On 2/6/2013 4:10 PM, Ralf Bader wrote: > WM wrote:
> The following is for other readers, not for you, because with you, hop and > malt is lost. > In Bridges' and Vita's book "Techniques of constructive analysis" a piece of > mathematical analysis is developed in Bishop style. This means primarily > that classical logic is replaced by intuitionistic logic. And Bishop style > analysis is, as the authors explain, a common core theory for other > variants of analysis: the classical one (Bishop + principle of excluded > third), the intuitionistic one (Bishop + Brouwer's continuity principle and > fan theorem), and of the Russian constructivist school (Bishop + > Church-Turing thesis/ Markov's principle). So, Bridges' and Vita's > constructive analysis is, they say, also valid intuitionistically (amomg > others). > Now, they define a real number x as a set of pairs (p,q) of rational numbers > with the following properties: p<=q for every (p,q) e x; [p,q] n [r,s] > (intersection of closed intervals of rationals) is nonempty for any two > (p,q), (r,s) e X; and for any positive rational eps, there is a (p,q) e x > with q-p<eps. Here, the understanding of set is different from the > classical one, but such a real x is necessarily a kind of infinite > collection. Some time later, Cantor's theorem is proved, namely, that to > any sequence (x_i) of reals there is a real not appearing in that sequence. > This is done by a (of course constructive) procedure of taking thirds of > intervals, starting from the unit interval and picking a third of it which > doesn't contain x_1, proceeding in this fashion with the following x_i and > collecting the pairs (p_i,q_i) from the intervals [p_i,q_i] obtained along > the way. So, in the sense of this Cantor's theorem, there are more than > countably many reals, intuitionistically. > So this kind of thing is possible intuitionistically, but from Mückenheim's > block-of-wood-pseudomathematics it is as infinitely far away as classical > analysis, and if Mückenheim tries to gain support for his crude views from > intuitionistic or constructivist side he is on a totally wrong track. > If one looks into other presentations of intuitionism, for example > Heyting's, this conclusion is confirmed. > >>>> Of course. That's why no uncoutable sets exist. >>> >>> Brouwer did not believe that all infinte sets are countable -- >>> your claims in that direction are simply false. >> >> I don't know what Brouwer believed. I know what he wrote > > but you don't understand it. >
And, by interpreting small pieces for the sole purpose of justifying an agenda WM does not respect it.
>> : Cantor's 2nd >> number class does not exist. >>> >>>>> And in van Dalen, p 118, a letter from Brouwer summarising his thesis: >>>>> "I can formulate: >>>>> 1. Actual infinite sets can be created mathematically, even >>>>> though in the practical applications of mathematics in the world >>>>> only finite sets exist." >>> >>>> Brouwer obviously had not the correct understanding of what actual >>>> infinity is, at least when writing that letter. Errare humanum est. >>> >>> I venture to suggest that Brouwer had a better grasp >>> of these matters than yourself. >> >> Maybe. But may also be that you have not a good grasp of his grasp. > > Of course, when discrepancies between Brouwer and, ummmh, you show up then > Brouwer is wrong and you are right. After all, you are The Greatest > Mathematicion Of All Times, hahahaha. >