Positive set‐operators of low complexity

The powerset operator, ??, is compared with other operators of similar type and logical complexity. Namely we examine positive operators whose defining formula has a canonical form containing at most a string of universal quantifiers. We call them ?‐operators. The question we address in this paper is: How is the class of ?‐operators generated ? It is shown that every positive ?‐operator Γ such that Γ(??) ≠ ??, is finitely generated from ??, the identity operator Id, constant operators and certain trivial ones by composition, ∪ and ∩. This extends results of [3] concerning bounded positive operators.     

Large transitive models in local ZFC

This paper is a sequel to Tzouvaras (Arch Math Log 49(5):571–601, 2010), where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and ${\Pi_1^1}$ -indescribable models, were considered. By analogy we refer to such models as “large models”, and the properties in question as “large model properties”. Continuing here in the same spirit we consider further large model properties, that resemble stronger large cardinals, namely, “elementarily embeddable”, “extendible” and “strongly extendible”, “critical” and “strongly critical”, “self-critical” and “strongly self-critical”, the definitions of which involve elementary embeddings. Each large model property ? gives rise to a localization axiom Loc ^? (ZFC) saying that every set belongs to a transitive model of ZFC satisfying ?. The theories LZFC ^?  = LZFC + Loc ^? (ZFC) are local analogues of the theories ZFC+“there is a proper class of large cardinals ψ”, where ψ is a large cardinal property. If sext(x) is the property of strong extendibility, it is shown that LZFC ^sext proves Powerset and Σ₁-Collection. In order to refute V = L over LZFC, we combine the existence of strongly critical models with an axiom of different flavor, the Tall Model Axiom (TMA). V = L can also be refuted by TMA plus the axiom GC saying that “there is a greatest cardinal”, although it is not known if TMA + GC is consistent over LZFC. Finally Vopěnka’s Principle (V P) and its impact on LZFC are examined. It is shown that LZFC ^sext  + V P proves Powerset and Replacement, i.e., ZFC is fully recovered. The same is true for some weaker variants of LZFC ^sext . Moreover the theories LZFC ^sext +V P and ZFC+V P are shown to be identical.     

Totally non‐immune sets

Let be a countable first‐order language and be an ‐structure. “Definable set” means a subset of M which is ‐definable in with parameters. A set is said to be immune if it is infinite and does not contain any infinite definable subset. X is said to be partially immune if for some definable A, is immune. X is said to be totally non‐immune if for every definable A, and are not immune. Clearly every definable set is totally non‐immune. Here we ask whether the converse is true and prove that it is false for every countable structure whose class of definable sets satisfies a mild condition. We investigate further the possibility of an alternative construction of totally non‐immune non‐definable sets with the help of a subclass of immune sets, the class of cohesive sets, as well as with the help of a generalization of definable sets, the semi‐definable ones (the latter being naturally defined in models of arithmetic). Finally connections are found between totally non‐immune sets and generic classes in nonstandard models of arithmetic.     

Classification of non‐well‐founded sets and an application

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti‐well‐founded (awf) sets, i. e., non‐well‐founded sets whose all maximal ∈‐paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets with 2 and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈‐sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbá?ek concerning nonstandard Boffa universes of sets is considerably extended.     

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All P₂{\Pi_2} consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ₀-Collection and minus ?{\in} -induction scheme. ZFC+ “there is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and P₁¹{\Pi_1^1} -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form Loc(ZFC+f){Loc({\rm ZFC}+\phi)} are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.     

A Functional Interpretation of Cycloid Processes: Application to the Study of Asymptotic Behaviour

We show that the classic Chapman–Kolmogorov equations of certain Markovian transition semigroups on finite state spaces have a formal analogy, of a homologic nature, in terms of cycloids ₁, ..., _B, and positive numbers w₁, ..., w_B. The collection _k ,w _k completely determines a Markov process {_n}, called a cycloid process, admitting an invariant probability distribution, and decomposes its distribution Prob(_n = , _{n + 1} = ) into a linear expression. The latter is further used in the study of the asymptotic behaviour of the cycloid process.     

Forcing and antifoundation

It is proved that the forcing apparatus can be built and set to work in ZFCA (=ZFC minus foundation plus the antifoundation axiom AFA). The key tools for this construction are greatest fixed points of continuous operators (a method sometimes referred to as “corecursion”). As an application it is shown that the generic extensions of standard models of ZFCA are models of ZFCA again.