Many years ago, Takeuti constructed a first order theory of ordinal numbers which we shall denote by ORD(T) and proved that ZF+(V=L) could be interpreted in ORD(T). Let F(x) designate the unique set correlated to the ordinal number x by the specific well ordering of the universe of ZF that can be defined when V=L is added to the axioms of ZF. Takeuti showed that a certain formula S(x,y) belonging to the language of ORD(T)-in which y,z are the only variables occuring free-expresses the fact that "the set F(y) is an element of the set F(z) in ZF". This was the basis for the interpretation and it was interesting because Takeuti's axioms for ORD(T) made ORD(T) an extension of Peano's arithmetic (with the finite ordinal numbers playing the role of non-negative integers). It would then be possible to regard ZF+(V=L) as a sub-theory of (a suitably extended) arithmetic, much as arithmetic has long been rgarded as a sub-theory of ZF.

Now, if we tried to "embed" a richer set theory such as "ZF+some large cardinal axioms" into an extension of arithmetic by proceeding in this way, we would run into the problem that V=L is inconsistent with many of these large cardinal axioms. However fewer of these axioms may be inconsistent with V=OD. Let G(x) designate the unique set correlated to the ordinal number x by the specific well-ordering of the universe of ZF* that can be defined when V=OD is added to the axioms of ZF* (where ZF* denotes ZF+some large cardinal axioms).

My question is this: Could there exist (or has there been constructed) a first order theory ORD(OD) such that (1) ORD(OD) is a theory of ordinal numbers that extends Peano's arithmetic (2) there is a formula of ORD(OD)-in which y,z are the only variables occuring free- that expresses the sentence " the set G(y) is an element of the set G(z) in ZF*"?

If so, we could interpret ZF*+(V=OD) in ORD(OD). The language of these first order theories of ordinal numbers should be thought of as containing terms and formulae representing recursive functions and predicates of ordinal numbers. Among these are the the recursive functions and predicates of finite ordinal numbers (non-negative integers) which belong to the language of the sub-theory-Peano's arithmetic. This is what is meant by saying that these theories of ordinal numbers are extensions of arithmetic.

  • $\begingroup$ How does the Takeuti theory compare with the theory SO, the theory of sets of ordinals, introduced by Koepke and Koerwien, as in section 5 of this paper: math.uni-bonn.de/people/koepke/Preprints/…? $\endgroup$ – Joel David Hamkins Jun 14 '13 at 0:40
  • $\begingroup$ @Garabed, I added some paragraph breaks to the question - I hope you don't mind. $\endgroup$ – Noah Schweber Jun 14 '13 at 2:13
  • $\begingroup$ It seems as though the theory SO should be interpretable in Takeuti's original theory ORD(T) and vice-versa. It is not clear to me whether Koepke and Korwien have also proved that a theory such as ORD(OD) cannot exist even if our theories of ordinal numbers are allowed to contain terms representing arithmetical or analytical predicates of ordinal numbers. The predicates represented by terms in SO-like those in ORD(T)-appear to be limited to just the ones which are recursive. $\endgroup$ – Garabed Gulbenkian Jun 15 '13 at 17:44
  • $\begingroup$ Thanks, Noah, for making my text alot more readable. This is just what I wanted to do but did not know how to do. $\endgroup$ – Garabed Gulbenkian Jun 15 '13 at 17:49
  • $\begingroup$ My apologies for a mistake. The predicates of ordinal numbers that I called "analytical" are actually called "hyperarithmetical" in the literaure. $\endgroup$ – Garabed Gulbenkian Jun 15 '13 at 22:54

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