Waterstones On Formally Undecidable Propositions of Principia Mathematica and Related Systems

Author: Nick

Rating: 5

Review: Excellent Product - Good Value

Author: Paul Vjecsner

Rating: 3

Review: As indicated in two other reviews of mine here, my comprehension of Goedel's work is opposite to the general one. My marking three stars regardless for this book is motivated by his extensive influence, but also by his fair admission later in life that his thesis could amount to hocus-pocus. Indeed, I see it as one of the prominent mistakes in logical history, and I shall endeavor to explain as best I can. It should suffice to consider his Section 1, an outline of his proposed proof. Although that section is brief, it already foreshadows an oppressingly complex logical symbolism for statements that in my view can be made much clearer using ordinary language. The symbolism, to be sure, is intended to establish a formal language, whose meaning is to be decided separately. This will be seen one of the problems. For now, let me give the principal statement Goedel contended to be true but undecidable (neither provable nor disprovable): "This statement is unprovable." He symbolized it (p.40) as: "~Bew[R(n);n]". Font limitations made me slightly change it; the tilde "~" means "not", "Bew" is a German abbreviation for "provable", and within brackets "R(n)" says "Statement n" and "n" stands for the full statement. Goedel proceeds: "...supposing...~Bew[R(n);n] were provable, it would also be correct; but that means...that...~Bew[R(n);n] would hold good, in contradiction to our initial assumption. If, on the contrary, the negation of ~Bew[R(n);n] were provable, then [its provability] would hold good. ~Bew[R(n);n] would thus be provable [in contradiction to the unprovability it states], which again is impossible." (I corrected some errors within brackets.) So since both ~Bew[R(n);n] and its negation are unprovable, it is undecidable, and Goedel continues (p.41): "...it follows at once that ~Bew[R(n);n] is correct, since...certainly unprovable (because undecidable). So the proposition which is undecidable in the system...turns out to be decided by metamathematical considerations." "Metamathematical", in excusing the contradiction, designates the above formal system void of assigned meaning, whereas the statement discussed is to have meaning. Not quite a lucid argument. Overlooked, furthermore, is a contradiction using the same reasoning as in the preceding. Coupled with the preceding finding that ~Bew[R(n);n] CANNOT be proved unprovable (for if so proved, it would be contradicted), can in contradiction be that it CAN be proved unprovable. For if it were instead provable, it would again be contradicted. The statement in question thus becomes a paradox, rather than true, similar to paradoxes like the "liar", mentioned by Goedel (p.40). He strangely adds to it the footnote: "Every epistemological [paradox] can likewise be used for a similar undecidability proof." The "liar", however, is, like all paradoxes, not a true statement, as required, but one harboring a contradiction. (I deal in my book with, and offer solutions to, paradoxes more fully, including Goedel's resulting one, without naming him.) There occurs, further, another huge blunder in the alleged proof. The undecidability is said to apply to some of mathematics; in the above formula, ~Bew[R(n);n], the "n" refers to a number, with this justification by Goedel (p.38): "For metamathematical purposes it is naturally immaterial what objects are taken as basic signs, and we propose to use natural numbers for them." Adding (p.39): "Metamathematical concepts and propositions thereby become concepts and propositions concerning natural numbers..." How so? In one breath he proposes using natural numbers as immaterial signs, and in the next breath the material concerns natural numbers! The fallaciousness can indeed be made clear by considering our statement, ~Bew[R(n);n], interpreted as "This statement is unprovable." As noted, in ~Bew[R(n);n] the "n", now a number, is to name the whole statement, inside which it is also used in "Statement n..." But whether or not the statement is named by a number, the point is that the name must refer to the intended content of the statement to correspondingly function, not to the usual number possibly represented. Therefore the statement, or anything else similarly used, has nothing to do with numbers, or mathematics generally.