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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 ..."
Abstract

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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
Naming and Diagonalization, from Cantor to Gödel to Kleene
"... We trace selfreference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ðD, typeð Þ, f gÞ, where D is a nonempty set; for every a 2 D, which is a name of a kary function, fag: Dk! D ..."
Abstract
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We trace selfreference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ðD, typeð Þ, f gÞ, where D is a nonempty set; for every a 2 D, which is a name of a kary function, fag: Dk! D is the function named by a, and typeðaÞ is the type of a, which tells us if a is a name and, if it is, the arity of the named function. Under quite general conditions we get a fixed point theorem, whose special cases include the fixed point theorem underlying Gödel’s proof, Kleene’s recursion theorem and many other theorems of this nature, including the solution to simultaneous fixed point equations. Partial functions are accommodated by including ‘‘undefined’ ’ values; we investigate different systems arising out of different ways of dealing with them. Manysorted naming systems are suggested as a natural approach to general computatability with many data types over arbitrary structures. The first part of the paper is a historical reconstruction of the way Gödel probably derived his proof from Cantor’s diagonalization, through the semantic version of Richard. The incompleteness proof–including the fixed point construction–result from a natural line of thought, thereby dispelling the appearance of a ‘‘magic trick’’. The analysis goes on to show how Kleene’s recursion theorem is obtained along the same lines.