[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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