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Topic:  Is a rhombus a kite? 
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Subject:  Is an integer rational? (was RE: Is a rhombus a kite?) 
Author:  Alan Cooper 
Date:  Jun 6 2005 
> On May 20 2005, Mathman wrote:
> > No joking: There is no good reason to *exclude*
> > the rhombus any more than there is reason to
> > exclude the integers from the set of rational numbers,...
> Strictly speaking however, the integers are not a subset of the
> rationals  though they do map onetoone onto such a subset. A
> rational number is the set of all equivalent pairs of integers (with
> equivalence defined as (a,b) equ. (c, d) iff. ad = bc)
Well this makes me wonder what are *the* integers?
Surely their image in the rationals does satisfy all of the axioms required of
an instance of the integer number system.
In fact, as for any axiomatic system, even when the axioms define a structure
that is unique up to isomorphism there may be many distinct instances of that
structure. (There are at least a couple of models of the integers constructible
within ZermeloFrankel set theory and any of these could be used to generate a
model of the rationals by taking ordered pairs, but there are also other models
of both number systems that might be constructed from other choices of
"fundamental" system.)
To my mind, 'Mathman' is right, and if an embedding exists then it is correct to
say that one structure "is contained in" (or even "is a subset of") the
other.
Alan
P.S.
(back to Rhombuses and Kites
 well actually Squares and Rectangles)
gsw also wrote: > Kids always complain
> when you tell them that a square is a rectangle, and usually
> want to add an exclusive clause to the definition of rectangle when
> you point out to them that the normal definition includes squares.
Has anyone studied or considered why kids naturally tend to do this even though
they have no difficulty with the idea that a cat is an animal? Is it based on
something fundamental about how we perceive shapes, is it because there is only
one named subclass of rectangles, or is it just a result of how the material is
first presented to them?
 
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