Give me ONE digit position of any real that isn't computable up to that digit position!
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From: MoeBlee on 10 Jun 2010 12:49 On Jun 9, 8:21 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > The closest that I've seen to such a poster is Nam > > Nguyen, who does appear to be respected more than most > > posters challenging standard theory. > > Nam's considerably more coherent than guys like Herc and AP, but I > wouldn't say that he's respected in any reasonable sense of the word. Nguyen is somewhat (I wouldn't say 'considerably') more coherent than the two horribly incoherent postgers you just mentioned. But Nam's main problem is that he's just so horribly mixed up about matters in mathematical logic while he is just as stubborn to not be corrected as are the two other posters you mentioned. MoeBlee
From: MoeBlee on 10 Jun 2010 12:50 On Jun 10, 12:09 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Jesse's arguments are more cohesive than those of Herc and AP, but I > wouldn't call his arguments respected, reasonable, or at times even > logical at all. You wouldn't, but I do. MoeBlee
From: FredJeffries on 10 Jun 2010 13:51 On Jun 9, 4:50 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > OK then. Here's what I'd like to see. > > In short, I want to see a counterexample to Little's > statement that: > > "The vast majority of posts arguing against ZFC or its > theorems in sci.math are actually from incompetent > cranks." > A couple days ago Herb posted: > What would be instructive would be to see at what point in Cantor's > proof that |S| < |P(S)| that the proof fails in NF(U). Maybe YOU could start the ball rolling towards a counterexample by responding to that.
From: Leland McInnes on 10 Jun 2010 14:33 On Jun 9, 7:50 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 9, 1:04 pm, Leland McInnes <leland.mcin...(a)gmail.com> wrote: > > > On Jun 8, 3:57 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > What I'd love to see is a poster who _fully_ > > > understands ZFC, perhaps as much or even more than > > > MoeBlee understands the theory -- yet still doesn't > > > believe that ZFC is the best theory. > > I think you > > should ask yourself rather more carefully exactly what you're looking > > for this mysterious poster to say, and then I think you'll start to > > understand why, despite finding plenty of people who meet your stated > > criteria, none of them actually meet your desired criteria.- Hide quoted text - > > OK then. Here's what I'd like to see. > > In short, I want to see a counterexample to Little's > statement that: > > "The vast majority of posts arguing against ZFC or its > theorems in sci.math are actually from incompetent > cranks." > > Thus, I want to see a post arguing against ZFC that's > from a competent poster. I want to see a thread which > starts with a poster discussing a theory other than > ZF(C), but I don't want the responses to have the air > of "You're wrong and everyone else here is right." I > don't want the OP to be called by a five-letter insult, > or to be accused of not understanding ZFC. I don't want > the OP to be accused of having an IQ below 90. > > I want to see a peaceful discussion in which the OP > discusses why they work in the alternate theory, while > the others post why they prefer ZFC to the theory, and > so on. I'm not sure what "arguing against ZFC" would look like; at worst it does a more than adequate job as a formal system from which to construct mathematics to model the world. I could imagine arguing a preference for another system. I doubt you'll get much catharsis out of seeing that though, since it's likely to start and end very quickly with a response along the lines of "yes, that does look interesting, but not really useful to my particular field". Just in case I'll get the ball rolling on a discussion of synthetic differential geometry/smooth infinitesimal analysis as compared to classical calculus. The prime difference here is that SDG supposes a non-punctiform continuum -- you can't pick out individual points from it -- which squares nicely with the intuitive notion of a continuum as an indivisible whole that it ultimately incapable of being described in terms of discrete/distinguishable points. This is, of course, incompatible with classical calculus and the classical real line which is nothing but distinguishable points. Now, to have such a notion of a continuum, you need to make some sacrifices, like, for instance, forgoing the law of excluded middle. On the upside you can develop a purely synthetic geometric calculus in which standard results fall out naturally as simple algebra involving infinitesimals. Indeed, you can build up a very natural differential geometry in which, for example, thinking about a vector field on a manifold as infinitesimal flow across the manifold versus an infinitesimal element of a symmetry group of the manifold is just a matter of lambda abstraction. So there we go; some positive comments about a theory that is not formulatable in ZFC (I suppose it might be, but it would not be pretty). I doubt you'll see many comments though, because I doubt I've said much that is especially controversial.
From: Transfer Principle on 10 Jun 2010 16:25
On Jun 10, 10:51 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > A couple days ago Herb posted: > > What would be instructive would be to see at what point in Cantor's > > proof that |S| < |P(S)| that the proof fails in NF(U). > Maybe YOU could start the ball rolling towards a counterexample by > responding to that. The set theorist Randall Holmes was a webpage on the NF(U) theories: math.boisestate.edu/~holmes/holmes/nf.html On that page, we scroll down and see the following: "Cantor's paradox of the largest cardinal: Cardinal numbers are defined in NF as equivalence classes of sets of the same size. The form of Cantor's theorem which can be proven in Russell's type theory asserts that the cardinality of the set of one-element subsets of A is less than the cardinality of the power set of A. Note that the usual form (|A| < |P(A)|) doesn't even make sense in type theory. It makes sense in NF, but it isn't true in all cases: for example, it wouldn't do to have |V| < |P(V)|, and indeed this is not the case, though the set 1 of all one-element subsets of V is smaller than V (the obvious bijection x |-> {x} has an unstratified definition!)." So we can see what's going on here. NF(U) is based on the idea of a Stratified Comprehension and types on the variables. In particular, in the formula "xey," y must be of a higher type than x. Thus, a set A and its power set P(A) usually don't even have the same type. But if we have "xey" and "xez," the y and z can be the same type. In particular, we can let y be P(x) and z be {x}, so that P(x) and {x} can have the same type. This is why Holmes often refers to the singleton subsets above. According to Holmes, Cantor's proof does show that P1(A), the set of singleton subsets of A, does have a smaller cardinality than P(A). But P1(A) often doesn't have the same size as A, and Holmes gives an explicit example of such a set -- the set V of all sets. Such sets are called "non-Cantorian." George Greene pointed out earlier that the idea of having sets that aren't the same size as their set of singleton subsets (which I admit is a very counterintuitive concept) renders NF(U) not worth considering except theoretically. But then again, there may be some posters who consider Cantor's theorem to be even more counterintuitive than card(P1(V)) < card(V), and such posters should be given the choice of having a set theory with non-Cantorian sets. And if NF(U) is still undesirable for those posters, then they can look for still other theories with a maximal (universal) set. All I want is for everyone to have the opportunity to _choose_ a set theory that best reflects his own intuition. |