[FOM] powerset {0} = {0,{0}} in intuitionistic set theory?
Peter LeFanu Lumsdaine
plumsdai at andrew.cmu.edu
Mon Oct 19 19:17:24 EDT 2009
> Hello,
>
> I've recently done some formalization of set theory. I wasn't sure
> whether I should use classical or intuitionistic ZF. So I decided to
> start with intuitionistic ZF and see how far I'd get.
>
> My axioms are essentially those of IZF at http://plato.stanford.edu/entries/set-theory-constructive/axioms-CZF-IZF.html
> . I introduce symbols for empty set, pair, union, powerset, and {x
> in A | phi(x)} after proving that those sets exist uniquely.
>
> Then, writing 0 for the empty set, I tried to prove
> (*) powerset {0} = {0,{0}}
> and was surprised to find that (*) seems to be equivalent to
> excluded middle.
>
> Intuitively, I feel that (*) has to be true. Now I wonder whether
> that makes me a classical mathematician or whether there are
> intuitionistic set theories in which (*) holds.
If your conception of set includes anything approaching the standard
separation axiom, then I'd say that your belief in (*) does indeed
make you a classical mathematician (or at least, classically-
inclined). :-)
Arguing informally, a separation axiom typically says "any nice
logical proposition can be used as a criterion for subset
membership". (*) says "membership is decidable for subsets of {0}".
Putting these together gives "every nice logical proposition is
decidable", which either makes you pretty classical or else severely
limits what kind of separation axiom you can accept.
A little more formally: Separation axiom schemes are typically of the
form
"for any set A, there is a set { x \in A | \phi(x) }"
for some class of formulas \phi(x). In the usual full separation
axiom (eg in IZF), all formulas \phi are allowed; in CZF and related
theories, it's restricted to e.g. bounded formulas, as described at
the Stanford Encyclopedia entry you link to.
If \phi is any nice proposition, then consider { x \in {0} | \phi }.
(*) implies that 0 is either in this subset or not in it, and hence
that \phi is either true or false. So if you accept (*), then any
formula with which you can invoke separation must be decidable.
This sounds pretty classical. It can be made more constructively
acceptable by turning it on its head: only assert separation for
formulas which are decidable! This axiom, "decidable separation", is
considered (though not at great length) in Moerdijk and Joyal's book
"Algebraic Set Theory"; I don't know where else, if anywhere, it's
discussed. If I recall correctly, it's extremely weak unless one has
further principles asserting decidability of some class of propositions.
Best,
-Peter.
--
Peter LeFanu Lumsdaine
Carnegie Mellon University
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