Re: Euclidean Geometry in Schools




On 26/06/2005 15:59, Timothy Murphy wrote:

> William Elliot wrote:
>
>> Construction is so messy. Let's instead teach axioms.
>> The reals are a complete Archimedean ordered field.
>
> What exactly does "complete" mean here?

Every Cauchy series converges.

.

Relevant Pages

  • Re: How many real numbers are there?
    ... the elements of my set T satisfy the ZF axioms. ... The construction of the real numbers does not depend on the specific ... THEREFORE the cardinality of the reals is countable!! ... The construction of the reals (ie, a complete ordered field) does ...
    (sci.math)
  • Re: The real numbers, and general comments
    ... But if you want to talk about the uncountability of the reals, ... should know that the construction of the reals (whether by Dedekind cuts ... ZF is a set of axioms for set theory. ...
    (sci.math)
  • Re: Every open interval contains a rational?
    ... In this sense Cantor's is a construction of real numbers from rational. ... Construction is preferred by some because a set of axioms may contain one which is not true. ... It's just that a proof based upon a specific way of defining the reals will not be a proof for someane who has seen a different way of defining them. ...
    (sci.math)
  • Re: Real Analysis: Construct Reals vs. Axioms
    ... reals in some Analysis texts, ... authors go to great extends to make a "construction" for the reals? ... that such a structure *can* be constructed, and to use the axioms to ...
    (sci.math)
  • Re: How many real numbers are there?
    ... the elements of my set T satisfy the ZF axioms. ... The construction of the reals (ie, a complete ordered field) does ... THEREFORE, the "reals" (ie, one particular complete ordered field, ...
    (sci.math)
Loading