details re Z-R |- PA consistent
in [Math]
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From: Transfer Principle on 28 Jul 2010 00:01 On Jul 24, 11:44 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 19, 9:15 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > I'd love to do this without "putting people in boxes." I'd love to > > defend certain posters and have it be seen not as "putting > > people in boxes," but as adhering to perfectly reasonable criteria > > for choosing which posters to defend, since -- as I repeat -- it is > > neither possible nor desirable to for me defend _everyone_. > <anti-anti-Cantorian sermon> > So you are trapped in an anti-Cantorian universe where it's all zero- > sum games and you can't pay Paul without stealing from Peter. I'd love it if I could break out of this trap of zero-sum games. I'd love it if I could pay both Peter and Paul. But the real question is, will Peter or Paul let me? Case in point -- in another thread, there is a dispute between Archimedes Plutonium and Iain Davidson, and I was called upon to settle it. I tried to give a positive-sum response and say that both AP and Davidson were right. But this response was rejected, because AP wants me to say that AP is right and Davidson is wrong, while Davidson wants me to say that Davidson is right and AP is wrong. Furthermore, Tonio laughed at me for giving such a response. So Jeffries wants me to avoid zero-sum games, but the only response that AP, Davidson, and Tonio will accept from me is a zero-sum response. So, out of curiosity, what would Jeffries recommend that I post in this situation? I want to avoid resorting to the zero-sum universe, but AP and Davidson are looking for a zero-sum response from me. I will not post in that other thread until Jeffries gives me a suggestion, then I'll decide whether to go along with that suggestion or post something else there. > YOU are living evidence of the moral bankruptcy of the anti-Cantorian > position, a system where whatever is not explicitly permitted is > forbidden. I don't necessarily believe that everything that isn't explicitly permitted is forbidden, but I admit that I might as well since I defend posters who do. Indeed, I have pointed out several times before that one distinguishing trait of those whom Jeffries calls "anti-Cantorian" (and others call five-letter insults) is that they believe that whatever isn't explicitly permitted is forbidden -- they want more control over what can be proved in their theory. On the other hand, those whom he calls "anti-anti-Cantorian" believe the opposite -- that whatever isn't explicitly forbidden is permitted. They delight in proving unexpected results from a small set of axioms (or schemata). And never the twain shall meet? But I still believe that they can. I'd like to believe that there can be a theory which doesn't forbid that which isn't explicitly permitted, but still allows for, say, infinite sets that work differently from those discovered by the three mathematicians mentioned by Jeffries (Bolzano, Dedekind, Cantor). Indeed, the poster Tony Orlow desires infinite sets that work differently from standard infinite sets.
From: FredJeffries on 28 Jul 2010 12:29 On Jul 27, 1:09 pm, Loadmaster <da...(a)tribble.com> wrote: > FredJeffries wrote: > > These different systems are also indicated by the quarrel "Do the > > natural numbers start at 0 or at 1?" Well, counting starts at 1 (most > > of the time) [...] > > Unless you consider the possibility that the collection of > objects (whatever that is) that you are counting can be > empty, in which case counting must being at zero. But if the collection is empty, one (pun intended) doesn't START counting at all. It is the result of the (non-started) counting which is zero.
From: FredJeffries on 28 Jul 2010 12:54 On Jul 27, 8:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 24, 3:49 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Jul 19, 8:46 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > i.e., we define N_n recursively as: > > > N_1 = {1} > > > N_(n+1) = {meN | Eab (aeN_n & beN_n & (a+b=m v ab=m))} > > These different systems are also indicated by the quarrel "Do the > > natural numbers start at 0 or at 1?" Well, counting starts at 1 (most > > of the time) but computation is better served by having a bit pattern > > of all 0's as the base. > Jeffries and his preference for 0eN. How ever did you derive THAT as my preference? Didn't I just say "counting starts at 1"? Haven't I at sundry times in divers posts pointed out that there is a difference between what I refer to as Counting Numbers and Information Numbers? Isn't it I who delight in pointing out that Peano's original axioms start with 1? This is another (trivial) example of your drawing a verifiably wrong conclusion about me. So why then should I have any confidence in your other conclusions?
