[FOM] V = WF costs nothing
Colin McLarty
colin.mclarty at case.edu
Wed Feb 6 17:30:00 EST 2008
>From Thomas Forster <T.Forster at dpmms.cam.ac.uk> Date Mon, 4 Feb
2008 22:53:59 +0000 (GMT)
wrote:
> The point is that there might be facts about
> large collections that cannot be presented (``spun'')
> as facts about wellfounded sets. Mightn't there?
Well certainly yes. But none can concern the existence of
mathematical structures up to isomorphism.
Hirschorn mentioned how Kunen already pointed out in his textbook
that the theory (ZF - Foundation + AC) proves every group is
isomorphic to a well-founded group, and every topological space is
isomorphic to a well-founded topological space. Of course Kunen
just meant these as typical examples, but he did not bother to say
typical *of what*.
Indeed that theory proves every structure definable in Bourbaki's
entire (higher order) theory of structures is isomorphic to a
well-founded one. And we can extend Bourbaki's theory by allowing
quotient types as well, and in many other ways, without losing this fact.
Personally I favor a categorical statement of the full generality:
(ZF - Foundation + AC) proves that the category of all sets and
functions is equivalent to the category of well-founded sets and
functions. This implies the claim about Bourbaki structures and
goes far beyond it. I think this is the *right* way to explicate
the fullest sense of "same structures" but I do not plan to lay that
out here.
Anyway, essentially all the claims about mathematical structures
that mathematicians normally entertain fit into this framework, and
concern structure up to isomorphism, so that the Well Founding axiom
makes no difference to them.
Notice this includes many paradigmatically set theoretic claims:
The standard form of the Continuum Hypothesis concerns the continuum
up to isomorphism (all of its subsets are finite, countable, or
equipollent with it). Measurability (as in measurable cardinal) is
an isomorphism invariant property (existence of a non-principle
ultrafilter complete to the cardinality of the set).
Well-founding makes no difference to any of those. Again, for
contrast, V=L does make a difference, most famously precluding
measurable sets in that sense.
> Church gave us a consistent set
> theory which says there is a universal set.
But this theory makes other changes as well. It restricts other
means of set formation. So it does not bear centrally on the
question of whether or not to adjoin V=WF to Zermelo set theory.
> Don't get me wrong: I'm not knocking the idea
> that everything can be encoded in Th(WF).
And I am not saying everything can be. I am saying that all claims
about structures up to isomorphism (which are the bread and butter
of mathematics) are provably decided in exactly the same way by (all
consistent extensions of) Zermelo set theory plus choice on one hand and
Zermelo plus choice and V=WF on the other.
best, Colin
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