[FOM] Infinitesimal calculus

[Rob Arthan]

The classical real numbers can be developed very nicely using decimal  
expansions (or base N for any N) as shown in:

@article{behrend56,
	author="F.A. Behrend",
	title="{A Contribution to the Theory of Magnitudes and the  
Foundations of Analysis}",
	journal="Mathematische Zeitschrift",
	volume=63,
	pages="345-362",
	year=1956}

The ordering relation is easy to define and easy to prove complete;  
the definition of addition requires a little bit of thought about how  
"carrying" works if you have to work from left to right. This gives a  
complete ordered abelian group - the additive group of real numbers.  
Multiplication is then be defined by analysing the group of order- 
preserving homomorphism of the additive group.

Regards,

Rob.

On 27 May 2009, at 01:24, joeshipman at aol.com wrote:

> I disagree. Real numbers can be explained to high school students
> perfectly soundly in terms of successive rational approximations
> (decimal expansions, binary expansions, continued fractions, etc.) to
> points on a geometric line. When this is done, they have a consistent
> intuition for "what real numbers are", an intuition which makes a  
> proof
> of the least upper bound property quite easy to understand. There is  
> no
> philosophical difficulty at this level, unless you try to get across
> the point that physical space might not have a structure that is
> precisely captured by this definition.
...