[FOM] CH and mathematics

Colin McLarty colin.mclarty at case.edu
Tue Feb 5 19:53:11 EST 2008

Thanks to both James Hirschorn for correcting some sloppiness, and
Thomas Forster for correspondence.  To be clear, when we say 

> "structure" in the formal sense (i.e. a triple (A,S,I),
> where A is the domain, S the signature and I the 
> interpretation)

the signature can be higher order.  I.e. it can involve power types. It
can also involve quotient types.  As Hirschorn puts it, then

> Theorem (ZF - Foundation + AC). Every structure is isomorphic 
> to a structure in WF.

As pointers to the proof I will respond on list to Forster (after very
helpful off-list correspondence).  First, ZF-Foundation already proves
every subset or powerset or quotient of a well-founded set is
well-founded. And so every function between well-founded sets itself has
a well-founded graph.  Actually, this depend on not using some perverse,
non-well-founded definition of ordered pair, but so be it.

That is the step that fails radically for constructible sets in place of
well founded.  

Then, second, with the axiom of choice every set is isomorphic to a
well-founded set.  (The analog of this second step of course *does* hold
for constructible sets.)

As Hirschorn says this is categorical thinking: Altogether (provably in
Zermelo set theory plus choice) the category of WF sets and functions is
*equivalent* in the technical categorical sense to the category of all
sets and functions in V.  

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