# $0\ne 1$ as real numbers

###### Theorem.

The real numbers 0 and 1 are distinct.

There are four relatively common ways of constructing the real numbers. One can start with the natural numbers^{} and augment it by adding solutions to particular classes of equations, ultimately considering either equivalence classes^{} of Cauchy sequences^{} of rational numbers or Dedekind cuts of rational numbers. One can instead define the real numbers to be the unique (up to isomorphism^{}) ordered field with the least upper bound property. Finally, one can characterise the real numbers as equivalence classes of possibly infinite^{} strings over the alphabet $\{0,1,2,3,4,5,6,7,8,9,.\}$ satisfying certain conditions. We offer a proof for each characterisation.

###### 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^{} $(x,x,\mathrm{\dots})$. 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 $\mathbb{R}$.
∎

###### 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

${0}_{\mathbb{R}}$ | $$ | ||

${1}_{\mathbb{R}}$ | $$ |

The proof that ${0}_{\mathbb{Q}}\ne {1}_{\mathbb{Q}}$ is similar to the previous proof. Observe that $$. 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. ∎

###### 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. ∎

###### Decimal strings.

If one defines

$0$ | $=\overline{(0,0,0,0,\mathrm{\dots})}$ | ||

$1$ | $=\overline{(1,0,0,0,\mathrm{\dots})}$ |

then since neither defining string ends with a tail of 9s and the strings differ in one position, their equivalence classes are distinct. ∎

Title | $0\ne 1$ as real numbers |
---|---|

Canonical name | 0ne1AsRealNumbers |

Date of creation | 2013-03-22 15:23:15 |

Last modified on | 2013-03-22 15:23:15 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 11 |

Author | mps (409) |

Entry type | Theorem |

Classification | msc 54C30 |

Classification | msc 26-00 |

Classification | msc 12D99 |

Related topic | DecimalExpansion |