[FOM] Higher Order Set Theory [Ackermann Set Theory]
Robert M. Solovay
solovay at Math.Berkeley.EDU
Fri Mar 11 21:47:27 EST 2005
In a recent posting to FOM Joe Shipman writes:
OK, so Ackermann set theory is equiconsistent with ZF, but
what is the consistency strength of "there exists (V_\alpha,
V_\beta) where \alpha is inaccessible and V_\alpha is an
elementary substructure of V_\beta" ?
"There are more than continuum many inaccessible cardinals"
certainly works, because then at least two have the same
theory, and the corresponding ranks satisfy that the lower is
an elementary substructure of the higher.
In reply Nate Ackerman writes:
I believe it suffices to have a single inaccessible, although I don't
remember the proof off the top of my head (but I don't think it is that
hard).
Let us call the principle under discussion SP. [So SP asserts:
"there exists (V_\kappa, V_\beta) where \kappa is inaccessible and
V_\kappa is an elementary substructure of V_\beta"]
In fact the consistency strength of SP is quite a bit larger than
Shipman asserts. I will give fairly sharp upper and lower bounds later in
this letter [they are somewhat technical] but first here are some easily
stated results:
1) In ZFC + SP we can prove the existence of an inaccessible limit
of inaccessibles. {So Shipman's proof must have an error. I will point out
the troublesome step below.}
2) ZFC proves: If alpha is Mahlo, then V_alpha is a model of ZFC +
SP.
I will first prove 1) and 2). Then I will state [and then prove]
the more precise bounds previously alluded to. Then I will take up
Shipman's proof.
Proof of 1): We work in ZFC + SP. First, V_beta thinks there is
an inaccessible. So V_kappa thinks there is an inaccessible. So there is
an inaccessible less than kappa.
Suppose toward a contradiction that kappa is not the sup of the
inaccessibles less than kappa. Let eta be this sup. Then V_kappa thinks
that eta is the sup of the inaccessibles. But then so does V_beta. This is
absurd since kappa is an inaccessible greater than eta known to V_beta.
So kappa is an inaccessible limit of inaccessibles. This proves
1).
We turn to 2). Let alpha be a Mahlo cardinal. Then the set of
gamma < alpha such that V_gamma is an elementary submodel of V_alpha is a
club in alpha. So the set, A, of inaccessible cardinals gamma such that
V_gamma is a proper elementary submodel of V_alpha is stationary in alpha.
Let kappa be the least member of A and beta the second member. Then kappa
and beta instantiate the fact that SP holds in V_alpha. Claim 2) is now
clear.
To sharpen 1) I need to introduce the theory ZM. Roughly ZF:
inaccessible = ZM: Mahlo.
Precisely, ZM will be obtained by adding the following scheme of
axioms to ZFC [one axiom for each n in omega]:
The class of all ordinals alpha such that [(a) alpha is
inaccessible and (b) V_alpha is a Sigma_n elementary submodel of V] is
unbounded in the class OR of all ordinals.
The sharpened form of 1) asserts: Let kappa, beta instantiate SP.
Then V_kappa is a model of ZM.
Let's prove this. Let kappa, beta instantiate SP. Of course,
V_kappa is a model of ZFC. Towards a contradiction let the new axiom with
index n fail in V_kappa. Then certainly, V_beta thinks that kappa is
inaccessible and a Sigma_n elementary submodel of V. So V_kappa thinks
there are such alpha. Let eta be a sup of the set of alpha < kappa which
are inaccessible and give Sigma_n elementary submodels V_alpha. Since the
axiom fails in V_kappa, eta < kappa. But then eta has the same property in
V_beta. This is absurd since kappa is > eta and is inaccessible and with
V_kappa a Sigma_n elementary submodel of V_beta.
For the sharpened form of 2) I need to introduce a slight
strengthening of ZM. Introduce a new predicate Sat. Add the [finitely
many] Tarski axioms which says that Sat codes up the satisfaction relation
for the language of set-theroy. In the various schemes of ZM allow the
predicate Sat to appear. {In particular for the new scheme asserting the
existence of many inaccessilbes interpret "Sigma_n elementary submodel" to
refer to the language with Sat added.}
Call this enlarged theory Sem(ZM). Then there is no difficulty
adapting our proof of 2) to show: Sem(ZM) proves SP.
Finally, the false step in Shipman's discussion is this: Suppose
that kappa < gamma are inaccessibles such that V_kappa and V_gamma have
the same first order theories. Then there is no reason to suppose that
V_kappa is an elementary submodel of V_gamma since the notion of
elementary submodel refers to sentences with parameters chosen from
V_kappa.
--Bob Solovay
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