Re: Relations as primitive

On Jan 11, 12:21 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:
> Marshall wrote:
> > On Jan 11, 12:54 am, David BL <davi..._at_iinet.net.au> wrote:
>
> >>[...]
>
> > The mathematical idea of functions is the set of ordered
> > pairs, where the left item in the pair is a member of the
> > domain and the right item is a member of the range.
> > The domain can then be considered also to be ordered
> > pairs, so that + then looks like
>
> > {((1,1),2), ((2,2),4) ... }
>
> > And so on. This is a consequence of the foundational
> > approach in which sets are the basic building blocks.
> > (And ordered pairs are defined in terms of sets, via
> > Kuratowski or whatever.)
>
> > However there is another way to go, which I first
> > encountered in ch. 4 of TTM, and which I don't see
> > discussed much or at all in mathematical circles,
> > and which I find immensely appealing. The approach
> > is to work directly with relations. So + looks like
>
> > {(x=1, y=1, z=2), (x=2, y=2, z=4) ...}
>
> > In point of fact, there is almost nothing of significance
> > to this idea; it's equivalent to the usual formulation.
> > But I nonetheless find the idea appealing of thinking
> > about the foundations of mathematics in terms of
> > taking relations as primitive rather than sets.
>
> Relations are sets.

Sure.

Relations are a specific kind of set. The usual foundational approach is to take (general) sets as primitive. However we could instead take a more specific kind of set as primitive, namely relations.

Marshall Received on Sat Jan 12 2008 - 00:31:56 CET