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#Interpretations_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.