Vaughan, Your quote runs two of my paragraphs together. .Certainly the point about N does not need Isbell's argument. Colin On Thu, Jun 6, 2013 at 6:00 PM, Vaughan Pratt <pratt@cs.stanford.edu> wrote:
On 5/25/2013 8:47 AM, Colin McLarty wrote:
Of course is also follows that NxN=N. But it does not follow, and in fact it is refutable, that the projection functions are the identity function 1_N. Isbell's argument is on p. 164 of my copy of CfWM (1998).
Why do you need Isbell's long argument, or even any monoidal structure on Set, to obtain a contradiction here? Just use that NxN is a product and observe that the pair (3,4) in NxN (as a map from 1 to NxN) would have to be both 3 and 4 (as maps from 1 to N) when the projections are the identity.
The inconsistency found by Isbell works even when the projections seem quite reasonable, namely when taken to be the three projections from the ternary product XxYxZ whose elements are triples (x,y,z) (as the necessary meaning of identity of associativity a: Xx(YxZ) --> (XxY)xZ). One can in fact consistently equip Skel(FinSet) with such structure. Isbell shows that extending this to infinite sets breaks down, namely by creating additional equations not encountered with finite sets due to interference between binary and ternary product resulting from the identification of N with NxN.
One can get close to a skeleton of Set using tau-categories as per section 1.493 of Cats & Alligators; N becomes the ordinal omega, whose square is a distinct object albeit still isomorphic to omega. The full subcategory of finite ordinals is (isomorphic to) Skel(FinSet).
Vaughan Pratt
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