Peter Webb wrote: > "Gerry Myerson" <firstname.lastname@example.org> wrote in message > news:email@example.com... > > In article <firstname.lastname@example.org>, > > "Peter Webb" <email@example.com> wrote: > > > >> "Gerry Myerson" <firstname.lastname@example.org> wrote in message > >> news:gerry-D682F9.email@example.com... > >> > In article <firstname.lastname@example.org>, > >> > "Gene Ward Smith" <email@example.com> wrote: > >> > > >> >> I didn't see any signs, as far as I had gotten, that he even knows > >> >> anything about modern set theory. Does he? > >> > > >> > I don't know. > >> > > >> > I reject astrology, even though I don't know anything about modern > >> > astrology (I don't even know if there is such a thing). I reject > >> > "creation science" and "intelligent design," even though I haven't > >> > read any recent writings of their advocates. I don't have to; I > >> > know where they're going, and I know they're never going to get > >> > anywhere useful, going in that direction. > >> > > >> > I personally don't put set theory in the same category as astrology > >> > or creation science. > >> > >> Doesn't this undermine your whole analogy? Why didn't you pick an > >> orthodox > >> theory like Evolution, Special Realtivity or Plate Techtonics as being > >> the > >> theory he is attacking? (Set theory is every bit as well accepted as any > >> of > >> these other topics). Because he looks less of a crank if you compare him > >> to > >> attacking astrology than him attacking (say) the Theory of Evolution, > >> even > >> though this is a much closer analogy? > > > > In its day, phlogiston was a well-accepted theory. Alchemy was orthodox. > > So was spontaneous generation. Bright people, not cranks, spent > > a lot of time and effort trying to prove the parallel postulate. > > It's not unknown in the history of science, even of mathematics, > > for very good people to do a lot of work that makes later generations > > scratch their heads and say, why did they go down that blind alley? > > why did they even bother to think about those things? > > > > It may be wrong to say that today's set theorists are yesterday's > > phlogiston theorists - but is it crankish? > > > > A poor analogy. Theories about the physical world are largely evaluated on > the basis of their predicitive capacity, and it is on this basis that these > theories have been proved wrong. Mathematics is based upon other crietria, > including consistency and the ability to generate interesting theorems. > Beleiving in phogiston was not crankish in the 17th century, but maintaining > that you can't bisect an angle with a straightedge and compass is most > certainly crankish - about as crankish as maintaining that the computable > reals are uncountable. >
What Norman means by this is that the computable numbers are not effectively countable. This is quite correct. There is nothing crankish about it, it is just that his way of expressing himself is somewhat eccentric.
> I'm no expert either, but lets go back to a quote from before: > > ---------- > Most mathematicians reading this paper suffer from the impression that the > `computable real numbers' are countable, and that they are not complete. As > I mention in my recent book, this is quite wrong. Think clearly about the > subject for a few days, and you will see that the computable real numbers > are not countable , and are complete. Think for a few more days, and you > will be able to see how to make these statements without any reference to > `infinite sets', > ------------------- > > The proof that computable numbers are countable (using the standard > definition of computable numbers) takes about 3 lines. In typical crank > style, he simply states that this result is false for reasons that are > obvious if you think about it.
He doesn't accept that the word "countable" is meaningful unless it means "effectively countable", so he thinks the result has to be re-interpreted if it is to be meaningful. By saying the computable numbers are complete and not countable, he means: every effectively Cauchy sequence of computable numbers has a computable limit, and the computable numbers are not effectively countable. It would perhaps be better if he translated his claims into our language so it became clear he's saying something we already agree with, but there's nothing unrespectable about this position.
> I do wonder how many of his students bought his new book proving that the > computable reals are uncountable, and whether they know more about set > theory as a result.
He doesn't prove it. He states that he prefers to use the field of computable numbers rather than the field of real numbers because he doesn't think the latter field really exists, although this won't affect any of the arguments in the book, and then he mentions his view that the computable numbers are complete but not countable without giving an argument for it. It would be better if he made clear that, when translated into widely used language, this result is something we all already accept, and that what he's really advocating is a redefinition of key terms. He just mentions it as a throwaway line. I don't think his students are going to get too confused by it.