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1 THE INEVITABILITY OF LOGICAL STRENGTH by Harvey M. Friedman Department of Mathematics The Ohio State University May 31, 2005 An extreme form of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing headaches such as the Godel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - mathematicians only use induction for natural statements that actually arise. If logicians would tailor their formal systems to conform to the naturalness of normal mathematics, then various logical difficulties would disappear, and the story of the foundations of mathematics would look radically different than it does today". Here we present some specific results that sharply refute aspects of this viewpoint. 1. EFA = I S 0 (exp) and logical strength. For our purposes, we say that a theory in many sorted free logic has logical strength if and only if the system EFA = I S 0 (exp) is interpretable in it. I will elaborate after EFA is presented. We introduced EFA = exponential function arithmetic, in [Fr80]. It was also used in the exposition of my work on Translatability and Relative Consistency, in [Sm82]. Since then, this system has often been referred to as I S 0 (exp). See, e.g., [HP98], p. 62. Also see [HP98], p. 405, second paragraph, referring to our presentation. We will use the terminology EFA here. First we present PFA = I S 0 . Here PFA stands for “polynomial function arithmetic”. The language of PFA is based on 0,S,+,•, £ ,=. The variables are intended to range over nonnegative integers.
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2 The S 0 formulas are the formulas of PFA defined as follows. i) every atomic formula of PFA is S 0 ; ii) if j , y are S 0 then so are j , j y , y y , j y , j y ; iii) if j is S 0 and x is a variable not in the term t of PFA, then ( $ x £ t)( j ) and ( " x £ t)( j ) are S 0 . In iii), the expressions are treated as abbreviations. The nonlogical axioms of PFA are as follows. 1. The axioms of Q. 2. ( j [x/0] ( " x)( j j [x/Sx])) j , where j is S 0 . The nonlogical axioms of Q are Q1. Sx = 0. Q2. Sx = Sy x = y. Q3. x ≠ 0 ( $ y)(x = Sy). Q4. x + 0 = x. Q5. x + Sy = S(x + y). Q6. x • 0 = 0. Q7. x • Sy = (x • y) + x. Q8. x £ y ( $ z)(z + x = y). This presentation is slightly different than that given in [HP98]. There £ is not taken as a primitive, but instead is defined by Q8. Also there the terms t in bounded quantification are required to be variables. We now present EFA. The language of EFA is 0,S,+, , £ ,*,=. The new symbol * is a binary function symbol standing for exponentiation. The S 0 (exp) formulas are the formulas of EFA defined as follows.
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