Date: Feb 6, 2013 11:41 AM Author: Alan Smaill Subject: Re: Matheology 203 WM <mueckenh@rz.fh-augsburg.de> writes:

> 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???

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