Re: Well Ordering the Reals


> >
> >
> 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.


> 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.


> 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.

> 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!

Albrecht

.

Relevant Pages

  • 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: Well Ordering the Reals
    ... >> Hi Albrecht - ... > 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)
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