1
CHAPTER 5
INDEPENDENCE OF EXOTIC CASE
5.1. Proposition C and Length 3 Towers.
5.2. From Length 3 Towers to Length n Towers.
5.3. Countable Nonstandard Models with Limited
Indiscernibles.
5.4. Limited Formulas, Limited Indiscernibles, x
definability, Normal Form.
5.5. Comprehension, Indiscernibles.
5.6.
Π
0
1
Correct Internal Arithmetic, Simplification.
5.7. Transfinite Induction, Comprehension, Indiscernibles,
Infinity,
Π
0
1
Correctness.
5.8. ZFC + V = L, Indiscernibles, and
Π
0
1
Correct
Arithmetic.
5.9. ZFC + V = L + {(
∃κ
)(
κ
is strongly kMahlo)}
k
+
TR(
Π
0
1
,L), and 1Con(SMAH).
5.1. Proposition C and length 3 towers.
In sections 5.1 – 5.9 we show that Proposition A implies
the 1consistency of SMAH (ZFC with strongly Mahlo
cardinals of every specific finite order). The derivation
is obviously conducted in ZFC. With some detailed
examination, we see that this derivation can be carried out
in the system ACA’ used in Chapter 4. For a detailed
discussion of RCA
0
and other subsystems of second order
arithmetic, see [Si99].
We actually show that the specialization of Proposition A
to rather concrete functions implies the 1consistency of
SMAH.
We use the following very basic functions on the set of all
nonnegative integers N.
DEFINITION 5.1.1. We define +,,•,
↑
,log as follows.
1. Addition. x+y is the usual addition.
2. Subtraction. Since we are in N, xy is defined by the
usual xy if x
≥
y; 0 otherwise.
3. Multiplication. x•y is the usual multiplication.
4. Base 2 exponentiation. x
↑
is the usual base 2
exponentiation.
5. Base 2 logarithm. Since we are in N, log(x) is the floor
of the usual base 2 logarithm, with log(0) = 0.
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DEFINITION 5.1.2. TM(0,1,+,,•,
↑
,log) is the set of all
terms built up from 0,1,+,,•,
↑
,log, and variables v
1
,v
2
,...
.
DEFINITION 5.1.3. Each t
∈
TM(0,1,+,,•,
↑
,log) gives rise
to infinitely many functions, one of each arity that is at
least as large as all subscripts of variables appearing in
t, as follows. Let the variables of t be among v
1
,...,v
k
, k
≥
1. Then we associate the function f:N
k
→
N given by
f(v
1
,...,v
k
) = t(v
1
,...,v
k
)
where t is interpreted according to Definition 5.1.1.
DEFINITION 5.1.4. BAF (basic functions) is the set of all
functions given by terms in 0,1,+,,•,
↑
,log, according to
Definition 5.1.3.
It is very convenient to extend TM(0,1,+,,•,
↑
,log) with
definition by cases, to get an alternative description of
BAF.
DEFINITION 5.1.5. ETM(0,1,+,,•,
↑
,log) is the set of
“extended terms” of the following form:
t
1
if
ϕ
1
;
t
2
if
ϕ
2
∧
¬
ϕ
1
;
...
t
n
if
ϕ
n
∧
¬
ϕ
1
∧
...
∧
¬
ϕ
n1
;
t
n+1
if
¬
ϕ
1
∧
...
∧
¬
ϕ
n
.
where n
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 Fall '08
 JOSHUA
 Math, Formulas, Naive set theory, Mathematical logic, Firstorder logic

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