Andrew Swan
We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.
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We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.
We prove that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional logic IPC. More generally, we show that IZF has the de Jongh property with respect to every intermediate logic that is complete with respect to a class of finite trees. The same results follow for CZF.
In recent years the question of whether adding the limited principle of omniscience, LPO, to constructive Zermelo-Fraenkel set theory, CZF, increases its strength has arisen several times. As the addition of excluded middle for atomic formulae to CZF results in a rather strong theory, i.e. much stronger than classical Zermelo set theory, it is not obvious that its augmentation by LPO would be proof-theoretically benign. The purpose of this paper is to show that CZF +RDC+ LPO has indeed the same strength as CZF, where RDC stands for relativized dependent choice. In particular, these theories pr…
By Easton's theorem one can force the exponential function on regular cardinals to take rather arbitrary cardinal values provided monotonicity and Koenig's lemma are respected. In models without choice we employ a "surjective" version of the exponential function. We then prove a choiceless Easton's theorem: one can force the surjective exponential function on all infinite cardinals to take arbitrary cardinal values, provided monotonicity and Cantor's theorem are satisfied, irrespective of cofinalities.
Can a positive function on R have zero Lebesgue integral? It depends on how much choice one has. Keywords: Lebesgue integral; Zermelo--Fraenkel theory; Feferman-Levy model
We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice a surjectively modified continuum function $θ(κ)$ can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton's Theorem is in sharp contrast to the situation in ZFC, where for singular cardinals $κ$, the value of $2^κ$ is strongly influenced by the behaviour of the continuum function below. Our construction can roughly be described as follows: In a ground model $V \models ZFC + GCH$ with a "reasonable" function $F: Card \rightarrow Card$ on the infinite cardinals, a class forcing ${\mathb…
We show that in the theory ZF + DC + for every cardinal λ, the set of infinite subsets of λ is well-ordered (i.e., Shelah's AX4), the θ-function measuring the surjective size of the powersets P(κ) can take almost arbitrary values on any set of uncountable cardinals. This complements our results from [FK16], where we prove that in ZF (without DC), any possible behavior of the θ-function can be realized; and answers a question of Shelah in [She16], where he emphasizes that ZF + DC + AX4 is a reasonable theory, where much of set theory and combinatorics is possible.
In this paper, we build Fidel-structures valued models following the methodology developed for Heyting-valued models; recall that Fidel structures are not algebras in the universal algebra sense. Taking models that verify Leibniz law, we are able to prove that all set-theoretic axioms of ZF are valid over these models. The proof is strongly based on the existence of paraconsistent models of Leibniz law. In this setting, the difficulty of having algebraic paraconsistent models of law for formulas with negation using the standard interpretation map is discussed, showing that the existence of mod…
According to Cantor, a set is a collection into a whole of defined and separate (we shall say distinct) objects. So, a natural question is ``How to treat as `sets' collections of indistinguishable objects?". This is the aim of quasi-set theory, and this problem was posed as the first of present day mathematics, in the list resulting from the Congress on the Hilbert Problems in 1974. Despite this pure mathematical motivation, quasi-sets have also a strong commitment to the way quantum physics copes with elementary particles. In this paper, we discuss the axiomatics of quasi-set theory and sketc…
The standard treatment of sets and definable classes in first-order Zermelo-Fraenkel set theory accords in many respects with the Fregean foundational framework, such as the distinction between objects and concepts. Nevertheless, in set theory we may define an explicit association of definable classes with set objects $F\mapsto\varepsilon F$ in such a way, I shall prove, to realize Frege's Basic Law V as a ZF theorem scheme, Russell notwithstanding. A similar analysis applies to the Cantor-Hume principle and to Fregean abstraction generally. Because these extension and abstraction operators ar…
Recently, the Elementary Process Theory (EPT) has been developed as a set of fundamental principles that might underlie a gravitational repulsion of matter and antimatter. This paper presents set matrix theory (SMT) as the foundation of the mathematical-logical framework in which the EPT has been formalized: Zermelo-Fraenkel set theory (ZF), namely, cannot be used as such. SMT is a generalization of ZF: whereas ZF uses only sets as primitive objects, in the framework of SMT finite matrices with set-valued entries are objects sui generis, with a one-by-one set matrix [x] being identical to the …
Independence of premise for intuitionistic Zermelo-Fraenkel set theory
针对Frittaion等人提出的直觉主义策梅洛-弗兰克尔集合论中前提独立性是否为可容许规则的问题，通过适配集合论语境的Friedman-Dragalin翻译变体，证明该规则成立。
Necessary and Sufficient Conditions for Proving Choice in Zermelo-Fraenkel Set Theory
本文在ZF集合论中提出无需选择公理证明特定集合族存在选择函数的条件，通过分离公理方案构建候选集并证明其非空，最终建立必要且充分条件。
We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E'). This extends Zermelo's 1930 theorem about second order ZFC.
