Everything can be defined using sets. Functions and ordered pairs can be defined as sets. For example, the ordered pair $(a, b)$ can be defined as:
$$ (a, b) := \{ \{a \}, \{a, b\}\} $$
We don't have to think of these things in terms of sets (and most mathematicians don't), but we do define them as such.
Is there a simpler or more intuitive proof that the least upper bound property is equivalent to the greatest lower bound property?
A linear continuum in topology is any set along with an order relation such that
(i) the order relation has the least upper bound property
(ii) if $x < y$ there exists an element $z$ such that $x < z$ and $z < y$.
A subset $A$ of $\mathbb{R}$ is inductive if
(i) $1 \in A$
(ii) given $x \in A$, we have $x + 1 \in A$
Define $\mathbb{Z}_+$(the set of positive integers) is the intersection of all inductive subsets of $\mathbb{R}$.
$\mathbb{Z}$ (all integers) is the union of $\mathbb{Z}+$, the element $0$, and the negatives of $\mathbb{Z}+$
Inductive principle: if $A$ is an inductive set of positive integers, then $A = ℤ+$. This is pretty easy to prove using the construction above.
Every nonempty subset of $\mathbb{Z}_+$has a smallest element.
A set $A$ is finite if there is a bijective correspondence of $A$ with some section of the integers. That is, $A$ is finite if it is empty or if there is a bijection $f : A \rightarrow \{ 1, \ldots, n \}$ for some $n$. In the former case we say $A$ has cardinality $0$; in the latter we say it has cardinality $n$.