Copyright © James R Meyer 2012 - 2017 www.jamesrmeyer.com
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 or as a PDF file at English translation of Gödel’s original proof, PDF file.
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:
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 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.
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).
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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It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.
There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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Copyright © James R Meyer 2012 - 2017