Toby Kenney wrote:
I think the following is a better proof:
Toby, thanks for that. Peter J. suggested a proof in a separate email to me, let me absorb and compare them.
Your suggestion of relaxing the condition to ts <= t seems to require changing from ordinary objects to partially-ordered objects. I think the initial one should be just the free monoid on two generators, with some partial order. You probably want to insist that s and t be inflationary.
Quite right, I forgot to specify that s and t were monotone (which implies inflationary), and I also left out that 0 is the empty sup. (According to J&M in "Algebraic Set Theory" 1995, specifying inflationary rather than monotone for the class as a whole gives the hereditarily transitive sets, i.e. the von Neumann ordinals. Monotone gives a larger class and a weaker notion of ordinal which nonetheless is inflationary as a consequence of monotonicity and initiality.)
If you´re considering partially-ordered objects, then this won´t be isomorphic to N^2. I´m not sure whether there will be any isomorphism in the topos, even without requiring it to preserve order. You could alternatively consider a condition like t=sts. I´m not really sure what you mean by your claim that t is the ordinal w - I think the object you describe isn´t even totally ordered - how can you compare t^2(s(0)) and t^2(0)? If you take a condition like t=sts to make it totally ordered, you still have a decreasing chain t > ts > ts^2 > ...
The conditions I forgot to mention, 0 <= x and t monotone, should give t^2(0) <= t^2(s(0)).
It looks like this should be something like the interval in N x Z, lexicographically ordered, going from (0,0) (so without the pairs (0,-n)). If this is still an attempt to construct the reals, then I don´t see any sensible way to describe addition on an infinite lexicographic product of copies of Z (the integers).
If both coordinates still have complemented subobjects, in particular if {0} x w and w x {0} are both complemented subobjects, then I should throw in the towel, consistent with Peter's advice. Are they? If w x {0} is not a complemented subobject then we're moving in the right direction: the next step is to define a suitable final coalgebra. Vaughan