Date: Feb 6, 2013 11:41 AM
Author: Alan Smaill
Subject: Re: Matheology 203

WM <> writes:

> On 6 Feb., 13:32, Alan Smaill <> wrote:
>> WM <> writes:
>> > On 6 Feb., 04:47, Ralf Bader <> 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???

You inserted in the quote "{{eg numbers like 0.9999...}}", which is
seriously misleading.

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.

> 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.

Brouwer did not believe that all infinte sets are countable --
your claims in that direction are simply false.

>> 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.

He understood the difference between a mathematical claim being false,
and it being (self)-contradictory.

> Regards, WM

Alan Smaill