Understanding Dedekind Cuts: How to Recognize a Cut

[mikeyBoy83]

I'm trying to wrap my head around these Dedekind cuts. The definition is straightforward but I'm a little confused about the downward closure part.

$x \in Q$ and $y<x \Longrightarrow y \in Q$

Does that mean that this is not a cut because it is bounded below?

{$x \in Q : x>1 \wedge x<2$}

Clear this up for me please.

 

[stevendaryl]

I'm trying to wrap my head around these Dedekind cuts. The definition is straightforward but I'm a little confused about the downward closure part.

$x \in Q$ and $y<x \Longrightarrow y \in Q$

Does that mean that this is not a cut because it is bounded below?

{$x \in Q : x>1 \wedge x<2$}

Clear this up for me please.

No, it's not a cut. Another way to think of it is that a cut splits the set of all rationals into two nonempty pieces: A "left" set, $L$, and a "right" set, $R$, where

• Every rational $x$ is either in $L$ or $R$.
  
• If $x$ is in $L$, and $y$ is in $R$, then $x < y$
  
• The real associated with the pair $L,R$ is the unique number $r$ that is greater than or equal to every element of $L$ and less than or equal to every element of $R$

A Dedekind cut is just such an $L$.

 

[Martin Rattigan]

stevendaryl wrote:

The real associated with the pair $L,R$ is the unique number $r$ that is greater than or equal to every element of $L$ and less than or equal to every element of $R$

Since Dedekind cuts are used to construct the reals I think it would be better to say that the real number $r$ is the cut, or preferably that $L$ as defined by OP is an extended real number.