1
5.2. From length 3 towers to length n
towers.
In this section, we obtain a variant of Lemma 5.1.7 (Lemma
5.2.12) involving length n towers rather than length 3
towers of infinite sets. However, we only assert that the
sets in the length n tower have at least r elements, for
any r
≥
1. Thus we pay a real cost for lengthening the
towers.
Because the sets in the tower are finite and not infinite,
certain indiscernibility properties of the first set in the
tower must now be stated explicitly as additional
conditions. See Lemma 5.2.12, iii), viii). These
indiscernibility properties can of course be obtained from
the usual infinite Ramsey theorem by taking a subset of the
infinite A
⊆
N from Lemma 5.1.7  but then we would only
have a tower of length 3.
We will apply Lemma 5.1.7 with f arising from term
assignments. Thus Lemma 5.2.12 uses g and not f.
Recall the definition of the language L (Definition 5.1.8).
In order to avoid having to write too many parentheses in
terms and formulas of L, we use the following two standard
precedence tables.
↑
•
+,
¬
∧
,
∨
→
,
↔
DEFINITION 5.2.1. Let t be a term of L. We write #(t) for
the maximum of: the subscripts of variables in t, and the
number of occurrences of the symbols
01+•
↑
()v
1
v
2
,...log
We count log as a single symbol. Note that for all n
≥
0,
{t: #(t)
≤
n} is finite.
DEFINITION 5.2.2. Let
ϕ
be a quantifier free formula in L.
We write #(
ϕ
) for the maximum of: the subscripts of
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variables in
ϕ
, and the number of occurrences of the
symbols
01+•
↑
()=<
¬
∧∨→↔
v
1
v
2
,...,v
r
log
in
ϕ
. Note that for all n
≥
0, {
ϕ
: #(
ϕ
)
≤
n} is finite.
DEFINITION 5.2.3. For all r
≥
1, let
β
(r) be the number of
terms t in L with #(t)
≤
r. We fix a doubly indexed sequence
t[i,r] of terms in L, which is defined if and only if r
≥
1
and 1
≤
i
≤
β
(r). For each r
≥
1, the sequence t[i,r], 1
≤
i
≤
β
(r), enumerates the terms t with #(t)
≤
r, without
repetition.
DEFINITION 5.2.4. For all r
≥
1, let
γ
(r) be the number of
quantifier free formulas
ϕ
in L with #(
ϕ
)
≤
r. We fix a
doubly indexed sequence
ϕ
[i,r] of quantifier free formulas
in L, which is defined if and only if r
≥
1 and 1
≤
i
≤
γ
(r). For each r
≥
1, the sequence
ϕ
[i,r], 1
≤
i
≤
γ
(r),
enumerates the quantifier free formulas
ϕ
with #(
ϕ
)
≤
r,
without repetition.
We adhere to the convention of displaying all free
variables (and possibly additional variables). Thus
t(v
1
,...,v
n
) and
ϕ
(v
1
,...,v
m
) respectively indicate that all
variables in the term t are among the first n variables
v
1
,...,v
n
, and all variables in the quantifier free formula
ϕ
are among the first m variables v
1
,...,v
m
.
Note that all terms t[i,r] have variables among v
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 Fall '08
 JOSHUA
 Math, Sets, X1, Countable set, Lemma

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