
Re: Matheology 203
Posted:
Feb 6, 2013 11:41 AM


WM <mueckenh@rz.fhaugsburg.de> writes:
> On 6 Feb., 13:32, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: >> WM <mueck...@rz.fhaugsburg.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

