Fathoming Gödel: An Examination of Kurt Gödel's 1931 Paper: On Formally Undecidable Propositions of Principia Mathematica and Related Systems 1

By James Spinosa

A short e-book with a very long title Fathoming Gödel: an Examination of Kurt Gödel’s 1931 Paper: On Formally Undeciable Propositions of Principia Mathematica and Related Systems 1. The title of my e-book includes the title of Kurt Gödel’s 1931 paper, and the title of his 1931 paper includes the title of Bertrand Russell and Alfred North Whitehead’s monumental work Principia Mathematica, the title of which harkens back to Isaac Newton’s Philosophiae Naturalis Principia Mathematica. I remember in high school someone lent me Ernest Nagel and James R. Newman’s book Gödel’s Proof published in 1958. At the time I was quite impressed and thought that if I kept studying assiduously I would one day understand the complex mathematics present in that text. It seems that Gödel’s proofs have had an aura of mystery around them for quite sometime. In high school I believed that Gödel had discovered something profound about the nature of reality and about the limits of human understanding. Now I think he is a fraud.

• Language: English
• Publisher: Lulu.com
• Release date: Dec 31, 2015
• ISBN: 9781329765740

Book preview

Fathoming Gödel: An Examination of Kurt Gödel's 1931 Paper: On Formally Undecidable Propositions of Principia Mathematica and Related Systems 1

Fathoming Gödel: An Examination of Kurt Gödel’s 1931 Paper: On Formally Undecidable Propositions of Principia Mathematica and Related Systems I

Copyright © 2015 by James Spinosa

All rights reserved.  This book or any portion thereof may not be reproduced or used in any manner whatsoever without the express written permission of the publisher except for the use of brief quotations in a book review or scholarly journal.

ISBN 978-1-329-76574-0

Vinculum Publishing

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Dedicated to Steven G. Spinosa: The Spinner Cares

Introduction

Fathoming Gödel is an examination of Kurt Gödel’s 1931paper: On formally undecidable propositions of Principia Mathematica and related systems I.

The translation being used for this critique of Gödel’s1931 paper is by Martin Hirzel.  It is available at: http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf.  The translation is dated November 27, 2000.  The paper is 22 pages in length, and Hirzel states, This document is a translation of a large part of Gödel’s proof . . . This translation omits all footnotes from the original, and only contains sections 1 and 2 (out of four).[1]  The omitted sections are entitled Generalizations and Implications for the nature of consistency.  The two translated sections are entitled: 1−Introduction, which is two pages in length and 2−Main Result, which is 15 pages in length.  The Main Result is divided into seven subsections: Definitions, Gödel-numbers, Primitive recursion, Expressing metamathematical concepts, Denotability and provability, Undecidibility theorem and Discussion.  In order to more easily understand this short e-book Fathoming Gödel, it is highly recommended that the reader download Martin Hirzel’s translation of a large part of Gödel’s proof.  The reader should keep the translation handy as he will probably have many occasions to refer to it in order to clarify many of the notions put forth in Fathoming Gödel.  One of the advantages of having the translation handy is that in moments of frustration the reader can always mutter to himself it’s not just gibberish; it’s gibberish squared.  Actually, it is essential that the reader have a translation of Gödel’s 1931 paper handy in order for Fathoming Gödel to make any sense at all.  Perhaps, no book has been as dependent as Fathoming Gödel is on an outside source for disclosing its meaning.  The dependence of (Herbert) Marshall McLuhan’s Gutenberg Galaxy on outside sources (plural) to disclose its meaning is similar, yet not quite the same.

The reason the words Principia Mathematica are underlined in the title of Kurt Gödel’s 1931 paper is because the words refer to the title of a book by Bertrand Russell and Alfred North Whitehead.  The reason the words Principia Mathematica are not underlined in the subtitle of my eBook Fathoming Gödel is that they were not underlined in my marketing image and metadata; that blunder and perhaps others caused errors to occur in the specially produced NCX/table of contents (an eBooks table of contents). Perhaps, now is the appropriate moments to elaborate on the mathematical symbols employed in this eBook. For several reasons, certain mathematical symbols may not be reproduced by the various formats (and file opening programs) with which this eBook may be reconstituted. Instead, the mathematical symbol may be replaced with a small square or some other type of figure, which indicates a specialized symbol has not been reproduced. To obviate this potential problem, the meaning of many of the mathematical symbols used in this eBook are enclosed in parentheses after the symbol. For example, the mathematical symbol ~ is called a tilde, in this eBook, its meaning will be not, and it will be written as ~ (not). In other words the symbol called a tilde followed by the word not in parentheses means not. Thus, if a small square is reproduced instead of a tilde in some reader’s version of this eBook, he will know that the small square is meant to represent a tilde. To avoid confusion, it should be noted the term ~ (not) means not; it does not mean not, not. The mathematical symbol ˄ is called a modified letter up arrowhead, in this eBook, its meaning will be and, and it will be written as ˄ (and). Again to avoid confusion, the term ˄ (and) means and; it does not mean the confusing phrase and, and. Likewise, the mathematical symbol ˅ is called a modified letter down arrowhead, in this eBook, its meaning will be or, and it will be written as ˅ (or). The mathematical symbol Ǝ is called a Latin, capital letter, reverse E, in this eBook, its meaning will be there exists, and it will be written as Ǝ (there exists). The mathematical symbol Ǝ is most often reproduced as a reverse, capital letter E without any serifs, but it may be less difficult for many formats to produce a reverse E with serifs. In logic and mathematics there is a symbol that can be described as an upside down, capital letter A, in this eBook, its meaning will be for all, and it will be written as Ƒ (for all). The failure of many attempts to reproduce the for all symbol as some variation of an upside down, capital letter A in a manner that would be compatible with most if not all eBook platforms was disappointing.  These failures lead to the idea to abandon the upside down, capital letter A as a typographic character. So in its stead this eBook will use the capital letter F with a hook Ƒ. Unfortunately, its use gives some of the text a slightly cuneiform appearance. The symbol used in logic and set theory used to denote the idea that an item is a member or an element of a set looks similar to the head of a three-pronged pitchfork lying on its side. It looks somewhat similar to the currency symbol for the Euro. So the currency symbol for the Euro € has been appropriated to represent the logic and set theory symbol for is an element of the set or is a member of the set.  Gödel often uses a term he does not bother to define.  At first, I erroneously reproduced it in this text as the arithmetical dot used to signify multiplication.  When I more closely examined Martin Hirzel’s translation with a magnifying glass, I concluded it was not the dot used to represent multiplication.  I now believe it is the dot used in the place of parentheses in logic and set theory.  The period . seems quite similar in size and positioning to the dot used in logic and set theory in the place of parentheses so the period will be used to represent the dot in this text.

The conclusion reached in Fathoming Gödel is that Gödel’s 1931 paper is a shell game.  It is based on several errors that are well camouflaged.  Some shortcomings in the paper are openly admitted although they are downplayed, and errors are also produced in an effort to force a particular conclusion.  This critique is limited to Gödel’s first incompleteness theorem since that is the point at which Martin Hirzel’s translation ends.

Torkel Franzén, in his book Gödel’s Theorem: An Incomplete Guide to