Dedekind Cuts, Constructing R

The idea of dedekind cuts is the following. Consider the real line:

The idea here is to think of A as all valid rational approximations of 2. We wish we could say the suprema of A is 2 but we can't. But what if we say that R includes all of these suprema?

Cut

A subset A⊆Q is a cut if it satisfies the following:

  1. A≠∅ and A≠Q (strict, non-trivial subsets)
  2. If a∈A and q∈Q is such that q<a, then q∈A (all lower terms at the 'sup' are in the set)
  3. If a∈A then ∃q∈A such that q>a (there is no rational maximum 'approximation')
Cr={t∈Q|t<r} is always a cut

The set Cr described is always a cut for r∈Q.

Proof
Let r∈Q be arbitrary. We'll prove all the conditions:

  1. Clearly r−1<r so r∈Cr and thus we have an element (so Cr≠∅). Further we know that r+1∉Cr so Cr≠Q.
  2. Let a∈Cr and q∈Q where q<a. Since a∈Cr then a<r. Thus then q<r, so clearly q∈Cr
  3. If a∈Cr then choose q=a+r2. Clearly q∈Q while q>a and still q<r so then q∈A as needed.
    ☐
Examples/Nonexamples of Dedekind Cuts

a. S={t∈Q|t≤2} is not a Dedekind cut because 2 is a rational maximum.
b. T={t∈Q|t2<2∨t<0} is a Dedekind cut (as we'll use variations of)
c. U={t∈Q|t2≤2∨t<0} is still a Dedekind cut
For in-depth explanations of these see [[AbbotSolns.pdf#page=246]].

R as a construction

Define R={A⊆Q:A is a cut}

Recall the ∃R theorem:

∃R

∃R

∃ an ordered field R which contains Q which satisfies the the axiom of completeness.

Where the axiom of completeness:

axiom of completeness

The Axiom of Completeness

Every nonempty set of real numbers that is bounded above has a least upper bound.

The idea here is that every irrational number will be identified by a cut set A, where since the cut has a least upper bound, then R contains it as well.

References \

  1. [[Abbott Real Analysis.pdf#page=298]]