We introduce the concept of inverse powerset by adding three axioms to the Zermelo-Fraenkel set theory. This extends the Zermelo-Fraenkel set theory with a new type of set which is motivated by an intuitive meaning and interesting applications. We present different ways to extend the definition of cardinality and show that one implies the continuum hypothesis while another implies the negation of the continuum hypothesis. We will also explore the idea of empty sets of different cardinalities which could be seen as the empty counterpart of Cantor's theorem for infinite sets.
Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory
本研究提出四值悖论相容且悖论不完全的集合论系统BZFC，通过分析ZFC公理并引入反经典性公理，证明单个非经典集合足以产生所有非经典集合，且与ZFC双向可解释，并应用于塔尔斯基语义。
In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe $\mathrm{V_{ex}}(A)$ in which the axiom of choice in all finite types ${\sf AC}_{\sf FT}$ is realized, where $A$ stands for an arbitrary partial combinatory algebra. This construction furnishes 'inner models' of many set theories that additionally validate ${\sf AC}_{\sf FT}$, in particular it provides a self-validating seman…
We show that the existence of a weakly compact cardinal over the Zermelo-Fraenkel's set theory is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.
Extensional realizability and choice for dependent types in intuitionistic set theory
该研究在直觉主义集合论的外延通用可实现性框架下，证明其可验证多个强度超出有穷类型选择公理的依赖类型选择原理，且添加这些选择原理不会改变构造性与直觉主义策梅洛-弗兰克尔集合论的算术部分。
We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic inductive definitions". We give applications to formal topology as well as a predicative justification of this principle.
More on the Boolean Prime Ideal Theorem
研究布尔素理想定理相关内容，通过证明，得出策梅洛 - 弗兰克尔集合论与依赖选择公理等的一致性。
The theory of finitely supported algebraic structures is related to Pitts theory of nominal sets (by equipping finitely supported sets with finitely supported internal algebraic laws). It represents a reformulation of Zermelo Fraenkel set theory obtained by requiring every set theoretical construction to be finitely supported according to a certain action of a group of permutations of some basic elements named atoms. Its main purpose is to let us characterize infinite algebraic structures, defined involving atoms, only by analyzing their finite supports. The first goal of this paper is to defi…
We give an accessible presentation to the foundations of nominal techniques, lying between Zermelo-Fraenkel set theory and Fraenkel-Mostowski set theory, and 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 theoretical significance and practical application.
Choice and independence of premise rules in intuitionistic set theory
本文证明大量直觉主义集合论在自然数有限类型对应规则下封闭，且形如$\exists y^σ\\\\, φ(y)$的语句具有存在性性质，尤其适用于${\sf CZF}$和${\sf IZF}$，采用融合泛型可实现性与真值的方法。
We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical primitives. The Anti-foundation Axiom plays a significant role in our development, since among other of its features, its replacement for the Axiom of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic interpretations. These interpretations also depend on such allied notions for sets as pictures, graphs, decorations, labelings and various mappings that we use. A syntax and semantics of operators acting on sets is developed. Such features enable construction of a theory of non-well-founded sets that we use to…
Chain Conditions and the Axiom of Choice
该研究在无选择公理的策梅洛-弗兰克尔集合论中，于多数学领域语境下确立了可数有限集并集可数断言的等价命题，还挖掘了其背后的范畴论原理。
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.
