[FOM] Intermediate value theorem (and ASD)]]

Vaughan Pratt pratt at cs.stanford.edu
Fri May 22 01:55:29 EDT 2009

On 5/20/2009 2:48 PM, Arnon Avron wrote:
> On Sat, May 16, 2009 at 03:07:33PM -0000, Paul Taylor wrote:
>> ASD, a calculus that DOES NOT USE THE POWERSET, or any sets
> So it uses something else, which is no less problematic
> (like A->B or A^B or whatever). Nobody can do
> what cannot be done.

Stone can.

Assuming X is a set, what structure do you impute to 2^X?  If the 
structure of a set, then 2^2^X will be yet bigger.  But if instead you 
organize 2^X as a complete atomic Boolean algebra or CABA, and if 2^2^X 
consists of the complete ultrafilters on 2^X, then 2^2^X ordered by 
inclusion is a discrete poset whose underlying set is isomorphic to 
(i.e. in bijection with) the set X.

If X is a complete semilattice, so are 2^X, 2^2^X, etc. where 2^X 
denotes the set of complete semilattice homomorphisms to the two-element 
chain (qua semilattice), organized as a complete semilattice.  All the 
even entries in this list are isomorphic to each other, as are all the 
odd entries.

If X is a finite-dimensional vector space over the field K, then so are 
K^X, K^K^X, and so on, and again these are isomorphic in the same way.

These are just a few instances of the very rich subject of Stone (more 
generally Pontrjagin etc.) duality.  By taking into account the 
structure created and/or destroyed in the process of forming K^X where K 
is the pertinent dualizing object, often but by no means always 2 (it is 
the additive group T of reals mod 1 in the case of locally compact 
abelian groups, the case of Pontrjagin duality), one can avoid at least 
some of the problems caused by neglecting that structure when forming 
power sets.

Chu spaces constitute a framework embedding all of these dualities and 
far more.

The above (including Chu spaces) is entirely for concrete Stone duality. 
  Paul or Andrej will need to explain which parts are retained in the 
abstraction to ASD as I have yet to see a list of axioms for ASD that my 
20th century thought patterns are capable of parsing.  I'm certain 
however that it must avoid the tendency of powerset to grow an unbounded 
tower of exponentials, without which the whole point of Stone duality 
would be lost!

Vaughan Pratt

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