Re: Is continuum completely filled up?

Here's a paradox: The reals are a complete ordered field. If you extend
the reals by including infinitesimals, the resulting extension field is no
longer complete. By adding things to a complete ordered field, we get an
incomplete ordered field.

How does adding infinitessimals make *R incomplete? I haven't figured
out where that should fail.

.

Relevant Pages

  • Re: Analysis with the smallest number...
    ... calculus when you can just accept the real number system for what it ... There are no infinitesimals in the real number system, ... ordered field that contains infinitesimals is most certainly not "within ... reals, by definition, are an Archimedean field. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... the reals by including infinitesimals, the resulting extension field is no ... incomplete ordered field. ... And What do you label to that basket? ... We can reason about it as concept using sinbols. ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... but unless their own work makes such infinitesimals useful, ... among standard mathematicians is that the infinite sets are "useful," ... field that is important to the solution of differential equations. ... the Axiom of Infinity in order to prove that a complete ordered field ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... incomplete ordered field. ... by saying that the field is incomplete. ... infinitesimals is bounded above, but has no least upper bound. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... The reals are a complete ordered field. ... the resulting extension field is no ... Saying something is "infinite" is ...
    (sci.math)