details re Z-R |- PA consistent
in [Math]
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From: FredJeffries on 24 Jul 2010 14:44 On Jul 19, 9:15 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 16, 8:01 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Jul 14, 3:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > Thus, how can I tell who really are the open-minded posters? > > You can't. Get over it. > > > Instead, I want to consider those who shut out their opponents ideas > > > to be closed-minded. > > You want to put people in boxes? Fine. > > What I want is to be able to defend certain posters, but if I defend > everyone, then I'm really defending no one. Is that where finitistic reasoning leads you? So if I feed everyone, I am really feeding no one? But even that (as bad as it is) is not what you are doing. You are actually saying "In order to feed some, I must starve others." > So, I want to have some > criteria for determining which posters to defend. > > Is open-mindedness is a poor criterion for deciding which posters I > should defend? No. But you AREN'T using open-mindedness as a "criterion for deciding which posters I should defend". You're using CLOSE-MINDEDNESS as a criterion of which posters to CONDEMN. > Then fine. But there has be _some_ way for me to > decide which posters to defend. > > I'd love to do this without "putting people in boxes." I'd love it if > I > could 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. YOU are living evidence of the moral bankruptcy of the anti-Cantorian position, a system where whatever is not explicitly permitted is forbidden. You anti-Cantorians would drag us back to the 17th century age of absolutism. But just as the enlightenment freed us from political tyrants, it also freed Bolzano, Dedekind, Cantor, ... to be able to critically examine the properties of the trans-finite. </sermon>
From: Ross A. Finlayson on 24 Jul 2010 22:36 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: > > > On Jul 16, 8:01 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > On Jul 14, 3:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > Thus, how can I tell who really are the open-minded posters? > > > You can't. Get over it. > > > > Instead, I want to consider those who shut out their opponents ideas > > > > to be closed-minded. > > > You want to put people in boxes? Fine. > > > What I want is to be able to defend certain posters, but if I defend > > everyone, then I'm really defending no one. > > Is that where finitistic reasoning leads you? So if I feed everyone, I > am really feeding no one? > > But even that (as bad as it is) is not what you are doing. You are > actually saying "In order to feed some, I must starve others." > > > So, I want to have some > > criteria for determining which posters to defend. > > > Is open-mindedness is a poor criterion for deciding which posters I > > should defend? > > No. But you AREN'T using open-mindedness as a "criterion for deciding > which posters I should defend". You're using CLOSE-MINDEDNESS as a > criterion of which posters to CONDEMN. > > > Then fine. But there has be _some_ way for me to > > decide which posters to defend. > > > I'd love to do this without "putting people in boxes." I'd love it if > > I > > could 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. > > YOU are living evidence of the moral bankruptcy of the anti-Cantorian > position, a system where whatever is not explicitly permitted is > forbidden. > > You anti-Cantorians would drag us back to the 17th century age of > absolutism. But just as the enlightenment freed us from political > tyrants, it also freed Bolzano, Dedekind, Cantor, ... to be able to > critically examine the properties of the trans-finite. > </sermon> As a particular and unique example of a mapping from the natural integers onto (1-1) the unit interval of reals, the equivalency function gives similar free-thinkers a conscientious avenue past the otherwise constraining trans-finite, into the "uncountable". Nobody need use the trans-finite in the applied (asymptotics suffice). People regularly integrate from secondary school, where the equivalency function conveniently founds a nonstandard measure theory, where standard theory just defines (throws in) more terms. Set theory has been post-Cantorian since Zermelo, for that matter since Russell's antinomies there noted in the naive Mengenlehre, yet still those non-standard features Cantor noted (about infinity) for example his big Absolute infinity or "counting backwards" from a numerical infinity and etcetera have ready connotation in structure via symmetry. Combinatorics in set theory are perfect in the finite (although interesting features exhibit in the asymptotics), the future will find that real numbers and features defined by real numbers (like material objects and fields in continuum analysis) are polydimensional and that the geometric mutations in the large and small do observe mathematical structure, that is irrelevant to the trans-finite, and mute from it. Also Goedel's theory of incomplete theories is by extension of the same argument incomplete, thus lacking to describe completeness, or rather the inexistence so posited. That doesn't speak to the position of ultrafinitists whose position can be modeled by a monkey on the beach. Warm regards, Ross Finlayson
From: Loadmaster on 27 Jul 2010 16:09 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.
From: Transfer Principle on 27 Jul 2010 23:21 On Jul 24, 3:17 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 20, 7:45 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > On Jul 20, 7:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > Keep in mind that WM has said more than once that he rejects formal > > > axiomatization. > > I suppose we can come up with the following: declare the set > > N of (WM-)naturals to equal N_n for some natural number n, > Isn't this a circular definition? You are using a natural number n to > define the set of natural numbers? Recall that I mentioned this to respond to Jeffries and his comment on Kolmogorov complexity. But if natural numbers are defined in terms of complexity and complexity is defined in terms of naturals, then how do we escape circularity? But, as MoeBlee points out above (and I include his response in this post), WM _rejects_ formal axiomatization. Someone who rejects formal axiomatization is less likely to even _care_ about avoiding circularity than someone who accepts axioms. Since I'm attempting to represent WM's ideas, WM's desiderata have priority over _all_ other considerations -- otherwise, how can I claim that these concepts represent WM's ideas at all? Thus, Jeffries's concern about avoiding circularity is less relevant as the poster whose ideas I'm attempting to represent is unlikely to even care about avoiding circularity. Indeed, at this point I see nothing wrong with simply defining N in the standard way, then declaring by fiat the subset N_WM to be the set of all naturals that WM accepts, and this set N_WM can be defined in any manner, circularity be darned. > 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 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. And if one responds by saying that those who use ZFC don't oppose large cardinals in the same way that finitists oppose infinite sets or ultrafinitists oppose large finite sets, then let me bring up the mathematician K. Kunen. According to Kunen, there are some cardinals so large that their existence would contradict the Axiom of Choice (but are compatible with ZF). So there really are large cardinals that users of ZF_C_ oppose. And so let me apply Jeffries's statement to this analogy: > [S]omeone may say that [Kunen's cardinal] > 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... And so as soon as a user of ZFC can explain in what sense that Kunen's cardinal exists (even as ZFC proves that it doesn't exist), only then need an ultrafinitist accept that 10^500^500^500 (or whatever large finite cardinal) exists in some sense.
From: Transfer Principle on 27 Jul 2010 23:36
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. Once again, whether I consider 0eN or ~0eN depends on the preferences of the poster whose ideas I'm trying to represent. According to WM, N begins with 1, period. And so I must begin my naturals with 1 as well, if I am to have any success at all at representing _WM_'s ideas. Again, WM's desiderata have priority over all other considerations, including Jeffries and his preference for 0eN. Furthermore, ~0eN has its advantages as well. If we were to insist on starting with zero, we would have: > > N_0 = {0} > > N_(n+1) = {meN | Eab (aeN_n & beN_n & (a+b=m v ab=m))} but then N_n = {0} for all n. (If we were to allow exponentiation and consider 0^0 = 1, then N_n would no longer be trivial, but not everyone considers 0^0 to be 1.) Even in standard theory, there's a Dedekind-cut construction (recently mentioned by David Ullrich in another thread) which starts from N+ (i.e., N without 0) and constructs the positive fractions Q+ without having to worry about zero denominators, then the positive D-cuts R+ without having to worry about defining multiplication for negative reals. So this construction is somewhat neater than starting with N0 (i.e., N including 0). |