Theorem   The real numbers 0 and 1 are distinct.

Proof. [Cauchy sequences] This construction proceeds by starting with a standard model of Peano arithmetic, the natural numbers $\mathbb{N}$ , extending to $\mathbb{Z}$ by adding additive inverses, extending to $\mathbb{Q}$ by taking the field of fractions of $\mathbb{Z}$ , and finally defining $\mathbb{R}$ to be the set of equivalence classes of Cauchy sequences in $\mathbb{Q}$ for an appropriately defined equivalence relation.

There is a natural embedding $i\!:\mathbb{N}\to\mathbb{R}$ defined by sending a given number $x$ to the equivalence class of the constant sequence $(x,\,x,\,\dots)$ . Since $i$ is injective and $0$ and $1$ are elements of $\mathbb{N}$ , to prove that $0\ne 1$ in $\mathbb{R}$ we need only show that $0\ne 1$ in $\mathbb{N}$ .

The name $1$ is a label for the successor $S0$ of $0$ in $\mathbb{N}$ . One of the axioms of Peano arithmetic states that $0$ is not the successor of any number. Therefore $0\ne S0$ in $\mathbb{N}$ , and so $0\ne 1$ in $\mathbb{R}$ .

Proof. [Dedekind cuts] This construction agrees with the previous one up to constructing the rationals $\mathbb{Q}$ . Then $\mathbb{R}$ is defined to be the set of all Dedekind cuts on $\mathbb{Q}$ . Letting $x_{\mathbb{Q}}$ represent the name of an element of $\mathbb{Q}$ and $x_{\mathbb{R}}$ represent the name of an element of $\mathbb{R}$ , we define
The proof that $0_{\mathbb{Q}}\ne 1_{\mathbb{Q}}$ is similar to the previous proof. Observe that $0_{\mathbb{Q}} < 1_{\mathbb{Q}}$ . Since no number is less than itself, it follows that $0_{\mathbb{Q}}\notin 0_{\mathbb{R}}$ but $0_{\mathbb{Q}}\in 1_{\mathbb{R}}$ . Thus these Dedekind cuts are not equal.

Proof. [Ordered field with least upper bound property] Here the fact that $0\ne 1$ is a consequence of the field axiom requiring $0$ and $1$ to be distinct.

Proof. [Decimal strings] If one defines
then since neither defining string ends with a tail of 9s and the strings differ in one position, their equivalence classes are distinct.