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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 k-Mahlo)} k + TR( Π 0 1 ,L), and 1-Con(SMAH). 5.1. Proposition C and length 3 towers. In sections 5.1 – 5.9 we show that Proposition A implies the 1-consistency 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 1-consistency 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, x-y is defined by the usual x-y 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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2 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 n-1 ; t n+1 if ¬ ϕ 1 ... ¬ ϕ n . where n
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