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.