Re: Courage?

On 4/10/2005 12:02 AM, Ross A. Finlayson wrote:

> With the real numbers, it seems that an enumeration or well-ordering

I see the problem with the real numbers in that they are not enumerable
in so far, they are lacking any possibility of numerical representation
.. In other words, well-ordering of the reals fails not because of a
missing first element but because the real numbers are not really
numbers to enumerate. Nonetheless, I do not doubt that e.g. the solution
pi to a geometrical problem can be approached as close as possible
within the rationls.


> It's taught that that explanation is not to be used as the mechanism
> behind the analytical integral of standard classical analysis, and it's
> perhaps good that that is so, because it took a hundred years for
> textbook authors to agree on the limit or delta-epsilon.

Using this limit by Weierstrass contradicts to Cantor's definition of
the real numbers and prevents to cross the border into the continuum.

> In more
> modern times, there are some texts that are a return to the use of
> infinitesimals in the integral calculus directly.

Well, you perhaps refer to hyperreal numbers. I do not support any
numbers in excess of infinity.

Regards,
Eckard

.

Relevant Pages

  • Re: Probability of picking a positive rational number at random
    ... Requiring well-ordering just obfuscates what's going on. ... That iota is great among infinitesimals represents that summing ... In a theory like ZFC with well-orderable reals, ... except that there are reals that biject to finite ordinals, ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... > concept of a well-ordering itself that I'm concerned with. ... > 1) If you want a model of reals that even vaguely follows the ... The computable reals are those with some "finite expression of their ... As the infinitesimals are considered, and those as fluents to their ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... > concept of a well-ordering itself that I'm concerned with. ... > 1) If you want a model of reals that even vaguely follows the ... The computable reals are those with some "finite expression of their ... As the infinitesimals are considered, and those as fluents to their ...
    (sci.logic)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... generic extension of a set as just the set in the generic extension. ... There are only reals between zero and one, and everywhere reals, real ... In the hyperreals infinitesimals can be greater than zero in ... If the normal ordering of the positive reals was a well-ordering, ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... generic extension of a set as just the set in the generic extension. ... There are only reals between zero and one, and everywhere reals, real ... In the hyperreals infinitesimals can be greater than zero in ... If the normal ordering of the positive reals was a well-ordering, ...
    (sci.logic)