[FOM] Intuitionism and 0=1

Vaughan Pratt pratt at cs.stanford.edu
Thu Nov 8 13:34:42 EST 2007

Presumably it depends on how broadly one construes analysis.  Abelian 
groups nicely generalize arithmetic addition, while abelian categories 
do the same with order replaced by homset (instead of a simple yes-or-no 
answer to whether x <= y one obtains the group of homomorphisms from x 
to y).

For the former, one could say that Pontrjagin duality is predicated on 
0=1 as the defining characteristic of the circle group T (think of the 
real line rolled up into a circle of circumference 1 so as to make T the 
additive group of reals mod 1, aka the multiplicative group of complex 
numbers on the unit circle but I'll stick to the additive viewpoint 
below).  This is the duality of topological abelian groups obtained by 
taking the dualizing object as the circle group and restricting to 
locally compact groups.  (For topological groups it is essential that 
the group homomorphisms be continuous in order that
T -o T, the dual of T,  be no more than Z.)

For the latter, Peter Freyd in a posting some years ago to the 
categories mailing list abstracted the common essence of abelian 
categories and toposes to define a class AT of categories such that the 
abelian categories are those satisfying 0=1 (equivalently, the strongly 
connected AT categories) while the toposes are those for which 0 is 
strict (the only morphisms to 0 are isomorphisms; in an abelian category 
every object has a (unique) map to 0).

While one might not be accustomed to thinking of a category as an 
arithmetic, note that the cartesian closed category FinSet of finite 
sets and their functions can be understood as, and indeed is sometimes 
spoken of as, the natural numbers under addition, multiplication, and 
exponentiation, there being only one set of each cardinality up to 
isomorphism.  Thus we have on the one hand natural number arithmetic in 
which 0=1 would collapse the AT category FinSet (or any other topos) to 
a category equivalent to the category 1 (all homsets singletons, the 
categorical counterpart of inconsistency), and on the other the 
corresponding arithmetic of abelian categories obtained by adding the 
postulate 0=1 to the postulates for AT categories.

The latter phylogeny applied to Ab (the abelian category of abelian 
groups), or to locally compact abelian groups if that turns out to make 
for a smoother arithmetic (my crystal ball went cloudy here), is then 
recapitulated in the ontogeny of T, and furthermore in all its 
subgroups, in all of which 0=1.

(The finite subgroups of T are just the finite cyclic groups Z_n.  When 
Z_n is obtained as the quotient Z/nZ it is natural to consider it to be 
generated by 1, so that 1 + ... + 1 = n = 0, but when it is obtained as 
the subgroup of T of order n, i.e. n points distributed uniformly around 
  T viewed as a circle, it is more natural to consider it to be 
generated by 1/n, so that 1/n + ... + 1/n = 1 = 0, a perspective from 
which 0=1 not only in T but in every Z_n *qua* subgroup of T if not 
*qua* quotient of Z, bearing in mind that Pontrjagin duality makes Z and 
T mutual duals and that subobject and quotient object are likewise 
mutually dual concepts.)


Robert Black wrote:
> In Heyting arithmetic we can define absurdity as 0=1 and then get ex 
> falso sequitur quodlibet for free, since every sentence of the 
> language follows from 0=1 and the axioms of HA just using minimal 
> logic. (Yes, Neil, I know you don't approve of doing this, but you're 
> not going to deny the formal result ...)
> My question is: does this still hold once we go beyond arithmetic, in 
> particular, does it hold for systems of formalized intuitionistic 
> analysis? Dummett says in his book (p. 13) that it 'might not be so 
> obvious', but is it true?
> Robert
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