Logic and Language

Logic and Language

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

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7: Relations of Natural Numbers 24 to 46

Note that (provided you have JavaScript enabled) clicking on (show) will reveal further details, while clicking again will hide it. Also, clicking on (show Gödel’s) will reveal relevant parts of Gödel’s text (shown in green), while clicking again will hide it. Please note that older browsers may not display some symbols correctly.

This guide is intended to assist in attaining a full understanding of Gödel’s proof. If there is any difficulty in following any part of the proof, please contact me and I will try to help. And if you have any suggestions as to how this guide might be improved, please contact me. 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.

The number-theoretic relations are now becoming more complex as they correspond to more complex statements about formulas of the formal system **P**. It is not intended to cover every detail of these relations, but rather to concentrate on the main points and the thrust of the argument. It would be easy to get bogged down in such details and then fail to see the wood for the trees. As noted on the previous page, the names of the relations are mainly abbreviations of German words, see notes at the foot of this page for the words and the English translations.

These define number-theoretic relations that correspond to assertions as to whether a symbol is a free variable or a bound variable within a given symbol string.

24. **v Geb n,x** (show)

25. **v Fr n,x** (show)

26. **v Fr x** (show)

The functions 27-30 lead up the function 31 which corresponds to the concept of the substitution of a free variable by a symbol or symbol string of the formal system.

27. **Su x(n|y) **28. **k St v,x** 29. **A(v,x)** 30. **Sb _{k}(x v|y)** 31.

Note that, depending on what version of the translation you are using, **Sb** may be represented in this format:

which is the format using in Gödel’s original paper.

Assertions as to which numbers correspond to the axioms of the formal system

Functions 32 and 33 define some axioms of the formal system. Relations 34-42 inclusive are relations that use the previously defined relations/functions to define which Gödel numbers correspond to the axioms of the formal system.

32. **x Imp y, x Con y, x Aeq y, v Ex y**
(show)

33. **n Th x**
(show)

34. **Z–Ax(x)** (show)

35. **A _{1}-Ax(x)**,

37. **Q(z,y,v)** (show)

38. **L _{1}-Ax(x)** (show)

39. **L _{2}-Ax(x)** (show)

40. **R-Ax(x)** (show)

41. **M-Ax(x)** (show)

42. **Ax(x)** (show)

Proofs in the formal system

The relations 43-46 deal with defining the number-theoretic relations that correspond to the concepts of the rules of inference of the system, the concept of a proof-schema, and the concept of a formula being provable in the system.

43. **Fl(x y z)** (show)

44. **Bw(x)** (show)

45. **x B y** (show)

46. **Bew(x)** (show)

Below is a list of names used for various relations in the text, which are mostly abbreviations of German words; translations are provided below:

A | Anzahl | = number |

Aeq | Aequivalenz | = equivalence |

Ax | Axiom | = axiom |

B | Beweis | = proof |

Bew | Beweisbar | = provable |

Bw | Beweisfigur | = proof-schema |

Con | Conjunktion | = conjunction |

Dis | Disjunktion | = disjunction |

E | Einklammern | = include in brackets |

Elf | Elementarformel | = elementary formula |

Ex | Existenz | = existence |

Fl | unmittelbare Folge | = immediate consequence |

Flg | Folgerungsmenge | = set of consequences |

Form | Formel | = formula |

Fr | frei | = free |

FR | Reihe von Formeln | = series of formulae |

Geb | gebunden | = bound |

Gen | Generalisation | = generalization |

Gl | Glied | = term |

Imp | Implikation | = implication |

l | Lange | = length |

Neg | Negation | = negation |

Op | Operation | = operation |

Pr | Primzahl | = prime number |

Prim | Primzahl | = prime number |

R | Zahlenreihe | = number series |

Sb | Substitution | = substitution |

St | Stelle | = place |

Su | Substitution | = substitution |

Th | Typenerhohung | = type-lift |

Typ | Typ | = type |

Var | Variable | = variable |

Wid | Widerspruchsfreiheit | = consistency |

Z | Zahlzeichen | = number-symbol |

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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.

16th Mar 2015 Bishops Dancing with Pixies?

23rd Feb 2015 Artificial Intelligence

31 Mar 2015 Cranks and Crackpots

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

www.jamesrmeyer.com