Fraenkel Mostowski Research Articles

MA(ℵ0) restricted to complete Boolean algebras and choice

AbstractIt is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment of Martin's Axiom either in or in . In this direction, we provide a partial answer by establishing that the Boolean Prime Ideal Theorem in conjunction with the Countable Union Theorem does not imply restricted to complete Boolean algebras in . Furthermore, we prove that the latter (formally) weaker form of and the Δ‐system Lemma are independent of each other in .We also answer open questions from Tachtsis [16] which concern the status of restricted to complete Boolean algebras in certain Fraenkel–Mostowski permutation models of and we strengthen some results from the above paper.

Equivariant ZFA and the foundations of nominal techniques

Abstract We give an accessible presentation to the foundations of nominal techniques, lying between Zermelo–Fraenkel set theory and Fraenkel–Mostowski set theory, which has several nice properties including being consistent with the Axiom of Choice. We give two presentations of equivariance, accompanied by detailed yet user-friendly discussions of its theory and application.

ON RAMSEY’S THEOREM AND THE EXISTENCE OF INFINITE CHAINS OR INFINITE ANTI-CHAINS IN INFINITE POSETS

AbstractRamsey’s Theorem is naturally connected to the statement “every infinite partially ordered set has either an infinite chain or an infinite anti-chain”. Indeed, it is a well-known result that Ramsey’s Theorem implies the latter principle.In the book “Consequences of the Axiom of Choice” by P. Howard and J. E. Rubin, it is stated as unknown whether the above implication is reversible, that is whether the principle “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” implies Ramsey’s Theorem. The purpose of this paper is to settle the aforementioned open problem. In particular, we construct a suitable Fraenkel–Mostowski permutation model ${\cal N}$ for ZFA and prove that the above principle for infinite partially ordered sets is true in ${\cal N}$, whereas Ramsey’s Theorem is false in ${\cal N}$. Then, based on the existence of ${\cal N}$ and on results of D. Pincus, we show that there is a model of ZF which satisfies “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” and the negation of Ramsey’s Theorem.In addition, we prove that Ramsey’s Theorem (hence, the above principle for infinite partially ordered sets) is true in Mostowski’s linearly ordered model, filling the gap of information in the book “Consequences of the Axiom of Choice”.

An Extension of a Permutative Model of Set Theory

In this paper we present a new axiomatic model of set theory called the Extended Fraenkel Mostowski model. It is dened by replacing an axiom of the Fraenkel- Mostowski model with a consequence of it; the other axioms of the Fraenkel-Mostowski model are left unchanged in the new Extended Fraenkel-Mostowski model. We use the theory of groups to dene the sets both in the Extended Fraenkel- Mostowski and in the Fraenkel-Mostowski approach. Several algebraic properties of the sets in the Fraenkel-Mostowski model remain also valid in the Extended Fraenkel- Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel- Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel- Mostowski model. Mathematics Subject Classication

A study of substitution, using nominal techniques and Fraenkel–Mostowksi sets

Fraenkel–Mostowski (FM) set theory delivers a model of names and alpha-equivalence. This model, now generally called the ‘nominal’ model, delivers inductive datatypes of syntax with alpha-equivalence — rather than inductive datatypes of syntax, quotiented by alpha-equivalence. The treatment of names and alpha-equivalence extends to the entire sets universe. This has proven useful for developing ‘nominal’ theories of reasoning and programming on syntax with alpha-equivalence, because a sets universe includes elements representing functions, predicates, and behaviour. Often, we want names and alpha-equivalence to model capture-avoiding substitution. In this paper we show that FM set theory models capture-avoiding substitution for names in much the same way as it models alpha-equivalence; as an operation valid for the entire sets universe which coincides with the usual (inductively defined) operation on inductive datatypes. In fact, more than one substitution action is possible (they all agree on sets representing syntax). We present two distinct substitution actions, making no judgement as to which one is ‘right’ — we suspect this question has the same status as asking whether classical or intuitionistic logic is ‘right’. We describe the actions in detail, and describe the overall design issues involved in creating any substitution action on a sets universe. Along the way, we think in new ways about the structure of elements of FM set theory. This leads us to some interesting mathematical concepts, including the notions of planes and crucial elements, which we also describe in detail.

A general mathematics of names

We introduce FMG (Fraenkel–Mostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax—de Bruijn indices, FM sets, and name-carrying syntax—have a relation generalising to all sets and not only sets of syntax trees. We also give syntax-free accounts of Barendregt representatives, scope extrusion, and other phenomena associated to α -equivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U .

Nominal logic, a first order theory of names and binding

This paper formalises within first-order logic some common practices in computer science to do with representing and reasoning about syntactical structures involving lexically scoped binding constructs. It introduces Nominal Logic , a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping, for freshness of names, and for name-binding. Its axioms express properties of these constructs satisfied by the FM-sets model of syntax involving binding, which was recently introduced by the author and M.J. Gabbay and makes use of the Fraenkel–Mostowski permutation model of set theory. Nominal Logic serves as a vehicle for making two general points. First, name-swapping has much nicer logical properties than more general, non-bijective forms of renaming while at the same time providing a sufficient foundation for a theory of structural induction/recursion for syntax modulo α -equivalence. Secondly, it is useful for the practice of operational semantics to make explicit the equivariance property of assertions about syntax – namely that their validity is invariant under name-swapping.

About И-quantifiers

Gabbay and Pitts observed that the Fraenkel–Mostowski model of set-theory supports useful notions of “name-abstraction” and “fresh-name”. In order to understand their work in a more general setting we introduce the notions of И-units and И-relations in a regular category D. A И-relation is given by a functor A # (-):D→D and we show that in the case that D is a topos then A # (-) has a right adjoint [A](-) that can be thought of as an object of abstractions. We also explore the existence of a right adjoint to [A](-) and relate it to the “name swapping” operations considered as fundamental by Gabbay and Pitts. We present many examples of categories where this notions occur and we relate the results here with Pitts' Nominal Logic.

On quasi-amorphous sets

A set is said to be amorphous if it is infinite, but cannot be written as the disjoint union of two infinite sets. The possible structures which an amorphous set can carry were discussed in [5]. Here we study an analogous notion at the next level up, that is to say replacing finite/infinite by countable/uncountable, saying that a set is quasi-amorphous if it is uncountable, but is not the disjoint union of two uncountable sets, and every infinite subset has a countably infinite subset. We use the Fraenkel–Mostowski method to give many examples showing the diverse structures which can arise as quasi-amorphous sets, for instance carrying a projective geometry, or a linear ordering, or both; reconstruction results in the style of [1] are harder to come by in this case.

The independence of the Prime Ideal Theorem from the Order-Extension Principle

AbstractIt is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel–Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a ‘generic’ extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure.