Marco Forti

An Aristotelian notion of size

Abstract The naive idea of “size” for collections seems to obey both Aristotle’s Principle: “the whole is greater than its parts” and Cantor’s Principle: “1-to-1 correspondences preserve size”. Notoriously, Aristotle’s and Cantor’s principles are incompatible for infinite collections. Cantor’s theory of cardinalities weakens the former principle to “the part is not greater than the whole”, but the outcoming cardinal arithmetic is very unusual. It does not allow for inverse operations, and so there is no direct way of introducing infinitesimal numbers. (Sizes are added by means of disjoint unions and multiplied by means of disjoint unions of equinumerous collections.) Here we maintain Aristotle’s principle, instead halving Cantor’s principle to “equinumerous collections are in 1–1 correspondence”. In this way we obtain a very nice arithmetic: in fact, our “numerosities” may be taken to be nonstandard integers. These numerosities appear naturally suited to sets of ordinals, but they depend, for generic sets, on a “labelling” of the universe by ordinals. The problem of finding a canonical way of attaching numerosities to all sets seems to be worth further investigation.

Choice principles in hyperuniverses

It is well known that the validity of Choice Principles is problematic in non-standard Set Theories which do not abide by the Limitation of Size Principle. In this paper we discuss the consistency of various Choice Principles with respect to the Generalized Positive Comprehension Principle (GPC). The Principle GPC allows to take as sets those classes which can be specified by Generalized Positive Formulae, e.g. the universe. In particular we give a complete characterization of which choice principles (e.g. the Selection Principle and the Ordering Principle) hold in Hyperuniverses. Hyperuniverses are structures which arose independently in Non-well-founded Set Theory and in Mathematical Semantics of Concurrent Programming Languages and are hitherto the only existing models of GPC. Hyperuniverses are naturally endowed with a κ-compact uniform κ-topology and are uniformly isomorphic to their exponential space, i.e. the space of their closed subsets endowed with the Exponential Uniformity.

Cardinal invariants and independence results in the poset of precompact group topologies

We study the poset B(G) of all precompact Hausdorff group topologies on an infinite group G and its subposet Bσ(G) of topologies of weight σ, extending earlier results of Berhanu, Comfort, Reid, Remus, Ross, Dikranjan, and others. We show that if Bσ(G) ≠ ∅ and 2¦GG′¦ = 2¦G¦ (in particular, if G is abelian) then the poset [2¦G¦]σ of all subsets of 2¦G¦ of size σ can be embedded into Bσ(G) (and vice versa). So the study of many features (depth, height, width, size of chains, etc.) of the poset Bσ(G) is reduced to purely set-theoretical problems. We introduce a cardinal function Dede(σ) to measure the length of chains in [X]σ for ¦X¦> σ generalizing the well-known cardinal function Ded(σ). We prove that Dede(σ) = Ded(σ) iff cf Ded(σ) ≠ σ+ and we use earlier results of Mitchell and Baumgartner to show that Dede(N1) = Ded(N1) is independent of Zermelo-Fraenkel set theory (ZFC). We apply this result to show that it cannot be established in ZFC whether BN1(Z) has chains of bigger size than those of the bounded chains. We prove that the poset HN0(G) of all Hausdorff metrizable group topologies on the group G = ⊕N0 Z2 has uncountable depth, hence cannot be embedded into BN0(G). This is to be contrasted with the fact that for every infinite abelian group G the poset H(G) of all Hausdorff group topologies on G can be embedded into B(G). We also prove that it is independent of ZFC whether the poset HN0(G) has the same height as the poset BN0(G).

Axiomatic characterizations of hyperuniverses and applications

Hyperuniverses are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in computer science, for giving denotational semantics à la Scott‐de Bakker, and in mathematical logic, in order to show the consistency of set theories that do not abid***e by the “limitation‐of‐size” principle. We present correspondence between set‐theoretic properties and topological properties of hyperuniverses. We give existence theorems and discuss applications and generalizations to the non‐κ‐compact case.

A simple algebraic characterization of nonstandard extensions

We introduce the notion of functional extension of a set X, by means of two natural algebraic properties of the operator “�” on unary functions. We study the connections with ultrapowers of structures with universe X, and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler’s characterization of nonstandard (= complete elementary) extensions.

Operations, Collections and Sets within a General Axiomatic Framework

This paper is part of a general research programme on the Foundations of Mathematics, Logic and Computer Science, carried out since the early eighties at the Seminar directed by Ennio De Giorgi at the Scuola Normale Superiore in Pisa. In this context, Foundations are not intended to provide safe and unquestionable grounds to scientific activity, but rather to provide conceptual environments where this activity can be carried out naturally and without artificial constraints. Earlier proposals of such foundational theories appear in [3, 1] (see also [11]). Further investigations, along theses lines, have been carried out by various mathematicians, logicians and computer scientists since Spring 1994, starting from the “Basic Theories” introduced in [4] (see, e.g., [17, 6, 5, 7, 12, 13]).

Addendum and corrigendum Choice Principles in Hyperuniverses Annals of Pure and Applied Logic 77 (1996) 35–52

Abstract The proof of Lemma 5 in our paper “Choice Principles in Hyperuniverses” [3], contains an error. In the present note we show that the statement of that lemma is false and hence the Axiom of Choice fails in all κ-hyperuniverses, for uncountable κ. However, a weaker version of Lemma 5 can be proved, which implies that the Linear Ordering Principle holds in all κ-metric κ-hyperuniverses.

An axiomatization of partial n -place operations

We propose a general theory of partial n-place operations based solely on the primitive notion of the application of a (possibly partial) operation to n objects. This theory is strongly selfdescriptive in that the fundamental manipulations of operations, that is, application, composition, abstraction, union, intersection and so on, are themselves internal operations. We give several applications of this theory, including implementations of partial n-ary λ-calculus, and other operation description languages. We investigate the issue of extensionality and give weakly extensional models of the theory.

An Axiomatic Approach to Some Biological Themes

In this paper we continue the enterprise of elaborating an axiomatic framework suitable for modern Biology, started in 1,2,3. We isolate and propose to the attention of the scientific community a few axioms expressing some basic biological facts and their theoretical interpretations. We hope we can foster criticism and contributions from researchers of any field of the Natural Sciences. In fact, we conceive this paper as part of a general research programme on the Foundations of Science, originated in the Eighties by E. De Giorgi at the Scuola Normale Superiore, Pisa and carried on by several researchers after his death in 1996 (see 4,5,6,7,8). The aim of this programme is not to provide safe and unquestionable grounds to scientific activity, but rather to develop conceptual environments where this activity can be carried out naturally and without artificial constraints, and the notions considered in various disciplines can be presented in a simple and clear-cut way, so as to allow for both a deeper analysis by researchers in the field, and a proficuous interdsciplinary debate.

Comparison of the axioms of local and global universality

1. In 1965 D. SCOTT introduced in an unpublished paper (see [S]) an axiom strongly contradicting the Axiom of Foundation. After him several authors, e.g. BOFFA ([I], [2]), HAJEK ([S]), considered the possibility of consistently replacing Foundation by axioms which postulate a strong non-well-foundedness of the membership relation. The authors examined this problem ([3], [ S ] ) , through a different approach, i.e. giving a “Free Construction Principle” for sets, while researching at the seminar on Foundations of Mathematics held by Prof. E. DE GIORGI a t the Scuola Normale Superiore in Pisa. The axioms derivable from this principle are studied in detail in [4].