In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 14 Mrz., 13:59, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > > WM <mueck...@rz.fh-augsburg.de> writes: > > > On 14 Mrz., 12:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > > >> WM <mueck...@rz.fh-augsburg.de> writes: > > >> > According to standard matheology one can choose one element each of an > > >> > uncountable set of sets. That is as wrong. Compare Matheology § 225.
Only in WM's corrupted version of math is anyone ever forced to use the axiom of choice, though there is little reason to object to it. > > > > >> You can and do of course reject this axiom. > > > > >> To show something is self-contradictory, however, you need to use the > > >> reasoning principles of the system you want to show is > > >> self-contradictory, not your own beliefs. > > > > > The axiom belongs to the system. It says that elements can be chosen.
The AOC is NOT a required part of THE system, but is entirely optional, and there is not much important mathematics that cannot be done, though with much greater difficulty, without it.
> > > To choose immaterial elements, hmm, how is that accomplished in a > > > system that contains the axiom of choice?
One always chooses material elements. > > > > I can only repeat myself -- > > where is the *logical* contradiction there, in terms of classical > > mathematics? > > You will find it if you try to answer my question. Choosing means > defining (by a finite number of words) a chosen element (unless it is > a material object). No other possibility exists. > > > > Of course, you think it's false, and unimaginable, and whatever > > words you want to use. > > > > But you claim it's *self-contradictory*, don't you? > > > > And that's a whole different claim. > > Please look up what Zermelo wrote. (In Matheology § 225 you will find > the orginal German text.) It is always possible /to choose/ an element > from every non-empty set and to union the chosen elements into a set > S_1.
If that is what Zermelo said then he was wrong to say it because one does not ever "union" elements, only sets. > > This means: It is possible to have and to apply uncountably many > finite words in order to choose and in order to distinguish the > elements in S_1 (a set can have only distinct elements by axiom).
What axiom says that a set can only have "distinct" elements, and what does it mean for elements to be "distinct"? Certainly in the real line, the elements form a continuum, in which they are NOT distinct in any usual sense as there is never a next larger or next smaller real.
WM has frequently claimed that a mapping from the set of all infinite binary sequences to the set of paths of a CIBT is a linear mapping. In order to show that such a mapping is a linear mapping, WM must first show that the set of all binary sequences is a vector space and that the set of paths of a CIBT is also a vector space, which he has not done and apparently cannot do, and then show that his mapping satisfies the linearity requirement that f(ax + by) = af(x) + bf(y), where a and b are arbitrary members of a field of scalars and x and y are f(x) and f(y) are vectors in suitable linear spaces.
By the way, WM, what are a, b, ax, by and ax+by when x and y are binary sequences?
If a = 1/3 and x is binary sequence, what is ax ? and if f(x) is a path in a CIBT, what is af(x)?
Until these and a few other issues are settled, WM will still have failed to justify his claim of a LINEAR mapping from the set (but not yet proved to be vector space) of binary sequences to the set (but not yet proved to be vector space) of paths ln a CIBT.
Just another of WM's many wild claims of what goes on in his WMytheology that he cannot back up. --