Re: Well Ordering the Reals

albstorz@xxxxxx said:
>
> > >
> > >
> > Hi Albrecht -
> >
> > You seem to have some of the same concerns as I do, rising out of an
> > appreciation for the continuum, and the embeddedness of discrete sets in it. Is
> > that right?
>
> I think, your grasp of this things is not far away from mine.
>
>
> > It seems to me an element of a set is an atomic, indivisible thing,
> > in the context of the set.
>
> Here is very importent to know, what "in the context of the set" means.
> This is an aspect concerning the axiom of seperation in ZFC.

Ah, this seems to eb the axiom of subsets. I am not sure this is related,
offhand. I guess what I meant is that, while one may have a set of sets or
strings, each of which in itself is divisible into parts, in the context of the
larger set, those elements are considered as indivisible atoms.
>
>
> > Your drops are not indivisible, but rather
> > continuous and indistinct.
>
> According to the definition of sets by Georg Cantor, which is the only
> meaningful definition that I know, a set is the union of
> distinguishable real or abstract things. (I know, it's not a good
> translation or coverage of the intention of Cantor, but I think, you
> know this definition).
> I don't see why a drop of tea isn't a distinguishable thing - in real
> and also as an abstract idea of it.
> Now I found, that Cantor's definition isn't complete when I detect,
> that there are things, which hold the definition, but when I build up
> an union of them, they lose their property to be distinguishable. I
> named the property to be distinguishable the property of elementness
> (which is maybe not a good name).
> The weakest consequence which I had to consider is, that I should have
> a test to ensure the permanence of the elementness.
> Maybe, for example, a real number itself is a distinguishable thing
> like a drop of tea, but a coherent intervall of reals is just a blend.

Yes, this is an issue of continuum, and the discreteness of elements. The
continuum is not discrete, but points in the continuum are discrete in
themselves. There is definitely a difference between discrete sets, where
elements are clearly separated from each other, and continuous sets, where we
speak more about a range, or a measure, of the set. I strongly believe that the
countable/uncountable distinction is more about this apsect of things than
about counting persay, and that some dichotomy between types of sets will
always exist. Your tea is continuous, and not countable, but quite measurable.
>
>
> > So, set theory doesn't deal well with this idea. So,
> > how do you measure such drops? Perhaps in moles of molecules?
>
> Sets have not to be measured, their elements have to be counted. A set
> of teadrops consists of drops of tea as the axiom of extensionality
> (ZFC) states.

Well, I would view each drop of tea, not as an element, but as a set of
molecules, and the tea in the cup, not as a set of drops, but as the union of
the sets of molecules. I think that is the way to bring that into focus. Does
that make some sense? Certainly the drops can not be re-separated, but the
molecules are still individuals. It's the lack of atomic nature that makes the
drops not work.

>
> > Perhaps we should
> > consider the size of a space to be an exact number of points in it? I think
> > this is where I am going with this. There is something to be said about
> > considering the extent of a set. I suppose you should measure your set in
> > ounces, then fill it up some more, and add a little bit of sugar and a crumpet.
> > Have a nice tea. :)
>
>
> That's a good suggestion, I think. Thanks!
Take care and enjoy. Earl Grey is sounding good right about now, perhaps with a
drop of Darjeeling....
>
> Albrecht
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

  • Re: Computer being developed modeled after human brain
    ... I hold such views on a continuum. ... such a false distinction. ... They are classifications crated by the brain when applied to the ... The distinction doesn't exist in the universe. ...
    (comp.ai.philosophy)
  • Re: Well Ordering the Reals
    ... >>> appreciation for the continuum, and the embeddedness of discrete sets in it. ... >> a test to ensure the permanence of the elementness. ... >> like a drop of tea, but a coherent intervall of reals is just a blend. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > appreciation for the continuum, ... > I don't see why a drop of tea isn't a distinguishable thing - in ... > a test to ensure the permanence of the elementness. ... > like a drop of tea, but a coherent intervall of reals is just a ...
    (sci.math)
  • Re: Halloween is coming!
    ... Well, it isn't my cup of tea, but I can see a distinction. ... believe that for some gay men, the thought of sex with women is ... it doubles their chances for a date ...
    (rec.motorcycles)