Relations as primitive

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.

Marshall Received on Fri Jan 11 2008 - 13:47:40 CST