The first line expresses the first hope expressed above — consistency. The second line expresses the second hope expressed above — completeness. We thus see that the Mathematician’s Credo is very closely related to what Russell and Whitehead were aiming for. Their goal, however, was to set the Credo on a new and rigorous basis, with PM serving as its pedestal. In other words, where the Mathematician’s Credo merely speaks of “a proof ” without saying what is meant by the term, Russell and Whitehead wanted people to think of it as meaning a proof within PM.
Gödel himself had great respect for the power of PM, as is shown by the opening sentences of his 1931 article:
The development of mathematics in the direction of greater exactness has — as is well known — led to large tracts of it becoming formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica (PM) and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann). These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i. e., reduced to a few axioms and rules of inference.
And yet, despite his generous hat-tip to Russell and Whitehead’s opus, Gödel did not actually believe that a perfect alignment between truths and PM theorems had been attained, nor indeed that such a thing could ever be attained, and his deep skepticism came from having smelled an extremely strange loop lurking inside the labyrinthine palace of mindless, mechanical, symbol-churning, meaning-lacking mathematical reasoning.
Miraculous Lockstep Synchrony
The conceptual parallel between recursively defined sequences of integers and the leapfrogging set of theorems of PM (or, for that matter, of any formal system whatever, as long as it had axioms acting as seeds and rules of inference acting as growth mechanisms) suggested to Gödel that the typographical patterns of symbols on the pages of Principia Mathematica — that is, the rigorous logical derivations of new theorems from previous ones — could somehow be “mirrored” in an exact manner inside the world of numbers. An inner voice told him that this connection was not just a vague resemblance but could in all likelihood be turned into an absolutely precise correspondence.
More specifically, Gödel envisioned a set of whole numbers that would organically grow out of each other via arithmetical calculations much as Fibonacci’s F numbers did, but that would also correspond in an exact oneto-one way with the set of theorems of PM. For instance, if you made theorem Z out of theorems X and Y by using typographical rule R5, and if you made the number z out of numbers x and y using computational rule r5, then everything would match up. That is to say, if x were the number corresponding to theorem X and y were the number corresponding to theorem Υ, then z would “miraculously” turn out to be the number corresponding to theorem Z. There would be perfect synchrony; the two sides (typographical and numerical) would move together in lock-step. At first this vision of miraculous synchrony was just a little spark, but Gödel quickly realized that his inchoate dream might be made so precise that it could be spelled out to others, so he started pursuing it in a dogged fashion.
Flipping between Formulas and Very Big Integers
In order to convert his intuitive hunch into a serious, precise, and respectable idea, Gödel first had to figure out how any string of PM symbols (irrespective of whether it asserted a truth or a falsity, or even was just a random jumble of symbols haphazardly thrown together) could be systematically converted into a positive integer, and conversely, how such an integer could be “decoded” to give back the string from which it had come. This first stage of Gödel’s dream, a systematic mapping by which every formula would receive a numerical “name”, came about as follows.
The basic alphabet of PM consisted of only about a dozen symbols (other symbols were introduced later but they were all defined in terms of the original few, so they were not conceptually necessary), and to each of these symbols Gödel assigned a different small integer (these initial few choices were quite arbitrary — it really didn’t matter what number was associated with an isolated symbol).