∈
INF such
that fA
≠
N.
The Thin Set Theorem as written above is a statement of
IBRT in A,fA on (MF,INF). This specific statement is due to
the present author, who studied it for its significance for
Reverse Mathematics and recursion theory.
A variant of this statement was already introduced much
earlier in the literature on the (square bracket) partition
calculus in combinatorial set theory, in [EHR65]. In their
language, the Thin Set Theorem reads
(
∀
n <
ω
)(
ω
→
[
ω
]
n
ω
).
The n indicates a coloring of the unordered ntuples from
the
ω
to the left of
→
, the lower
ω
indicates the number of
colors, and the
ω
in [ ] indicates the cardinality of the
“homogenous” set. But here [ ] indicates a weak form of
homogeneity  that at least one color is omitted.
The mathematical difference between this square bracket
partition relation statement and the Thin Set Theorem is
that the former involves unordered tuples, whereas the
latter involves ordered tuples. However, see Theorem 1.4.2
below for an equivalence proof in RCA
0
. Also see [EHMR84],
Theorem 54.1. It was immediately recognized that this
square bracket partition relation follows from the usual
infinite Ramsey theorem, which is written in terms of the
round parenthesis partition relation
(
∀
n,m <
ω
)(
ω
→
(
ω
)
n
m
).
Experience reveals that when the Thin Set Theorem is stated
exactly in our formulation above (with ordered n tuples),
mathematicians who are not experts in the partition
calculus, do not recognize the Thin Set Theorem’s
connection with the partition calculus and combinatorial
set theory. They are struck by its fundamental character,
and will not be able to prove it in short order. They
apparently would have to rediscover the infinite Ramsey
theorem, and in our experience, long before they invest
that kind of effort, they demand a proof from us.
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The Thin Set Theorem  as an object of study in the
foundations of mathematics  first appeared publicly in
[Fr00], and in print in [FS00], p. 139. There we remark
that it trivially follows from the following well known
Free Set Theorem for N.
FREE SET THEOREM. Let k
≥
1 and f:N
k
→
N. There exists
infinite A
⊆
N such that for all x
∈
A
k
, f(x
1
,...,x
k
)
∈
A
→
f(x
1
,...,x
k
)
∈
{x
1
,...,x
k
}.
The implication is merely the observation that if A obeys
the conclusion of the Free Set Theorem, then A\{min(A)}
obeys the conclusion of the Thin Set Theorem (min(A) is not
a value of f on (A\{min(A)})
k
).
The Free Set Theorem is easily obtained from the infinite
Ramsey theorem in a well known way. Choose infinite A
⊆
N
such that the truth value of f(x
1
,...,x
k
) = y depends only
on the order type of x
1
,...,x
k
,y, provided x
1
,...,x
k
,y
∈
A.
If f(x
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 Fall '08
 JOSHUA
 Math, Set Theory, Topology, Empty set, Metric space, set theorem, Thin Set Theorem

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