(both) its completeness (and/or, conversely, its categorical consistency) by way of its own axioms. Which means that Hilbert’s program was impossible as stated since there’s no way the second point can be rationally combined with assumption-1 as long as the system of axioms is indeed finite – otherwise you have to add an infinite series of new axioms, beginning, I guess, with Pappus’s! Likewise, Gödel’s incompleteness theorem reveals that neither <i>Principia Mathematica</i>, nor any of other consistent system of recursive arithmetic, could decide whether every proposition, and/or its negation, was provable within that system itself.
Yet beyond Hilbert’s faux pas concerning Pappus, one should note that Gödel’s theorem itself, in some realistic sense, then actually supports Hilbert’s basic idea of a deeper, more inclusive, ‘meta-logical’ foundation as a ‘Gödelian mapping’ that ‘covers’ all mathematics and geometry. Indeed, it was Hilbert’s student Gentzen who used a Gödel mapping ‘orders’ of ‘trans-infinite’ systems of numbers to actually prove Gödel’s theorem: so truly meta-logically validates ordinary arithmetic. In any case, though this conclusion also loosely conforms to Russell’s logistic ideations, it at once demonstrates a vast improvement over his criticism of Cantor’s proof for an infinite series of cardinal numbers – which, after-all, is the point of Cantors arguments in the sense that some ‘axiom of continuity’ like Archimedes’s is required to generate a infinite field of real numbers. Which makes Hilbert’s Pappian faux pas seem almost trivial in comparison – as I’d like to know how Russell expected to find some ‘greatest cardinal
Pages: 1 2 3 4 5 6 7
