Undergrad 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 &lt; 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.