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From: Keith Thompson <kst-u@mib.org>
Newsgroups: sci.logic,sci.math,comp.theory,comp.ai.philosophy
Subject: Re: Cantor was wrong about cardinality --- Reals are defined inconsistently
Date: Mon, 16 Sep 2019 15:29:03 -0700
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André G. Isaak <agisaak@gm.invalid> writes:
[...]
> Intervals are sets. In case you're not familiar with basic set theory, 
> {A, B}, {B, A}, and {A, A, B} are all the exact same set. There is no 
> order defined over the elements of a set. If you want to talk about the 
> first element or second element of a set, you'd actually need to define 
> some sort of ordering relation. For countable sets, this is trivially 
> easy to do. For uncountable sets, though, it is not. So unless you can 
> propose some way of actually determining the order of the elements in a 
> nondenumerably infinite set, this argument doesn't hold up.
>
> For example, in the interval (0, 1) can you explain how you would 
> determine the position of (e.g.) 0.25? Assume it is the nth element in 
> this set. solve for n.

So by "ordering", you mean, for example, a mapping between the
members of the set and the positive integers?  Of course no such
mapping is possible for the real interval (0, 1).  We can define
an ordering that lets us determine n, but it will miss some members
of the interval (in fact <handwave> almost all of them</handwave>).

I would have thought of an "ordering" as a relationship such that
for any two members X and Y of the set, exactly one of X<Y, X=Y,
or X>Y is true -- which is certainly the case for sets of reals.
But that kind of ordering doesn't help us determine the "first"
member of the set.

I agree with your point.  I just wanted to be clear about the
terminology.

-- 
Keith Thompson (The_Other_Keith) kst-u@mib.org  <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */