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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 15:46:52 -0700
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André G. Isaak <agisaak@gm.invalid> writes:
[...]
> I already replied to this post, but wanted to provide an alternative 
> response in case you don't follow my earlier one.
>
> [A, B] is defined as all real numbers x such that A <= x <= B
>
> (A, B] is defined as all real numbers x such that A < x <= B
>
> What's important to realize here is that you don't get (A,B] by starting 
> with [A,B] and 'removing the first point'. These are simply two 
> different intervals. There is no operation involved in which you get one 
> from the other.

I disagree with that last statement, and would be interested in a
demonstration that I'm wrong about that.

[A,B] is a set of real numbers.  (A,B] can be constructed by taking
[A,B] and removing a single element, namely A, and similarily vice
versa.  All members of (A,B] are members of [A,B], and there is
exactly one real number that is a member of [A,B] and not of (A,B].

Of course this doesn't imply that there's a "first element" of (A,B].
There clearly isn't.  (Pete has correctly recognized a contradiction,
but has grabbed onto the wrong end of it.)

-- 
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 */