We introduce the forcing model of IZFA (Intuitionistic Zermelo-Fraenkel set theory with Atoms) for every Grothendieck topology and prove that the topos of sheaves on every site is equivalent to the category of 'sets in this forcing model'.
Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$ either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. Thus we mount an analysis of the categorical large cardinals. This mathematical analysis leads naturally to philosophical issues concerning structuralism and realism, including especially the tension between categoricity and reflection. Ultimately we ident…
In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory $\text{ZFC}^-$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\text{ZFC}^-$ that are bi-interpretable, but not isomorphic---even $\langle H_{ω_1},\in\ra…
Models of Set Theory: Extensions and Dead-ends
本文研究了ZF集合论任意模型的扩展，提出新的构造方法，并证明某些ZF模型无法通过适当扩展得到。
We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$. This shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas. This is an improved version of our paper in the Proceedings of the American Mathematical Society, which used a weak version of the Axiom of Choice for the same result.
Existence of bases implies the axiom of choice, a foundation-free proof
在无基础公理的ZF集合论中，证明每个向量空间有基蕴含选择公理，且该结果可推广至含原子的集合论。
In Feferman's work, explicit mathematics and theories of generalized inductive definitions play a central role. One objective of this article is to describe the connections with Martin-Lof type theory and constructive Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of the relationship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky's univalence axiom. A further goal is to investigate the strength of intuitionistic theories …
On cohomology of locally profinite sets
本文在Zermelo-Fraenkel集合论中构造了首个非递降且忠实平坦的交换环之间映射的例子，涉及局部紧致集的上同调结构。
Mazurkiewicz Sets and Containment of Sierpiński-Zygmund Functions under Rotations
基于 Kharazishvili 的工作，构造了在不同方向包含或不包含 Sierpiński-Zygmund 函数的 Mazurkiewicz 集，并证明其能否表示为两个该函数的并集与 ZFC 独立，同时提出 Hamel 函数的未解问题。
Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My conjecture is that this method works well because mathematicians use only those predicates to form sets, which yield closed hereditary consistent predicate extensions. And for every open formula they use in the process of constructing of a certain (special) set (bottom up), we can always find an "almost-closed" formula (i.e. a parameter-free formula with only the f…
Amorphous sets and dual Dedekind finiteness
本文研究非晶集合与对偶戴德金有限性的关系，证明存在非晶集合其幂集对偶戴德金无限，否定Truss问题，并推广Goldstern结果，证明严格非晶集合的幂集的任意有限次幂对偶戴德金有限。
The principle of open determinacy for class games---two-player games of perfect information with plays of length $ω$, where the moves are chosen from a possibly proper class, such as games on the ordinals---is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it im…
The Reverse Mathematics of Analytic Measurability
本文研究Lusin定理的反数学强度，证明分析集的Lebesgue正则性等价于Σ¹₁-IND over ATR₀，而完整定理等价于Π¹₁-CA₀ over ATR₀。
Existence Theorem for Cumulative Universe Towers and Its Applications
本文构建了 Grothendieck 宇宙的累积塔，通过归纳-递归定义宇宙编码与秩，证明其在依赖积、依赖和等下的封闭性及严格累积性，并在 ZF 集合论上确立了高阶类型论的可靠集合论模型存在性定理。
Chiron is a derivative of von-Neumann-Bernays-Gödel (NBG) set theory that is intended to be a practical, general-purpose logic for mechanizing mathematics. Unlike traditional set theories such as Zermelo-Fraenkel (ZF) and NBG, Chiron is equipped with a type system, lambda notation, and definite and indefinite description. The type system includes a universal type, dependent types, dependent function types, subtypes, and possibly empty types. Unlike traditional logics such as first-order logic and simple type theory, Chiron admits undefined terms that result, for example, from a function applie…
A Countable, Dense, Dedekind-Complete Subset of $\mathbb{R}$ Constructed by Extending $\mathbb{Q}$ via Simultaneous Marking of Closed Intervals with Rational Endpoints
本文从基数的模型相对性出发，在 ZFC 与 Wang 的 Σ-模型中构造可数、稠密且 Dedekind 完备的实数子集，以反思不可数性证明与 CH。
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