|
|
Re: Matheology 203
Posted:
Feb 6, 2013 5:04 PM
|
|
WM wrote:
> On 6 Feb., 17:41, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>
>> > Unlike WM? Did I deny that???
>>
>> You inserted in the quote "{{eg numbers like 0.9999...}}", which is
>> seriously misleading.
>
> Sorry. But it is a well known fact that for every finite period there
> is a base in which this period can be expressed as a single symbol.
Indeed, Alan was only partially correct in considering your insertion as
misleading. As an indicator of your total ignorance, this insertion is to
the point.
>> Thus your quote in no way contradicts Ralf Bader's observation --
>> Brouwer in no way supports your claim that "There is no sensible way of
>> saying that 0.111... is more than every FIS".
>>
>> In fact Brouwer says the opposite here -- 0.1111... is created,
>> and it is *distinct* from any finite sequence.
>
> It is distinct, but it is not definable by creating it. Simply *try*
> it. You must fail.
What should he try? Something like writing up this sequence in full glory up
to its "end" on a piece of paper? You have been told almost infinitely
often during the years that everybody knows that this is impossible.
Nevertheless you repeat your idiotic proposals.
> Brouwer has been bisased by the general opinion
> that 0.111... in fact is an infinite sequence. Nevertheless it is
> wrong. But it has lasted several years until I have recognized it.
> Let's see how long it will take you.
The following is for other readers, not for you, because with you, hop and
malt is lost.
In Bridges' and Vita's book "Techniques of constructive analysis" a piece of
mathematical analysis is developed in Bishop style. This means primarily
that classical logic is replaced by intuitionistic logic. And Bishop style
analysis is, as the authors explain, a common core theory for other
variants of analysis: the classical one (Bishop + principle of excluded
third), the intuitionistic one (Bishop + Brouwer's continuity principle and
fan theorem), and of the Russian constructivist school (Bishop +
Church-Turing thesis/ Markov's principle). So, Bridges' and Vita's
constructive analysis is, they say, also valid intuitionistically (amomg
others).
Now, they define a real number x as a set of pairs (p,q) of rational numbers
with the following properties: p<=q for every (p,q) e x; [p,q] n [r,s]
(intersection of closed intervals of rationals) is nonempty for any two
(p,q), (r,s) e X; and for any positive rational eps, there is a (p,q) e x
with q-p<eps. Here, the understanding of set is different from the
classical one, but such a real x is necessarily a kind of infinite
collection. Some time later, Cantor's theorem is proved, namely, that to
any sequence (x_i) of reals there is a real not appearing in that sequence.
This is done by a (of course constructive) procedure of taking thirds of
intervals, starting from the unit interval and picking a third of it which
doesn't contain x_1, proceeding in this fashion with the following x_i and
collecting the pairs (p_i,q_i) from the intervals [p_i,q_i] obtained along
the way. So, in the sense of this Cantor's theorem, there are more than
countably many reals, intuitionistically.
So this kind of thing is possible intuitionistically, but from Mückenheim's
block-of-wood-pseudomathematics it is as infinitely far away as classical
analysis, and if Mückenheim tries to gain support for his crude views from
intuitionistic or constructivist side he is on a totally wrong track.
If one looks into other presentations of intuitionism, for example
Heyting's, this conclusion is confirmed.
>> > Of course. That's why no uncoutable sets exist.
>>
>> Brouwer did not believe that all infinte sets are countable --
>> your claims in that direction are simply false.
>
> I don't know what Brouwer believed. I know what he wrote
but you don't understand it.
> : Cantor's 2nd
> number class does not exist.
>>
>> >> And in van Dalen, p 118, a letter from Brouwer summarising his thesis:
>> >> "I can formulate:
>> >> 1. Actual infinite sets can be created mathematically, even
>> >> though in the practical applications of mathematics in the world
>> >> only finite sets exist."
>>
>> > Brouwer obviously had not the correct understanding of what actual
>> > infinity is, at least when writing that letter. Errare humanum est.
>>
>> I venture to suggest that Brouwer had a better grasp
>> of these matters than yourself.
>
> Maybe. But may also be that you have not a good grasp of his grasp.
Of course, when discrepancies between Brouwer and, ummmh, you show up then
Brouwer is wrong and you are right. After all, you are The Greatest
Mathematicion Of All Times, hahahaha.
--
"Die Natur hat schon häufig natürliche Zahlen zerlegt, zum Beispiel...die
acht Beine einer Spinne in die vier Himmelsrichtungen." Prof. Dr. W.
Mückenheim, Mathematikkoryphäe der "Hochschule Augsburg", am 01.10.09 in
de.sci.mathematik
|
|
