On 12/6/2013 8:53 PM, Virgil wrote: > 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. > > Nice! >
I can actually add a little more.
First, Cantor had been in the tradition of logical foundations and did call for treating the real numbers in the sense of the formalist domain assumptions. His construction was a justification for that view. His theological views probably push this into idealism or platonism, and the logical position would be one of mathematical realism (bivalent, mutually exclusive logic).
Second, while the order relation grounds the trichotomy, the infinitary classes differentiate the rational from the irrational. I know that is obvious; but, I left it unsaid.
Third, I would strongly recommend anyone to find a copy of Kelley's topology and read the metrization lemma in the section on uniform spaces. Logical identity is a non-numeric concept. In Kelley's proof, one needs both Cantor's fundamental sequences and Dedekind's cuts as a greatest lower bound. The given uniformity is merely a nested collection of relations meeting certain constraints. Those constraints correspond to the triangle inequality and, say, symmetry. But, they are constraints on relation products. They are not numeric.
The domain is partitioned using set differences, and, each difference is associated with a value from among diminishing inverses of powers of two. This is the "Cantorian" contribution. But, these are just "polygonal magnitudes" between elements of the domain. The multiplicities of these magnitudes are reduced to a single magnitude by greatest lower bounds. That would be the "Dedekind" contribution. Unless there are proofs which eliminate one step or the other, it seems that both men made independent, necessary steps toward our understanding of identity of real numbers in terms of a metric compatible with logical notions.
Fourth, we have become accustomed to "real analysis" in the sense of the isomorphism between sequences of naturals obtained through the power set and ordered in the sense of the Baire space. These relate to the usual order of real numbers through continued fractions, if not by some other means. But, if you try to do the construction of the reals within set theory by using a finite initial segment of ordinals to ground a modular number base, you can get to the Dedekind cuts; you just cannot "place" the Dedekind cuts in the set universe. After considering this problem, I realized that the usual construction leading to Dedekind cuts (depending on the natural number order) and the continued fraction relationship enables one to locate the Dedekind cuts with the power set of the finite ordinals (remember, this construction recreated a modular number base from a finite initial segment).
I point this last construction out because all of these constructions are slightly different. Dedekind cuts are like initial segments. Cantorian fundamental sequences are symmetric -- or, at least, nested -- convergences. The continued fraction convergence is like a two-player game choosing from the left side and from the right side alternately -- but, we understand our triangle inequalities in terms of absolute values. Each is different. And each adds to the difficulty of how we understand the real numbers in our set theory.