Re: Courage?

Eckard Blumschein wrote: 
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.


In a separate post, you posed the question of whether pi was a quantity,
and stated that you believed it was. To say that the reals are not "really"
numbers is to say that pi is not "really" a number (I'm not sure what you
mean by this expression; would it not be simpler to simply say "the reals
are not numbers" and perhaps to define a name describing what they are?).
I don't see how pi, or anything else, can be a "quantity" but not a "number".
Perhaps you could explain this?



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

.

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