1
4.2. Proof using Strongly Mahlo Cardinals.
Recall Proposition A from the beginning of section 3.1.
This is the Principal Exotic Case.
PROPOSITION A. For all f,g
∈
ELG there exist A,B,C
∈
INF
such that
A
∪
. fA
⊆
C
∪
. gB
A
∪
. fB
⊆
C
∪
. gC.
Recall the definitions of N, ELG, INF,
∪
., fA, in
Definitions 1.1.1, 1.1.2, 1.1.10, 1.3.1, and 2.1.
In this section, we prove Proposition A in SMAH
+
. It is
convenient to prove a stronger statement.
PROPOSITION B. Let f,g
∈
ELG and n
≥
1. There exist
infinite sets A
1
⊆
...
⊆
A
n
⊆
N such that
i) for all 1
≤
i < n, fA
i
⊆
A
i+1
∪
. gA
i+1
;
ii) A
1
∩
fA
n
=
∅
.
LEMMA 4.2.1. The following is provable in RCA
0
. Proposition
B implies Proposition A. In fact, Proposition B for n = 3
implies Proposition A.
Proof: Let f,g
∈
ELG. By Proposition B for n = 3, let A
⊆
B
⊆
C
⊆
N be infinite sets, where fA
⊆
B
∪
. gB, fB
⊆
C
∪
.
gC, and A
∩
fC =
∅
.
Note that C,gC are disjoint. Hence C,gB are disjoint. In
addition, A,fA are disjoint, and A,fB are disjoint. We now
verify the inclusion relations.
Let x
∈
A
∪
fA. If x
∈
fA then x
∈
B
∪
gB
⊆
C
∪
gB. If x
∈
A then x
∈
C
⊆
C
∪
gB.
Let x
∈
A
∪
fB. If x
∈
fB then x
∈
C
∪
gC. If x
∈
A then x
∈
C
⊆
C
∪
gC. QED
Recall the definition of f
∈
ELG from section 2.1: there
are rational constants c,d > 1 such that for all but
finitely many x
∈
dom(f), cx
≤
f(x)
≤
dx.
We wish to put this in more explicit form. Assume f,c,d are
as above. Let t be a positive integer so large that 1 + 1/t
< c,d < t, and for all x
∈
dom(f), x > t
→
cx
≤
f(x)
≤
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dx. Let b be an integer greater than t and max{f(x): x
≤
t}. Then for all x
∈
dom(f),
x > t
→
f(x)
≤
bx.
x
≤
t
→
f(x)
≤
b.
x
≤
b
→
f(x)
≤
b
2
.
Hence f
∈
ELG if and only if there exists a positive
integer b such that for all x
∈
dom(f),
x > b
→
(1 + 1/b)x
≤
f(x)
≤
bx.
x
≤
b
→
f(x)
≤
b
2
.
We now fix f,g
∈
ELG, where f is pary and g is qary.
According to the above, we also fix a positive integer b
such that for all x
∈
N
p
and y
∈
N
q
,
i. if x,y > b then
(1 + 1/b)x
≤
f(x)
≤
bx
(1 + 1/b)y
≤
g(y)
≤
by.
ii. if x,y
≤
b then f(x),g(y)
≤
b
2
.
We also fix n
≥
1 and a strongly p
n1
Mahlo cardinal
κ
.
We begin with the discrete linearly ordered semigroup with
extra structure, M = (N,<,0,1,+,f,g).
The plan will be to first construct a structure of the form
M* = (N*,<*,0*,1*,+*,f*,g*,c
0
*,.
..), where the c*’s are
indexed by N. This structure is non well founded and
generated by the constants 0*,1*, and the c*’s. The
indiscernibility of the c*’s will be with regard to atomic
formulas only. The first nonstandard point in M* will be
c
0
*.
While it is obvious that we cannot embed M* back into M, we
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 Fall '08
 JOSHUA
 Math, Philosophy of language, Order theory, Lemma, M**

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