0
$\begingroup$

Nearly fifty years ago Takeuti called attention to a phenomenon that occurs in connection with the construction of set theories such as ZF that result in a hierarchy of sets (indexed by ordinal numbers). The phenomenon is that the collection of theories (regarded as sets of sentences belonging to a formalized first order language) is much smaller than the collection of sets (belonging to the universe of ZF) which could serve as models for these theories.

More precisely, let L denote a classical first order language in which ZF is formalized. The only "non-logical" symbols that occur in ZF are the symbol for "membership" and the symbol for "equality". For each ordinal number z, let R(z) denote the set of all sets of rank z belonging to the universe of (some extension of) ZF. Assume that this extension of ZF is consistent, so that each particular set R(z) either is or is not a model of any particular sentence of L. The collection of sentences of L is denumerably infinite. Each consistent theory formalizable in L can be identified with the collecion of its theorems which are sentences of L. Clearly, the cardinal number of the collection of all such consistent theories cannot exceed the cardinal number of the continuum.

Consequently there exists at least one (consistent) theory T, formalizable in L, and an unbounded collection Q of ordinal numbers such that y belongs to Q if and only if the set R(y) is a model of the theory T. Takeuti called such theories as T "absolute" set theories and believed that only one could exist. That one would be the "ultimate" or "best possible" set theory, in which all the "meaningful" questions that could be asked about sets would be answered.

My questions are:

  • Have there been any further investigations of these so-called "absolute" set theories and if so, have these yielded any interesting mathematical theorems?
  • In particular, is it known whether more than one of them can or must exist?
$\endgroup$
  • 7
    $\begingroup$ There must be something missing. Clearly there are infinitely many such theories. For each $n \in \omega$ consider the statement, "There is a largest limit ordinal $\alpha$ and exactly $n$ ordinals above it." $\endgroup$ – Monroe Eskew Jul 6 '14 at 20:33
  • 3
    $\begingroup$ It's unclear what you mean by 'ordinal number $z$'. If you mean an object $z$ in a model $V$ of ZF such that $V \vDash$ '$z$ is an ordinal', then any extension of ZF is "absolute". However, it sounds that you mean something else. $\endgroup$ – François G. Dorais Jul 6 '14 at 20:37
  • 3
    $\begingroup$ I've never heard of this, and googling 'Takeuti "absolute theories"' yields no hits; could you give a citation? $\endgroup$ – Noah Schweber Jul 6 '14 at 21:14
  • 1
    $\begingroup$ "Absolute" set theory (or theories) are defined and discussed on pages 79-80 of the book "Foundations of Mathematics: Symposium Papers Commemorating the Sixtieth Birthday of Kurt Godel" which was published in 1969 by Springer Verlag. The article by Takeuti entitled "The Universe of Set Theory" appears on pages 74-128 of this book. After defining "absolute" set theories, Takeuti does not say much more about them in this article. That is why I am asking whether there has been any further work in tis area. $\endgroup$ – Garabed Gulbenkian Jul 8 '14 at 12:51
4
$\begingroup$

There is indeed a crucial point missing in your description. Takeuti's idea is described in his article as follows:

"Let us fix a language of set theory e.g. the first order language with one constant $\epsilon$. Let $\mathfrak{T}_\alpha$ be the set theory of $R(\alpha)$ w.r.t. this language. Since the creation of ordinals is endless and the cardinality of the set of all possible theories in our fixed language is bounded, some theory must appear endlessly many times among [the] $\mathfrak{T}_\alpha$s. We believe that only one theory will appear overwhelmingly densely among them. We wish to define the absolute set theory to be this theory."

(pg. 80, emphasis added)

The emphasized part is the whole piece that makes this meaningful, since as noted in the comments, plenty of theories will occur unboundedly often. Now, what does Takeuti mean by "overwhelmingly densely?"$^*$ This is the crucial point!

Note, by the way, that everything here takes place within a fixed model. So once we specify some appropriate and reasonably definable notion of "overwhelming density," each model will have an object it thinks of as "the absolute theory." In particular, there is no a priori reason to expect sentences of the fomr "$\varphi$ is in the absolute theory" to be provable in ZF(C), even after we've managed to define a notion of "absolute theory."

One common notion of "overwhelmingly dense" is club (closed unbounded): a class of ordinals $\mathcal{C}\subseteq ON$ is club if it is unbounded and for every $\alpha$, if $sup(\mathcal{C}\cap\alpha)=\alpha$ then $\alpha\in\mathcal{C}$. Now it is certainly true that at most one theory will occur club frequently in this sense. However, it is not obvious to me that there need be a theory with this property! For example, in ZFC any uncountable regular $\kappa$ can be partitioned into $\kappa$-many disjoint stationary sets, none of which can contain a club (since their complements are each stationary). So if we want a universe with choice in the background, it seems extremely odd to assume that partitioning the ordinals into continuum-many pieces will yield a club-containing piece (and in fact I suspect this is simply provably false, but I don't have time to work it out right now).

However, perhaps it is consistent with $ZF+DC$ that on any uncountable ordinal with uncountable cofinality, the filter generated by club sets is an ultrafilter ($ZF+AD$ does imply that the club filter on $\omega_1$ is an ultrafilter, but I don't believe it implies this for arbitrary ordinals); more generally, it may be consistent with $ZF+DC$ that any class partition of the universe into set-many pieces has a club-containing piece. And, of course, there's the fact that we're not partitioning the universe in some random way, so maybe some appropriate axioms about reflection properties would lead to the theory of $V$ itself being $V$'s absolute theory.

So I think where this leaves your question is:

  • The notion of "absolute set theory" depends on a notion of "overwhelming density," and it is not at all clear what this should be, especially if we want different models to have at least some agreement on their absolute theories.

  • This means that the question of whether such theories can exist is too vague. The simplest precise form of this question would be for "overwhelmingly dense" to mean "contains a club," in which case I suspect the answer is that it is consistent to have such a theory around a model of ZF, but I don't know how to prove it.

  • As to the question of whether people continued to look at these, I suspect the answer is somewhat yes, if indirectly, but I don't know. Certainly I've never heard someone explicitly talking about this idea, but on the other hand it seems closely related both to reflection and to inner model theory (e.g., $0^\#$ can be thought of as "the absolute theory of $L$ in $V$" if it exists).


$^*$ Takeuti actually does go on to talk about what "overwhelmingly dense" could mean; but I haven't had time to more than skim the paper, so I don't want to say something wrong about what he does.


Let me add, unrelatedly, that I think the ideas developed in this article in the first few pages, on the complexity of the continuum function, etc., are quite interesting! Basically, Takeuti asks "what is the (ordinal-)computability-theoretic complexity of the function $f: \alpha\mapsto \beta\iff 2^{\aleph_\alpha}=\aleph_\beta$?" (and related questions). I don't know how much this sort of thing has been pursued, since most of the ordinal computability I'm aware of takes place within $L$, but I'd be very interested to hear about it.

$\endgroup$
  • 1
    $\begingroup$ It's not consistent with ZF+DC that every partition of Ord into a set of (definable) classes has a club in one of the pieces. The reason is that DC (or even countable choice) implies that $\omega_1$ is regular, and so every club will contain ordinals of cofinality $\omega$ and ordinals of cofinality $\omega_1$. This doesn't quite kill the idea of taking "overwhelmingly dense" to mean containing a club. That's because "$z$ has cofinality $\omega$" might not be expressible as a first-order property in $R(z)$. $\endgroup$ – Andreas Blass Aug 2 '14 at 2:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.