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Naming and Diagonalization, from Cantor to Gödel to Kleene
 in Logic Journal of the IGPL, 22 pages, and on Gaifman’s website
, 2006
"... Gödel’s incompleteness results apply to formal theories for which syntactic constructs can be given names, in the same language, so that some basic syntactic operations are representable in the theory. It is now customary to derive these results from the fixed point theorem (also known as the reflec ..."
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Gödel’s incompleteness results apply to formal theories for which syntactic constructs can be given names, in the same language, so that some basic syntactic operations are representable in the theory. It is now customary to derive these results from the fixed point theorem (also known as the reflection theorem), which asserts the existence of sentences that “speak about themselves”. Let T be the theory and, for each wff φ, letpφqbe the term that serves as its name. Then the theorem says that, for any wff α(v) (with one free variable), there exists a sentence β for which: T ` β ↔ α(pβq) β is sometimes called the fixed point of α(v). All that is needed for the fixed point theorem is that the diagonal function, the one that maps each φ(v) toφ(p(φ(v)q)), be representable in T. The construction of β is more transparent if we assume that the functions is represented by a term of the language, diag(x). This means that the following holds for each φ(v): T ` diag(pφ(v)q) =pφ(pφ(v)q)q (Here ‘= ’ is the equality sign of the formal language; we use it also to denote equality in our metalanguage.) In other words, we can prove in T, for each particular argument, what the
Chapter 0: The Easy Way to Gödel’s Proof and Related Topics ∗
"... This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observati ..."
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This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observations on circularity and draw brief comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motives for it are made obvious. We suggest this as a more efficient way to teach the topic than what is found in the standard textbooks. For the sake of self–containment Cantor’s original diagonalization is included. A broader and more technical perspective on diagonalization is given in [Gaifman 2005]. In [1891] Cantor presented a new type of argument that shows that the set of all binary sequences (sequences of the form a0, a1,…,an,…, where each ai is either 0 or 1) is not denumerable ─ that is, cannot be arranged in a sequence, where the index ranges over the natural numbers. Let A0, A2,…An, … be a sequence of binary sequences. Say An = an,0, an,1, …, an,i, …. Define a new sequence A * = b0, b1,…,bn, … , by putting: bn = 1, if an,n = 0, bn = 0, if an,n = 1 Then, for each n, A * ≠ An, since the n th member of A * differs from the the n th member of An. Hence, A * does not appear among the Ai’s. A diagram of the following form, which appears already in Cantor’s original paper, illustrate the idea. The new sequence A * is obtained from the diagonal, by changing each of its values. The method came to be known as diagonalization. A0 = a0,0 a0,1... a0,n... A1 = a1,0 a1,1... a1,n...
The Easy Way to Gödel’s Proof and Related Matters
"... This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders the proof of the so–called fixed point theorem transparent. We also point out various historical details and make some observations on circularity and some comparis ..."
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This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders the proof of the so–called fixed point theorem transparent. We also point out various historical details and make some observations on circularity and some comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motive for arithmetization and what it should accomplish are made obvious. We suggest this as a way to teach the incompleteness results to students that have had a basic course in logic, which is more efficient than the standard textbooks. For the sake of self–containment Cantor’s original diagonalization is included. A broader and more technical perspective on diagonalization is given in [Gaifman 2005]. Motivated partly by didactic considerations, the present paper presents things somewhat differently. It also includes various points concerning natural language and circularity that appear only here. Consider an infinite sequence of sets of natural numbers: X0, X1,…, Xn, … Here ‘n ’ is an index running over all natural numbers. We can regard it as the number representing the set Xn. Let us use ‘X(y) ’ for ‘y ∈ X ’. Cantor considers the set X * of all n’s such that n ∉ Xn; in other words: X*(n) ⇔ not–Xn(n) Suppose that, for some k, X * = Xk; then for every n we have: Xk(n) ⇔ not–Xn(n) For n=k we get: Xk(k) ⇔ not–Xk(k) Contradiction. This shows that for every enumeration (by natural numbers) of sets of natural numbers, there is a set not in the enumeration. X * is defined by diagonalization, that is, by applying the predicate ‘Xn () ’ to its own representative number, n. The term derives from the following picture, a version of which is to be found in Cantor’s 1891 paper.