From: kramsay@aol.com (KRamsay)
Subject: Re: Zermelo set theory
Date: 19 Jan 1999 06:16:38 GMT
Newsgroups: sci.math
Keywords: Non-ZFC models of Set Theory
In article <2gzp7gul54.fsf@hera.wku.edu>,
Allen Adler writes:
|Where can I find a construction of a model of Zermelo set theory
|which is not a model of Zermel-Frankel set theory?
I don't know a reference offhand, but the sets of rank ,
Allen Adler writes:
|Tal Kubo and Keith Ramsay have given the example of the set of
|all sets of rank less than $\omega+\omega$, in which it is alleged
|that one can't prove that $\omega+\omega$ is well ordered.
It's a common question of a formal system to ask what's the smallest
ordinal which it can't prove exists. First-order Peano arithemetic,
for instance, can't show epsilon_0 is well-ordered, and from epsilon_0
being well-ordered is enough to show PA is formally consistent. We
wouldn't want to leave the impression that Z is weaker than PA in this
regard, however!
The problem is that while one can prove in Z that there is a well
ordering of order type w+w (e.g. nRm iff n,
Allen Adler wrote:
@kramsay@aol.com (KRamsay) writes:
@
@> In article <2gyamz10sj.fsf@hera.wku.edu>,
@> Allen Adler writes:
@
@> |(1) Where can I find a detailed account of Zermelo set theory
@> | and its models?
@>
@> I don't know. What I've read of it has all been in discussions of why
@> one has the axiom of replacement in ZF.
@
@After thinking about it a little, it occurred to me that people who
@work with "subsystems of analysis" (something I know very little about)
@work with models of set theory with very restricted versions of the
@replacement axiom, e.g. the replacement axiom schema holds for formulas
@satisfying some restricted condition with respect to the Levy hierarchy,
@e.g. Delta_1^2 or something like that. So maybe they talk about Zermelo
@set theory somewhere? I just tried math-sci by looking under titles
@containing "subsystems of analysis" but didn't find anything that looked
@like it might discuss Zermelo set theory systematically. Maybe someone
@else can do better?
Subsystems of analysis restrict Comprehension (and Choice)... yes,
to Pi-1-1, Delta-1-2 or so. The other axioms (on top of PA) are Extensionality
and full (set-variable) Induction... all in all rather weak as a set theory!
(Which it wasn't meant to be; its aim is to study not "sets" but "sets of
reals".) Note that it's a two-sorted theory; "model" means an M (|= PA)
_and_ a choice of some family of subsets of M to interpret the set variables.
A typical "weakness" of Z is: it cannot prove P, P = "every
wellordering is isomorphic to an ordinal" (because P is false in V_w+w --
all wellorderings of w are there, including those of type w+w, but w+w
is not -- and V_w+w |= Z). On the other hand, poor Analysis cannot even
formulate the notion "ordinal"!
Anyway... I posted a more detailed discussion in sci.logic a couple
of months ago (thread subject had "second-order arithmetic" in it, I think).
@> |(2) Does there exist a model of Zermelo set theory which satisfies
@> | the axiom of choice but which is not a model of Zermelo-Frankel
@> | set theory?
@>
@> If the axiom of choice is true, then the sets of rank the axiom of choice. Without assuming the axiom of choice, take any
@> model of ZF which does satisfy the axiom of choice (like Goedel's L
@> of some model) and take the submodel of sets which have rank relative to that model.
@
@This looks neat, but let me make sure I understand it. Suppose one has
@an element A of L_{omega+omega} whose elements are pairwise disjoint.
@Let B be an element of L which meets each element of A in a singleton.
@Let C be the union of A. Since the union axiom holds in Zermelo set theory,
@C also belongs to L_{omega+omega}, hence to L_alpha for some
@alpha < omega+omega. Therefore every constructible subset of C, in
@particular B, belongs to L_{alpha+1}, hence to L_{omega+omega}.
@So L_{omega+omega} satisfies choice. Is this argument correct?
@
@Naively, it looks as though the same argument shows that if V is a model
@of ZFC, V_lambda is a model of Zermelo set theory with choice for every
@limit ordinal lambda greater than omega. Is that correct? Or does one
@have to make additional assumptions, such as that the cardinailty of
@lambda is a regular cardinal?
Lambda limit and > w is correct.
E.g. ZF (or any consistent extension thereof) is not finitely
axiomatizable over Z; an easy proof is to show that for any statement Q
consistent with Z, there is a Q' that is a theorem of ZF + Q but not of
Z + Q. Sure enough, Q' is "there is a limit ordinal lambda, > w, such
that Q^V_lambda" (Q relativized to V_lambda).
(Q' = "Consis(Z + Q)" would also do... but the above seems simpler).
Roughly speaking, PA (essentially, Z(F) - Inf) "is about" V_w, the
hereditarily finite sets; Analysis is about the hereditarily countable sets;
Z is about V_w+w.
Ilias