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

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 fom Posts: 1,968 Registered: 12/4/12
Re: Matheology 400: Quantifier Confusion
Posted: Dec 6, 2013 9:01 PM

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.

>
>
> 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.

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