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.
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.
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
He checks the first number, 47, which turns out to be the Godel number
of 'c'; he then believes much more strongly that the seventh letter will be 69, the Godel number
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
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.