You'll notice that although GG still lists Cantor's "paradox" in his index, in the text he doesn't quite bring himself to say that there is such a thing.  Why not?  Because he has read Garciadiego's BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC 'PARADOXES,' which shows quite clearly that there is no such thing as Cantor's paradox, or the Burali-Forti paradox, or Russell's for that matter.  The so-called "set-theoretic paradoxes" were for the most part inventions of Russell, and not a single one from the period, comes out as anything but a meaningless formulation.The problem this creates for GG is that so-called "set theory" is nonsense, and not much worth wasting time on.  Apart from Cantor's own pathetic inability ever to define what a set is, the history is a farce of the blind leading the blind--trying to "avoid" formulations which are not paradoxes or anything else.  This is worth writing about?  Worth listing 1900 items in a bibliography, about?  It's sad, but a good study in how wastes of time and resources occur. So GG goes ahead and talks about these "paradoxes" as if they really were such, and about people's "responses" to them as if there was anything to respond to.  GG still hasn't quite weaned himself from the "paradoxes," although he cites Garciadiego and should have known better.  The gist of the book is that the "paradoxes" which led to Godel's argument (and those of the Intuitionists, the Logicists and Formalists as well as their successors), are not paradoxes at all--they are meaningless formulations. This undermines most, if not all, of twentieth-century mathematics, and in particular destroys Godel's very sloppy argument.Garciadiego cites Richard's own formulation of this "contradiction" (Richard's term) in a letter to Poincare. He also cites Richard reducing the argument to meaninglessness. What does this have to do with Godel? It's simple. For Godel, Richard's "paradox" means that truth in number theory cannot be defined in number theory. On this basis, he distinguishes truth from provability. He combines his idea of Richard's "paradox" with the idea that provability in number theory can be defined in number theory. He arrives at the conclusion that if all the provable formulae are true, there must be some true but unprovable formulae. However, since Richard's "paradox" is without meaning, since it has no logical content whatsoever and is simply letters pulled out of a bag, there is no basis in Godel's argument for distinguishing truth from provability. It turns out that there is no logical content in the idea that if all the provable formulae are true, there must be some true but unprovable formulae.People are having a hard time getting over the notion that Godel didn't do his homework, and has nothing to say, but really you have to grow up.  Get over it.  The problem is that Godel was a terrible scholar, and did not apply himself sufficiently to the details of the development of set theory.Garciadiego's book has implications for all twentieth-century mathematics. Here are just a few examples of horrendous errors which explain a lot about why mathematics today is regarded as the province of clowns.  For example, Brouwer based the idea of an infinite ordinal number on the idea that Cantor had proved well-ordering of the ordinal numbers. But not only did Cantor never prove this, but also, he never said he had done so, and never used the term infinite ordinal number.  Turing never examines the "paradoxes" in order to determine whether they are simply meaningless formulations. Thus, in an attempt to "prove that there is no general method for determining about a formula whether it is an ordinal formula, we use an argument akin tothat leading the Burali-Forti paradox, but the emphasis and the conclusion are different." As Garciadiego reveals, there is no Burali-Forti paradox. In the context of an attempt to prove the Trichotomy Law, START Start TRANSACTION WITH CONSISTENT SNAPSHOT; /* 2152 = 4f915966ee24e2f1247693944d3ffdea

The Search for Mathematical Roots, 1870-1940 PDF

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