Bill Halchin asks
I don't understand how the notion of a subobject classifier determines whether "el" does or does not belong to the subobject
U ----------> 1 V _| | | | true | | V chi V X --------> Omega If any element "belongs to U" in the arrows sense that it factors through U, ie there is a map from 1 (or somewhere) to U, then the composite with chi:X->Omega is equal to (the composite with) true. Conversely, if an element of X "belongs to U" in the logical sense that its composite with chi is equal to (the composite with)true, then we have a commutative square that can be compared to the pullback. For more about the logical consequences of this property of Omega, and in particular the mysterious formula a & F(a) = a & F(true) see "Geometric and Higher Order Logic ..." in TAC, v7 (2000) pp 284--338. For a presentation of set theory (or, as I call it, Zermelo type theory) in a form that is both the way that ordinary mathematicians use it, and directly related to the topos-theoretic way of saying things, see Section 2.2 of my book "Practical Foundations of Mathematics". Both accessible via my home page at http://www.dcs.qmw.ac.uk/~pt Paul 20-Aug-2001 18:17:29 -0300,5651;000000000000-00000012