In article <IZCdnUREjLrrHD_PnZ2dnUVZ_sKdnZ2d@giganews.com>, fom <fomJUNK@nyms.net> wrote:
> On 12/6/2013 1:11 PM, Michael F. Stemper wrote: > > On 12/05/2013 02:42 PM, fom wrote: > >> On 12/5/2013 2:28 PM, Michael F. Stemper wrote: > >>> On 12/05/2013 01:04 PM, Zeit Geist wrote: > >>>> On Thursday, December 5, 2013 12:49:37 AM UTC-7, fom wrote: > >>>> > >>>>> No. It also uses the fact that the listing may be > >>>>> > >>>>> arbitrarily given, > >>>>> > >>>>> http://en.wikipedia.org/wiki/Algorithmically_random_sequence#Interpretat > >>>>> ions_of_the_definitions > >>>>> > >>>>> > >>>>> > >>>> > >>>> Be careful here, the "arbitrarily chose" sequence does Not need to be > >>>> a "random" sequence. > >>>> It is just a sequence of Real Numbers who only properties is that it > >>>> IS a sequence of Real Numbers that "supposedly" contains all Real > >>>> Numbers. > >>> > >>> There is no need for the assumption that it contains all reals. We can > >>> prove that no sequence of reals contains all of them without having to > >>> first assume that it does. > > > >> I had to think about that. > > > > Now, you've caused me to think. Specifically, about the following > > paragraph: > > > >> Cantor's proof had been directed to an audience > >> who thought of "infinity" as a monolithic concept. > >> Proving that any list asserting to put the set > >> of reals in correspondence with the naturals was, > >> in fact, not a complete list demonstrated that > >> "infinity" could be viewed as a plural notion subject > >> to logical analysis. > > > > Unfortunately, even after sleeping on it and re-reading it, I still > > don't get it. Would you please expand on this? (For starters, did > > Cantor include the assumption that the list had all reals?) > > > > In "On a property of the set of real > algebraic numbers", he gives a proof > where a sequence of real numbers "given > by a law" can always be shown to be > incomplete. In his letters to Dedekind > prior to this published paper, he assumes > that an exhaustive sequence of real numbers > can be given as a well-ordered countable > listing. > > > >> Your statement reflects Hilbert's formalism. > > > > My inclinations are much more formalist than Platonist. > > > > There is a certain sense in which they are the same > because of the assumptions of identity on the formal > axiomatic domain. Usually when a statement like yours > is made, that reflects a strong tendency toward > conventionalism (everything begins with stipulation). > Then, the presumption of ideal truths one might associate > with Platonism is denied. > > To understand the earlier statement with regard to Cantor, > consider the period in question. Whereas Euclidean > geometry had been portrayed or presumed by many as a > physical theory of space, the demonstration that the > postulate of parallels was independent put geometric > intuition in question. Moreover, if one takes certain > statements from Dedekind at their word, static graphics > had been used for a number of geometric proofs. In > the present day, one would simply smile if given a > carefully done ruler and compass drawing (at least, > I would like to think so -- but, I have personal > experience to the contrary in a major university). > > Because of what had been happening in geometry, > mathematics was being "arithmetized". This was a > natural progression. During the initial debates > over infinity, practical minded scientists and > mathematicians developed numerical methods which > gave Cauchy the means of formulating the epsilon-delta > notion of limit. This simply became more prioritized > with the problems in geometry. > > At the same time, the relationships between number > systems became evident as notions like complex > numbers and quaternions were developed. Conventional > arithmetical systems could be extended as pairs or > quadruples. So, one has a simple kind of representation > theory developing. In terms of foundations, we now > understand that as the construction of the reals > from the naturals. > > But, this construction develops with the group most > motivated toward a logical foundation. With Dedekind > and Cantor one gets the full use of infinitary > class-based constructions (A Dedekind cut is much > more than an ordered pair by which an integer or > positive rational can be understood with respect to > a natural.). Classes are, historically, a topic > for logicians and not mathematicians. > > This class-based constructivism precedes Hilbert's > "Foundations of Geometry", although not by much. It > includes Frege's definition of natural numbers. > Later, one has Russell's predicative formalisms in > "Principia Mathematica" and some of Carnap's work. > > To understand the real difference, one must consider > the essential element in the non-formalist construction > of the reals -- at each stage, the order relation is > preserved. So, effectively, the relation of trichotomy > on the reals is inherited from the sequential order > on the naturals. It is *this* which grounds the > identity of individuals in the real number system and > not the presumed axioms of identity intrinsic to > semantic theory for a formalist deductive system. > > So, in a world full of mathematicians which did not > work entirely within axiom systems and did not really > think about how there could be a plurality of infinities, > Cantor's work had been rather surprising. In spite > of WM's portayals of theology, Cantor made his mathematical > justifications on the basis of the notion of an abstract > arithmetical system and provided an arithmetical calculus > with which to work.