as real numbers
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 satisfying certain conditions. We offer a proof for each characterisation.
This construction proceeds by starting with a standard model of Peano arithmetic, the natural numbers , extending to by adding additive inverses, extending to by taking the field of fractions of , and finally defining to be the set of equivalence classes of Cauchy sequences in for an appropriately defined equivalence relation.
There is a natural embedding defined by sending a given number to the equivalence class of the constant sequence . Since is injective and and are elements of , to prove that in we need only show that in .
The name is a label for the successor of in . One of the axioms of Peano arithmetic states that is not the successor of any number. Therefore in , and so in . ∎
This construction agrees with the previous one up to constructing the rationals . Then is defined to be the set of all Dedekind cuts on . Letting represent the name of an element of and represent the name of an element of , we define
The proof that is similar to the previous proof. Observe that . Since no number is less than itself, it follows that but . Thus these Dedekind cuts are not equal. ∎
Ordered field with least upper bound property.
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. ∎
|Title||as real numbers|
|Date of creation||2013-03-22 15:23:15|
|Last modified on||2013-03-22 15:23:15|
|Last modified by||mps (409)|