# [FOM] ontology

Carlos Gonçalves cgon1 at iscte.pt
Sun Jan 18 12:24:05 EST 2004

Hi Thomas,

Concerning isomorphism and identity. Consider the set of natural numbers
and denote it by N and consider the set of even numbers, denote it by E,
and the set of odd numbers, denote it by O.

Define  g = {(1,2),(3,4),(5,6)...}, then g is a bijection and it defines an
isomorphism between O and E, in what regards the relation <.

The relational structures for the sets E and O defined by the relation <
are isomorphic, however, the set E and O are different and ordered pairs
that are in each of the relations are not the same, therefore, we have that
these two structural relations are different particulars, hence, not identical.

Best Regards,

C. Pedro

At 10:14 16-01-2004 +0000, you wrote:

>My feeling is that Carlos's answer is ``obviously'' correct.  But there
>is a cost to this.  It flies in the face of the idea that identity of
>mathematical entities is isomorphism.  That's why i raised this example,
>as it forces us to examine this principle (that identity = isomorphims)
>more closely.
>
>       Thomas
>
>
>On
>Thu, 15 Jan 2004, Carlos Gonçalves wrote:
>
> > At 18:14 14-01-2004 +0000, you wrote:
> >
> > >An essay question:
> > >
> > >
> > >Are the finite ordinals the same mathematical objects as the finite
> > >cardinals?  Give reasons...                       [\aleph_0 marks]
> >
> > Hi Thomas,
> >
> > These are different objects, the notion of ordinal, in particular, appeals
> > to a notion of order (and of well ordered set) while a cardinal is,
> > following Cantor, obtained through an operation of double abstraction,
> both
> > from the order in which the elements of the set are given and from the
> > nature itself of those elements.
> >
> > C. Pedro
> >
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> >