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Topic: RE: Matheology 400: Quantifier Confusion
Replies: 188   Last Post: Dec 13, 2013 5:48 AM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Matheology 400: Quantifier Confusion
Posted: Dec 6, 2013 9:53 PM

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.

> >
> >
> > 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!
--

Date Subject Author
12/4/13 Tucsondrew@me.com
12/4/13 wolfgang.mueckenheim@hs-augsburg.de
12/4/13 fom
12/4/13 wolfgang.mueckenheim@hs-augsburg.de
12/4/13 fom
12/4/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Virgil
12/5/13 Tucsondrew@me.com
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Tucsondrew@me.com
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/6/13 Tucsondrew@me.com
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/8/13 wolfgang.mueckenheim@hs-augsburg.de
12/8/13 Virgil
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/6/13 Tucsondrew@me.com
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/7/13 Virgil
12/8/13 wolfgang.mueckenheim@hs-augsburg.de
12/8/13 Virgil
12/6/13 fom
12/6/13 Virgil
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/4/13 Virgil
12/4/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 fom
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 fom
12/5/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Virgil
12/5/13 Virgil
12/5/13 Tucsondrew@me.com
12/5/13 fom
12/5/13 Tucsondrew@me.com
12/5/13 fom
12/5/13 Michael F. Stemper
12/5/13 fom
12/5/13 Tucsondrew@me.com
12/5/13 fom
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Tucsondrew@me.com
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/6/13 Virgil
12/6/13 Michael F. Stemper
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/7/13 albrecht
12/7/13 fom
12/7/13 albrecht
12/6/13 fom
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 Virgil
12/7/13 fom
12/8/13 Virgil
12/8/13 fom
12/6/13 fom
12/6/13 Virgil
12/6/13 fom
12/7/13 ross.finlayson@gmail.com
12/5/13 Tucsondrew@me.com
12/5/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Virgil
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/6/13 albrecht
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 albrecht
12/6/13 Virgil
12/6/13 Tucsondrew@me.com
12/6/13 albrecht
12/6/13 albrecht
12/6/13 fom
12/6/13 Virgil
12/7/13 albrecht
12/8/13 albrecht
12/5/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/5/13 Tucsondrew@me.com
12/4/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/5/13 Virgil
12/5/13 ross.finlayson@gmail.com
12/9/13 William Hughes
12/9/13 wolfgang.mueckenheim@hs-augsburg.de
12/9/13 William Hughes
12/9/13 wolfgang.mueckenheim@hs-augsburg.de
12/9/13 Tucsondrew@me.com
12/10/13 wolfgang.mueckenheim@hs-augsburg.de
12/10/13 Tucsondrew@me.com
12/10/13 wolfgang.mueckenheim@hs-augsburg.de
12/10/13 Virgil
12/10/13 Tucsondrew@me.com
12/10/13 Virgil
12/9/13 William Hughes
12/10/13 wolfgang.mueckenheim@hs-augsburg.de
12/10/13 William Hughes
12/10/13 wolfgang.mueckenheim@hs-augsburg.de
12/10/13 William Hughes
12/10/13 wolfgang.mueckenheim@hs-augsburg.de
12/10/13 William Hughes
12/10/13 Virgil
12/11/13 wolfgang.mueckenheim@hs-augsburg.de
12/11/13 Virgil
12/11/13 Tucsondrew@me.com
12/11/13 fom
12/11/13 Tucsondrew@me.com
12/11/13 William Hughes
12/11/13 wolfgang.mueckenheim@hs-augsburg.de
12/11/13 William Hughes
12/11/13 wolfgang.mueckenheim@hs-augsburg.de
12/11/13 William Hughes
12/11/13 Virgil
12/11/13 Virgil
12/11/13 fom
12/11/13 Virgil
12/13/13 albrecht
12/10/13 Virgil
12/10/13 wolfgang.mueckenheim@hs-augsburg.de
12/10/13 Virgil
12/10/13 Tucsondrew@me.com
12/11/13 wolfgang.mueckenheim@hs-augsburg.de
12/11/13 Virgil
12/11/13 Tucsondrew@me.com
12/11/13 wolfgang.mueckenheim@hs-augsburg.de
12/11/13 Tucsondrew@me.com
12/11/13 wolfgang.mueckenheim@hs-augsburg.de
12/11/13 Tucsondrew@me.com
12/12/13 wolfgang.mueckenheim@hs-augsburg.de
12/12/13 Tucsondrew@me.com
12/12/13 Virgil
12/12/13 Virgil
12/12/13 Virgil
12/12/13 Virgil
12/12/13 Virgil
12/11/13 fom
12/11/13 Virgil
12/12/13 wolfgang.mueckenheim@hs-augsburg.de
12/12/13 Virgil
12/11/13 Virgil
12/11/13 Virgil
12/10/13 Virgil
12/11/13 William Hughes
12/11/13 Virgil
12/11/13 William Hughes
12/11/13 Virgil
12/11/13 William Hughes
12/11/13 fom
12/11/13 Virgil
12/10/13 Virgil
12/9/13 Virgil
12/9/13 Tucsondrew@me.com
12/9/13 William Hughes
12/9/13 Tucsondrew@me.com
12/9/13 Virgil
12/9/13 William Hughes
12/9/13 Virgil
12/9/13 Tucsondrew@me.com
12/10/13 William Hughes