fom
Posts:
1,035
Registered:
12/4/12
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Re: Matheology 203
Posted:
Feb 6, 2013 11:21 PM
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On 2/6/2013 4:10 PM, Ralf Bader wrote:
> WM wrote:
<snip>
> 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.
>
And, by interpreting small pieces for the sole
purpose of justifying an agenda WM does not
respect 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.
>
Or as Virgil would write: WMythematician
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