Gödel numbering

(20) Thus, if P is a particular proposition of the system it will have a number p associated with it, known as its Godel number. Godel then created a proposition G that says, loosely, "The proposition whose Godel number is g cannot be proved using the results of the system." The remarkable feature about G is that its Godel number actually is g.

Each statement of number theory, a sequence of symbols, acquires a Godel number by which it can be referred to and in this way statements of number theory can be understood on two different levels: as statements of number theory, and also as meta-mathematical statements about number theory (Hofstadter, 1999).

Specifically, recall that Godel's first theorem constructs a sentence P such that, as is provable in PM or a related system, P [left arrow][right arrow] ~Prov([P]), where Prov is a one-place "provability predicate" and enclosure in square brackets gives the Godel number of the formula enclosed.

Godel then proposed building Godel numbers for sentences made up of various variables and constants by writing down the Godel numbers of each individual variable or constant and then concatenating them to form the Godel number for the entire proposition by the following rule: g('a_1 & a_2 & a_3 & a_4...

The intuitive meaning of a formula Ta is that a is the Godel number of a true sentence in the language of arithmetic.

(Any such list can, of course, be described finitistically and can, by coding, be allocated a Godel number, since Fin(ZF) is a conventional first-order formal theory.) We wish to find a finite model of F whose domain is a finite set of hfpsets (or, equivalently, natural numbers).

Since every expression of PM is associated with a particular Godel number, a meta-mathematical statement about formal expressions and their typographical relations to one another may be construed as a statement about the corresponding Godel numbers and their arithmetical relations to one another.

([upper left corner]p[upper right corner] might be the numeral for the Godel number of p.) "[proves]" is a preposed verb phrase (of our language) meaning "is provable in the theory".

[DELTA] (g(A)) A [DELTA] (g([DELTA](g(A)) A)) I (g(A), g(B)) [DELTA](g(A)) [DELTA](g(B))) where g(A) is the numeral in the formal arithmetic that represents the Godel number of A under some Godel numbering and I(x, y) is the formal predicate that represents the metamathematical property of the wff whose Godel number is y following from the wff whose Godel number is x.

Ismael discusses several themes that are central to Hofstadter's work: Godel numbers, Escher's paradoxically self-referential artwork, and the core concept of a self-representational loop, all without a single reference to Hofstadter.

By shifting between the arithmetized "metasystem" and the Godel numbers assigned to them, he can discover holes in the completeness of the system.

However, suddenly h thinks of a new hypothesis that he had not previously thought of: perhaps the numbers are the Godel numbers spelling 'crimson'.