modeling Zermelo set theory at a countable stage of the constructible universe
- From: david.madore@xxxxxx (David Madore)
- Date: 19 Apr 2008 20:29:42 -0400
In what follows, I will attempt to prove the following fact in ZFC,
where P is an arbitrary statement (in the language of set theory):
If P holds in L (the constructible universe), then there
exists a countable ordinal \delta such that L_\delta (the
constructibles up to stage \delta) is a model of ZC (Zermelo
set theory, i.e., ZFC minus the Replacement scheme but with
full Separation) + P.
My (meta?-)questions are (1)whether my proof is correct,
(2)whether/how it can be simplified, (3)whether the way I'm doing it
is stupid and (4)to what extent the "naïve attempts" below are,
indeed, naïve (or whether they can be made to work). Besides these
meta-questions, I also ask a real (but probably stupid :-) question at
the bottom.
Before the proof which (I hope!) works, here are a couple of naïve
attempts:
Naïve attempt number 1: Apply some kind of pre-cooked reflection
principle for L (of the form "if L satsifies P then for some countable
\delta does L_\delta satisfy P", i.e., the same statement as above but
without ZC). This fails because ZC is not a single statement. And I
fail to see how a completely straightforward Löwenheim-Skolem argument
can work.
Naïve attempt number 2: Find a countable transitive model M of ZC+P,
and look at L^M. This fails because ZC does not prove that L
satisfies ZC. Nor is it clear that the "L satisfies ZC" scheme is
implied by ZC plus a finite number of theorems of ZFC; apparently the
latter *is* true, though, if I read correctly a theorem by
A. R. D. Mathias in his paper entitled "The Strength of Mac Lane Set
Theory", *Annals of Pure and Applied Logic*, 110 (2001) 107-234 (see
theorem 5.27 in <URL: http://www.dpmms.cam.ac.uk/~ardm/maclane.pdf >),
but I'm not at all sure how far this is relevant (that's part of my
meta-questions, I guess).
Now here's a proof which I believe works:
Let S be the theorem of ZFC (but not of ZC!) which is the conjunction
of the following three statements: (i)the Gödel operations of any two
sets exists, (ii)every set admits a closure under the Gödel operations
and (iii)for every ordinal \alpha, the (L_\beta) indexed by
\beta<\alpha exists. (See Jech, *Set Theory*, either (13.9) and lemma
13.3 in the old edition, or (13.12) in the new "third millennium"
edition.) Call a transitive set M satisfying S "adequate". The
essential property (Jech, *loc. cit.*) is that if M is transitive and
adequate and satisfies V=L then M is some L_\delta.
(We work in ZFC.) Assume P holds in L. Now, in L, the following
holds by reflection: there exists some \gamma such that V_\gamma
satisfies S + V=L + P + all the axioms of ZFC except the Separation
and Replacement schemes; but V_\gamma automatically satisfies full
Separation, so it satisfies ZC + S + V=L + P. Thus, in L there exists
some transitive set satisfying ZC + S + V=L + P, but then that set
exists in the universe (satisfying <...> is absolute). By
Löwenheim-Skolem, let E be a countable elementary submodel of it, and
let M be the transitive (Mostowski) collapse of E. Then M is
transitive and adequate (because it satisfies S) and countable and is
a model of ZC + V=L + P. So it is an L_\delta with \delta countable.
QED.
Finally, here's a real question:
What can we say (in ZFC) about the order the following ordinals (for m
and n natural numbers):
* the smallest ordinal \delta such that L_\delta satisfies ZC +
\Sigma_n Replacement
* the smallest ordinal \delta such that L_\delta satisfies ZC limited
to \Sigma_m Separation + \Sigma_n Replacement (with m>=n).
.
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