On THE TRANSITIVE HULL OF a Κ‐NARROW RELATION

Abstract

We will prove in Zermelo-Fraenkel set theory without axiom ofcboice that the transitive hull R' of a relation R is not much "bigger" than R itself. As a measure for the size of a relation we introduce the notion of 6-narrowness using surjective Hartogs numbers rather than the usul injective Hartogs values. T h e main theorem of this paper states that the transitive hull of a 6-narrow relation is *+-narrow. As a n immediate corollary we obtain that, for every infinite cardinal 6, the class HC6 of all 6-hereditary sets is a set with von Neumann rank e(HC6) 5 6 ' . Moreover, e(HC6) = I(+ if and only if I( is singular, otherwise Q(HCK) = 6. The statements of the corollary are well known in the presence of the axiom of choice (AC). To prove them without ACas carried through here is, however, much harder. A special case of the corollary (6 = W I , i.e., the class HCwl of all hereditarily countable sets) has been treated independently by T. JECH. 04A05, 03E20, 08A02, 08A65, 08BPO. Key words: Transitive hull of a relation, surjective Hartogs number, 6-hereditary sets, Peano algebras, infinitary languages. ~~~ ~~~ ')The main results of this paper were mentioned first in a talk ("Zur Konstruktion infinitiirer Sprachen L,,g ohne Auswahlaxiom") at the Mathematisches Forschungsinstitut Oberwolfach, West Germany, 25.-30.4.197T, and presented in detail -"Die Klasren Xis und HCs der Mengen von erblicher Kardinalzahl < s (in ZF ohne Auswahlaxiorn)"at the same place, 5.-11.4.1981.

On -hereditary Sets and Consequences of the Axiom of Choice

“…Thus, the present author has given up and adopted this notation, too. His attempt to avoid confusion by using the word "transitive hull" for R * (see e. g. [1]) has also not met much sympathy in the literature. So one just has to live with this terminological ambivalence hoping that "the meaning will be clear from the context"!…”

“…It is a theorem of ZF 0 that all classes H κ and HC κ are sets. Actually, one can prove much more (Diener [1]):…”

“…The proof of (3) ⇒ is clear. To cut the proof of (3) ⇐ short we will use a result of [1]: Assume that κ is regular. If x ∈ HC κ , then ∈ |Tc({x}) is * κ-narrow, and so is, by [1, Theorem 2.1], (∈ |Tc({x})) * = ∈ * |Tc({x}).…”

“…For any two classes A and B, A \ B denotes the set-theoretical difference. Given an arbitrary relation R, i. e., a class of ordered pairs a, b , R −1 is the converse relation, R | X is the restriction of R to class X, i. e., R | X := R ∩ (X × X), and R X is the domain-restriction of R to X. R[X] denotes the R-image of class X, fd(R) = dom(R) ∪ rng(R) is the field of R, and R • S is the relative product of relations R and S. For any two functions f and g, f · g denotes the composition, defined 1) e-mail: Servitor@mi.uni-koeln.de 2) For the reader's convenience we have tried to stick to Howard-Rubin's terminology as close as possible without giving up the terminology and notation of [1]. There is, however, a difference that should be mentioned here.…”

“…For convenience, we will introduce the following abbreviations: R e m a r k 1.7. In [1] we defined the notion of a surjective Hartogs value ℵ * (x) of an arbitrary set x as the least ordinal which is not an image of x. A relation was called (surjectively) κ-narrow if ℵ * (O R (x)) ≤ κ for every x.…”