This note concerns the equivalence between the three properties usually taken as an axiom in synthetic constructions of the real numbers. We start with the least upper bound property, call L, which is usually appeared in construction of the real numbers.
(L, least upper bound axiom): If
is a non-empty subset of
, and if
has an upper bound, then
has a least upper bound
, such that for every upper bound
of
, there holds
.
The second property, call C, is the completeness of reals.
(C, completeness axiom): If
and
are non-empty subsets of
with the property
for any
and
, then there is some
such that
for any
and
.
The third, also last, property, call AC, is the set of two results: the Archimedean property and Cantor’s intersection theorem. These two results often appear as consequences of the construction of reals.
Archimedean property: For any real numbers
and
with
, there exists some natural number
such that
.
Cantor’s intersection theorem: A decreasing nested sequence of non-empty, closed intervals in
has a non-empty intersection.
Our aim is to prove that in fact the above three properties (L), (C), and (AC) are equivalent. Our strategy is to show the following direction:
(L) ⟶ (C) ⟶ (AC) ⟶ (L).
(more…)