Date: Oct 5, 2017 7:49 PM
Author: Markus Klyver
Subject: Re: Dipshit Zelos Malum claims that ...

Den torsdag 5 oktober 2017 kl. 16:48:33 UTC+2 skrev Python:
> Markus Klyver wrote:
> > Den torsdag 5 oktober 2017 kl. 06:21:32 UTC+2 skrev Zelos Malum:
> ...
> >> They aren't equivalent because dedekinds cut requires that ALL
> >> rational numbers are included in the partition, as that is waht
> >> defines a fucking partition. How many times does this have to be said?

> >
> > This is not true, though. A Dedekind cut must necessarily contain all
> > elements but one of the elements of ?. Consider a cut with all rationals
> > strictly less and strictly greater than 0. Then 0 isn't in the cut.

>
> No. The Dedekind Cut for 0 is {x|x<0},{x|x>=0}, 0 is in the cut.
> This is the standard way to inject Q inside the set of cuts, the
> cut associated to a rational q is {x|x<q},{x|x>=q}
>
> (Wikipedia:It is a partition of the rational numbers into two non-empty
> sets A and B, such that all elements of A are less than all elements of
> B, and A contains no greatest element.)


I stand corrected. 0 will be in B. I was confused by Rudin's "Dedekind cuts", which aren't partitions of ?. Rudin does not have an upper cut, but his construction is of course equivalent.