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Topic: Matheology 203
Replies: 3   Last Post: Feb 6, 2013 4:06 PM

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mueckenh@rz.fh-augsburg.de

Posts: 14,578
Registered: 1/29/05
Re: Matheology 203
Posted: Feb 6, 2013 9:40 AM
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On 6 Feb., 13:32, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 6 Feb., 04:47, Ralf Bader <ba...@nefkom.net> wrote:
> >> According to Mückenheim, "There is no
> >> sensible way of saying that 0.111... is more than every
> >> FIS". Of the authorities you called upon, whom would you find capable of
> >> regardng this as a sensible assertion

>
> > Compare Matheology § 030:   We can create in mathematics nothing but
> > finite sequences, and further, on the ground of the clearly conceived
> > "and so on", the order type omega, but only consisting of equal
> > elements {{i.e. numbers like 0,999...}}, so that we can never imagine
> > the arbitrary infinite binary fractions as finished {{Brouwers Thesis,
> > p. 143}}. [Dirk van Dalen: "Mystic, Geometer, and Intuitionist: The
> > Life of L.E.J. Brouwer", Oxford University Press (2002)]

>
> van Dalen, unlike WM, is careful to note Brouwer's own note
> on "equal elements":
>
>    "Where one says 'and so on', one means the arbitrary
>     repetition of the same thing or operation, even though that thing or
>     operation may be defined in a complex way"
>
> thus justifying existence of expansions like 0.12121212...


Unlike WM? Did I deny that??? Of course even the existence of 0.
[142857] and every other periodic decimal fraction is possible
according to Brouwer. If you can't believe that this is covered by my
§ 030, then simply use the septimal system even if it is not an
optimal system.
>
> "arbitrary" sequences are a different matter.


Of course. That's why no uncoutable sets 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.

Just a question: Have you ever seen a Cantor-list where more than half
of the interesting sequences (a_j) of digits a_kj with k < j had
infinite length? Have you ever seen a Cantor-list with at least one of
the interesting sequences of digits having infinite length? No? Why
the heck do you believe that they play the crucial role in Cantor's
"proof"?

Try to imagine this "proof" without the obviously counterfactual
belief that irrelevant tails beyond a_jj play any role. What remains?

Regards, WM



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