Re: Well Ordering the Reals
- From: aeo6@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (Tony Orlow (aeo6))
- Date: 01 Dec 2005 00:02:21 GMT
albstorz@xxxxxx said:
>
> albstorz@xxxxxx wrote:
>
>
> 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?
>
> Just an spontan idea to make my argument clearer: What is a molecule
of
> tea? There are no such things.
> Oh, well, tea is made of molecules, most of which are water, and all
of which
are separable (barring chemical reactions - maybe we should talk about
atoms?)
> If you have a set of lines, is it also a set of points of which the
> lines consists? If you have a set of functions, is it also a set of
the
> letters and numbers which the function descripes?
>
These seem to be two separate situations. In the spatial example,
there is a
plane with a continuum of lines, each of which is a continuum of
points.
Certainly, in my mind, you have infinitely more points than you have
finite or
infinite lines, though not in standard theory. When it comes to
formulas, one
can break them down into primitive concepts like constants, variables
(typed or
untyped), and operators, and certainly, each of those is a member of a
language
which can be constructed from 1 or more symbols from an alphabet.
That's a
discrete set. If you have a set of functions, then within that set you
would
consider each function as atomic, even though you may see the function
as a
string, an ordered set of symbols, which can be further decomposed. I
am not
sure that presents a big problem for me, though there may be ways to
pretty it
up a bit.
>
>
> 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
>
>
>
--
Smiles,
Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
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