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583 An Account of Gödel’s Proof for Poets, Painters, and Art Historians A mathematical system is said to be complete if and only if all true statements in the system can be derived, using the accepted rewriting procedures, from the elementary propositions that found the system; a system is said to be consistent if no contradictions can be derived. There are infinite possibilities for selecting the axioms and rules of inference for a formal system; yet none of them can generate all the truths about the natural numbers. To show this, Gödel took Hilbert’s point that every formalization of a branch of mathematics is itself a mathematical object that can be manipulated using mathematical means (i.e., following standardized rewriting procedures). Gödel’s proofs depended on representing a formal system that purports to encompass arithmetic within arithmetic itself by assigning to any statement about numbers or the relationships between numbers a unique number (which has come to be called its Gödel number). Thus, he set about constructing a set of natural numbers that would mirror all statements about relationships between the natural numbers. We ask, “Would it be possible to formulate paradoxes such as Russell’s in a formal system?” (see “Logic and paradox” in the main text). If you wish to 3 Appendix Appendix Three 584 reconstruct the classical Liar’s Paradox, then you must build a formula Q that asserts that “the formal system, S, proves ~Q”—that is, that asserts that S proves not-Q (for ‘~’ is the logical symbol for negation). How can one possibly make a formula assert its own properties? How can one make a formula talk about formulas (for self-reference is required if Russell’s paradox, and other equivalent paradoxes, are to arise)? Normally, formulas of the first-order arithmetic offer assertions about natural numbers, not about propositions. Gödel’s method for making arithmetic formulas to refer to propositions was based on this simple but brilliant insight: to force the formulas to talk about themselves, Gödel introduced a numerical coding of formulas such that by manipulating the numerals themselves, one produces statements about mathematical statements. Gödel did this by assigning values to each of the logical connectives of Russell and Whitehead’s Principia Mathematica. For example, we might, using the assignments that Nagel and Newman used in their famous layperson’s account of Gödel’s proof, represent ‘~’ (the logician’s symbol for ‘not’) by the number 1, ‘V’ (i.e., the logician’s symbol for ‘or’) by the number 2, ‘⊃’ (the symbol for ‘implies’) by the number 3, ‘∃’ (i.e., the existential quantifier ‘there exists a’) by the number 4, ‘=’ (‘equals’) by the number 5, ‘0’ by the number 6, ‘s’ (for ‘the immediate successor of’ some number, the importance of which, in Whitehead and Russell’s calculus, can be traced to Giuseppe Peano) by 7, ‘(’ (i.e., ‘open parenthesis’) by the number 8, ‘)’ (i.e., ‘closed parenthesis’) by the number 9, and ‘,’ (i.e., comma) by the number 10. Arithmetical statements also include variables—of several types, in fact: numerical variables (i.e., variables that can take on numerical values), sentential variables (i.e., logical expressions or formulas) and predicate variables (i.e., variables that attribute properties to numbers or numerical expressions, e.g., P(x), where P can be ‘is prime’ or ‘is odd’ or ‘is even’).1 If for any number n, whenever P(n) is true P(n') is also true, then every number has property P. Numerical variables we represent by prime numbers that are greater than 10 (for the Principia Mathematica used ten logical signs); thus, x and y in a statement that contains two numerical variables might be assigned the numbers 11 and 13, respectively. (I am glossing over issues relating to the representation of variables— interested readers can consult Nagel and Newman’s Gödel’s Proof.) A statement like (∃x)(x = sy) would then be represented by 8 (the number we assign to the open parenthesis sign), 4 (for the existential qualifier), 11 (the number we have assigned ‘x’), 9 (the number we assign to close parenthesis statement), 8, 11, 5 (the number we assign to the sign ‘7’), 13, 9. This set of numerals, the series enclosed by ‘[’ and ‘],’ represents the given formula. But we would prefer to represent the string by a single, unique number. How can this be done in such a way that we can be certain that each...

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