More pathologies

[Lawrence Bryan]

Interesting, indeed. Let's try it again...


These pathological functions are a lot of fun, especially ones that  
can be defined simply. For example take a real function whose value  
for rational numbers a/b is 1/b (except for the rational, 0, whose  
value will be defined as 0) and whose value for irrational numbers is  
0. This function is continuous at every irrational point and (again  
except for 0) discontinuous at every rational point.

These kinds of things come about when intuitive notions are being  
defined rigorously. We all know what a straight line is, but how would  
one define it? Some mathematicians just gave up. In David Hilbert's  
Foundation of Geometry, for example, Chapter I, Section 1, "The  
Elements of Geometry and The Five Groups of Axioms", Hilbert calls  
forth three distinct sets of objects: "points", "lines", and "planes"  
as basic elements with no definition of them. The axioms that follow  
say something about the relationship of each of the these objects to  
one another. Mathematically the three objects could have been called  
"plunks", "pillycopples", and "bordnoys". Axiom I.1 would then have  
been written, "For every two plunks, A and B there exists a  
pillycopple  that contains each of the plunks A and B". The proofs of  
any of the theorems that follow need not have any relationship at all  
to what we intuitively think of as points, lines, or planes. For  
example Theorem 1, "Two pillycopples in a bordnoy either have one  
plunk in common or none at all."

Lawrence, fondly remembering his sojourn in Professor Swingle's  
topology class some forty plus years ago...


On May 9, 2008, at 10:21 AM, Henry wrote:

> No body!  Now that's what I call a pathology!
>
> From: Lawrence Bryan
> Sent: Friday, May 09, 2008 1:05 PM
> Subject: More pathologies
>
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