This page is keyboard accessible:
• Use Tab, Shift + Tab keys to traverse the main menu. To enter a sub-menu use the Right Arrow key. To leave a sub-menu use the Left Arrow or the Escape key.
• The Enter or the Space key opens the active menu item.
• To skip the menu and move to the main content, press Tab after the page loads to reveal a skip button.
• To get back to the top of the page anytime, press the Home key.
• For more information, click here: Accessibility   Close this tip.

Note: Full functionality of this web page requires JavaScript to be enabled in your browser.
 

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 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:

 

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.

 

 

Footnotes:

 


 

 


Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked (comments will be checked before appearing on this site). Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - this will only be used to notify you of replies to your comments - it will never be used for any other purpose, will never be displayed and does not require verification. Comments are common to the entire website, so please indicate what section of the site you are commenting on.

 

If you cannot see any comments below, it may be that a plug-in on your browser is blocking Disqus comments from loading. Avast anti-virus in particular is known to do this, especially with Internet Explorer and Safari. See Disqus Browser plug-in/extension conflicts or Why isn’t the comment box loading?.

 

 

Please wait for comments to load …  

 

The Lighter Side

 

NEWS

Lebesgue Measure

There is now a new page on Lebesgue measure theory and how it is contradictory.

 

 

Illogical Assumptions

There is now a new page Halbach and Zhang’s Yablo without Gödel which demonstrates the illogical assumptions used by Halbach and Zhang.

 

 

Peter Smith’s ‘Proof’

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.

 

 

Easy Footnotes

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).

 

 

O’Connor’s “computer checked” proof

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.

 

 

New page on Chaitin’s Constant

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.

 

Previous Blog Posts  

 

13 May 2015 Good Math, Bad Math?

 

30 Apr 2015 The Chinese Room

 

31 Mar 2015 Cranks and Crackpots

 

16th Mar 2015 Bishops Dancing with Pixies?

 

23rd Feb 2015 Artificial Intelligence

 

Links  

 

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:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

 

Printer Friendly

 

All pages on this website are printer friendly, and will print the main content in a convenient format. Note that the margins are set by your browser print settings.


Note: for some browsers JavaScript must be enabled for this to operate correctly.

 

Comments

 

Comments on this site are welcome, please see the comment section.

 

Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.

 

If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.

 

Feedback about site design would also be appreciated so that I can improve the site.

 


Copyright © James R Meyer 2012 - 2017  
www.jamesrmeyer.com