Since Gilmore showed that some theory with a positive comprehension scheme is consistent when the axiom of extensionality is dropped and inconsistent with it (see [1] and [2]), the problem of the consistency of various positive comprehension schemes has been investigated. We give here a short classification, which shows clearly the importance of the axiom of extensionality and of the abstraction operator in these consistency problems. The most difficult problem was to show the consistency of the comprehension scheme for positive formulas, with extensionality but without abstraction operator. In his unpublished thesis, Set theory in which the axiom of foundation fails [3], Malitz solved partially this problem but he needed to assume the existence of some unusual kind of large cardinal; as his original construction is very interesting and his thesis is unpublished, we give a short summary of it. M. Forti solved the problem completely by working in ZF with a free-construction principle (sometimes called an anti-foundation axiom), instead of ZF with the axiom of foundation, as Malitz did.This permits one to obtain the consistency of this positive theory, relative to ZF. In his general investigations about “topological set theories” (to be published), E. Weydert has independently proved the same result. The authors are grateful to the Mathematisches Forshungsinstitut Oberwolfach for giving them the opportunity of discussing these subjects and meeting E. Weydert during the meeting “New Foundations”, March 1–7, 1987.
Abstract. The notion of "ramifiability" (or "tree-property"), usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar "large cardinal" properties.Mathematics Subject Classification: 03E55, 04A10, 06A99.
Two distinct and apparently "dual" traditions of non-classical logic, three-valued logic and paraconsistent logic, are considered here and a unified presentation of "easy-to-handle" versions of these logics is given, in which full naive set theory, i.e. Frege's comprehension principle + extensionality, is not absurd.
This is a study of the extension, to directed sets, of the well-known notion of ramifiability (also called tree-property), usually applied to cardinals. This very natural extension plays a crucial role in some specific types of Cauchy-completions.Mathematics Subject Classification: 04A10, 03C55, 03355, 06A99.The initial interest for ramifiable directed sets comes from investigations about some specific Cauchy-completions, called Malitz-completions, which are explained in detail in [4, 51; the use of ramifiable cardinals in similar situations can however be found in earlier work, e.g. in [3]. We will study ramifiability for directed sets here in detail and put it in relation with important parameters and properties of directed sets. We will also recall the minimal necessary background about directed sets and Malitzfamilies, in order to make this paper as self-contained as possible, allowing ourselves to omit proofs that are straightforward or present in [4]. Our metatheory is ZFC (Zermelo-Fraenkel with axiom of choice), the ordinals and cardinals are the classical ones (see e. g. [l]), 1x1 will denote the cardinal of the set X and v+ will denote the successor cardinal of the cardinal v. A set X is called v-finite when 1x1 < v . Directed setsA directed set is an ordered set (0,s) such that each finite subset X has an upper bound (i.e. (3 E D)(Vz E X ) z 5 d ) ; inter alia, D # 8. Directed sets are used in topology (see [S]).For Y c D , we define: ( Y ] , := { d E D : ( 3 y E Y ) d < y } , [ Y ) , := { d E D : ( 3 y € Y ) y < d } .We will often just write ( Y ] instead of ( Y ] , , etc. We write also ( d ] for ( { d } ] . A subset Y C D is cofinal zn D iff ( Y ] , = D , and we denote by COFD the set of the cofinal subsets of D. Ramifiable Directed Sets 217From here on, D will always denote a non-trivial directed set, i.e. a directed set A function z : D -A will be called a D-net in A , and we adopt the notation A basic chain in D is an order-preserving a-net in D (for some ordinal a seen as a directed set), which image has no upper bound in D; in other words, it is a function z : a -D such that z = ( z p ) o C a , , D 5 0' iff zo 5 z p , and { x p : p < a } has no upper bound in D. We say that a is the order-type of x.The characteristic of D is the parameter 6~ (already introduced in [4]), for which we can give several (equivalent) definitions: without maximum element. L = ( Z d ) d E D , where zd := X(d).50 = the greatest cardinal 6 such that D is 6-directed (i. e. each &finite X
It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a well-known set theory based on an extended positive comprehension scheme.
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