Date: Feb 6, 2013 7:32 AM
Author: Alan Smaill
Subject: Re: Matheology   203

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...

"arbitrary" sequences are a different matter.

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."

> Regards, WM

Alan Smaill