A Step by Step Guide to Gödel’s Incompleteness Proof:
1: Part 1 of Gödel’s paper
Part 1 of Gödel’s paper
This part of Gödel’s paper is an introduction and is not intended to be completely rigorous, so it should not be read with such expectations, and should be read as expository material that lays out an overview of the paper. This guide is intended to be read alongside the English translation of Gödel’s original proof which can be viewed online at English translation of Gödel’s original proof.
In the paragraph beginning “The development of mathematics…”, Gödel sets out his claim that all formal systems are incomplete. Then, in the paragraph beginning “Before going into details… ”, Gödel sets out the basic ideas underlying his proof; the principal ideas are as follows:
The definition of a formal system
A formal system consists of a definite set of symbols. The formal system includes definitions that define which combinations of the symbols are valid formulas of the system. A proof (or proof-schema, as Gödel puts it) in a formal system is simply a series of formulas, beginning with one or more axioms, where each formula in the series follows from one of more previous formulas by the rules of inference of the system - where the last formula is the formula that is proved by the proof-schema.
Note: Gödel refers to a formal system PM - this refers to Russell’s system ‘Principia Mathematica’. (Footnote: Russell, Bertrand, and Alfred North Whitehead, “Principia Mathematica”; 3 Volumes, 1910, 1912, and 1913, Cambridge University Press. ) The formal system P that Gödel actually uses in his proof includes typed classes in a similar fashion to that in ‘Principia Mathematica’ together with the Peano axioms. (Footnote: The Peano axioms were formulated by the Italian mathematician Giuseppe Peano. They constitute a formal definition of the fundamental properties of natural numbers. See for example, The Peano Axioms at Wolfram. )
The mapping of the formulas of a formal system to numbers
The symbols of the formal system are mapped to natural numbers, so that for every symbol of the formal system, there is a corresponding unique number. Using this mapping, then a formula of the formal system becomes mapped to a series of natural numbers, and a proof-schema becomes mapped to a finite series of series of natural numbers. And although Gödel does not state it here, in fact, his proof includes a method by which each such series of numbers is transformed into a single number, so that for each formula, and for each proof/proof-schema, there can be a unique corresponding natural number.
Mapping of relationships between formulas to relationships between numbers
Since relationships between formulas of the formal system, or between proof-schemas of the formal system, or between formulas and proof-schemas of the formal system, are all precisely defined notions, there can be corresponding relationships between the natural numbers that correspond to the formulas and proof-schemas of the formal system. Furthermore, if the correspondence is defined correctly, then if any such relationship between formulas/proof-schemas applies, then the corresponding relationship between natural numbers also applies. Also the inverse - if any relationship between natural numbers applies, then the corresponding relationship between formulas/proof-schemas also applies (note that if a natural number does not correspond to some symbol string of the formal system, then there can be no corresponding relationship, since there is not a corresponding formula/proof-schema).
Outline of a proof
The rest of the Part 1 of Gödel’s paper from the point “We now obtain an undecidable proposition…” is an outline description of a proof. But it is not an outline description of the method Gödel actually used in his proof, but of a different method. This has led to confusion for many people, because they are expecting the proof proper to follow that outline. For this reason, the reader is advised to ignore the rest of the introduction until one is fully cognizant with the actual proof that Gödel sets out in Part 2 of his paper.
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