From: FredJeffries on 28 Jul 2010 13:18 On Jul 27, 9:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 24, 11:44 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > YOU are living evidence of the moral bankruptcy of the anti-Cantorian > > position, a system where whatever is not explicitly permitted is > > forbidden. > > I don't necessarily believe that everything that isn't explicitly > permitted is forbidden, but I admit that I might as well since I > defend posters who do. > > Indeed, I have pointed out several times before that one > distinguishing trait of those whom Jeffries calls "anti-Cantorian" > (and others call five-letter insults) is that they believe that > whatever isn't explicitly permitted is forbidden -- they want > more control over what can be proved in their theory. On the > other hand, those whom he calls "anti-anti-Cantorian" I have never called anyone "anti-anti-Cantorian" except myself. > believe the opposite -- that whatever isn't explicitly forbidden > is permitted. They delight in proving unexpected results from > a small set of axioms (or schemata). > > And never the twain shall meet? > > But I still believe that they can. I'd like to believe that there > can be a theory which doesn't forbid that which isn't explicitly > permitted, but still allows for, say, infinite sets that work > differently from those discovered by the three mathematicians > mentioned by Jeffries (Bolzano, Dedekind, Cantor). Indeed, > the poster Tony Orlow desires infinite sets that work differently > from standard infinite sets. I got the "whatever is not explicitly permitted is forbidden" vs. "whatever is not explicitly forbidden is permitted" distinction from our friend Alexander Yessenin-Volpin. cf. his "On the Logic of the Moral Sciences" http://yessenin-volpin.org/onthelogic.pdf at page 11, his "principle of liberalism" vs "principle of despotism" Lest you take my attempt at hyperbole (which I tried to indicate by means of <anti-anti-cantorian sermon> tags) too seriously, please note that in the next paragraph he goes on <quote> The often encountered confusion of the concepts not permitted and forbidden, or not forbidden and permitted, is based on the assumption that every act is either permitted or forbidden and then only one of the two. But only complete methods, almost never encountered (in complex cases), satisfy this assumption. </quote> Further, please note that a clear, understandable mathematical definition (say of an Abelian Group) is an example a dogma which sets out what is permitted (to be an Abelian Group) and all else is forbidden (to be an Abelian Group). Fred Jeffries is IN FAVOR of clear, unambiguous mathematical definitions!!!!!
From: FredJeffries on 28 Jul 2010 14:05
On Jul 27, 8:21 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 24, 3:17 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > But this is a problem that comes up often: someone may say that > > 10^500^500^500 does not exist, but there must be some sense in which > > it does exist in order to be able to point to it and say that it > > doesn't exist... > > One analogy that I often bring up is: > > Large Cardinal : Mainstream :: Mainstream : Finitism > > that is to say, large cardinal theories extend the mainstream > theory by adding inaccessibles, etc., in the same way that the > mainstream theory extends the finitist theories by adding > infinite sets. I would say, rather, that a large cardinal axiom RESTRICTS the mainstream theory in that it declares that a certain cardinal definitely exists rather than leaving the existence an open question. Adding the commutativity axiom to the group definition RESTRICTS the notion of group. Every Abelian group is a group, but there are groups which are not Abelian. > > I say that the finitists shouldn't be forced to have infinite sets, > and the ultrafinitists shouldn't be forced to have large finite sets, > just as those who use ZFC shouldn't be forced to have large > cardinals as well. (If I understand correctly) in ZFC, one neither has nor doesn't have large cardinals. One of your hypothesized ZFC users doesn't (perhaps) use large cardinals, but she cannot prove that they don't exist. She also cannot avoid that there are models of ZFC in which there do exist (some varieties of) large cardinals. |