Saharon Shelah
We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can be generalized. E.g., there is a class of regular cardinals, and we can prove cardinal inequality.
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We deal with relatives of GCH which are provable. In particular we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice are assumed but some of the results gives us additional information even in ZFC.
We continue our investigation on pcf with weak form of choice. Characteristically we assume DC + P(Y) when looking and prod_{s in Y} delta_s. We get more parallel of theorems on pcf.
We prove in ZF that there is an inner product space, in fact, nicely definable with no orthonormal basis.
This paper can be viewed as a continuation of [KS09] that dealt with the automorphism tower problem without Choice. Here we deal with the inequation which connects the automorphism tower and the normalizer tower without Choice and introduce a new proof to a theorem of Fried and Kollár that any group can be represented as an automorphism group of a field. The proof uses a simple construction: working more in graph theory, and less in algebra.
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.
Solovay's inaccessible over a weak set theory without choice
在完全无选择公理的背景下，研究Zermelo集合论中$Σ^1_3$集Lebesgue可测性的相容强度，并建立类似于Solovay-Shelah定理的结果。
Set theory without choice: not everything on cofinality is possible
我们证明（ZF+DC），例如：如果mu =| H（mu）|则μ ^+是正则不可测的。这与mu = aleph_{omega}的可测量性结果相反，见Apter Magidor [ApMg]
We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems.
Sequential and distributive forcings without choice
本文研究无选择公理下集合论中$κ$-分配力迫与$κ$-序列力迫的等价性问题，证明该等价在弱选择公理下失效，给出拟正常性条件以刻画依赖选择公理的保持性，并分析两类力迫对不同选择公理的保持与违反情况。
A note on Łoś's Theorem without the Axiom of Choice
本文在不使用选择公理的前提下研究Łoś定理，证明其等价于超幂初等等价弱形式，并构造反例说明该弱形式不蕴含基本定理，同时表明可数选择等弱选择公理不能由Łoś定理推出。
We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in aleph_alpha for some (equivalently for every) ordinal alpha>0. Another central result here is, in this context: the number of models of a countable first order T of cardinality aleph_alpha is either >=|alpha| for every aleph_αor it has a small upper bound (close to beth_2).
We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey's or (appropriately phrased) Hindman's theorem; such sets may exist if one does not assume the Axiom of Choice. We obtain very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets.
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…
The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well-orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo-Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals (among other general instances) is provable without the axiom of choice.
We show that it is consistent relative to ZF, that there is no well-ordering of $\mathbb{R}$ while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we can assume that every projective hypergraph on $\mathbb{R}$ has a maximal independent set, among a few other things. For example, we get transversals for all projective equivalence relations. Moreover, this is possible while either $\mathsf{DC}_{ω_1}$ holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, "projectiv…
Frucht's Theorem without Choice
本文证明了Frucht定理在ZFA集合论中独立，展示其在部分置换模型中不成立但在另一些中成立，并指出Babai的更强版本在无选择公理时可能失效。
In set theory without the Axiom of Choice, we consider Ingleton's axiom which is the counterpart in ultrametric analysis of the Hahn-Banach axiom. We show that in $ZFA$, set theory without the Axiom of Choice weakened to allow "atoms", Ingleton's axiom does not imply the Axiom of Choice (this solves in $ZFA$ a question raised by van Rooij (1992). We also prove that in $ZFA$, the "multiple Choice" axiom implies the Krein-Milman axiom. We deduce that, in $ZFA$, the conjunction of the Hahn-Banach, Ingleton and Krein-Milman axioms does not imply the Axiom of Choice.
The Logic of Cardinality Comparison Without 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，给出了可同时推理集合布尔运算与基数比较的完备公理系统，明确了戴德金有限集和一般情形下的核心原理。
It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom of Choice than traditional elementary analysis. The claim is based on the observation that the hyperreals entail the existence of nonprincipal ultrafilters over N, a strong version of the Axiom of Choice, while the real numbers can be constructed in ZF. The axiomatic approach to nonstandard methods refutes this objection. We formulate a theory SPOT in the st-$\in$-language which suffices to carry out infinitesimal arguments, and prove that SPOT is a conservative extension of ZF. Thus the methods of…
In this paper, without the axiom of choice, we show that if a certain downward Löwenheim-Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of $\mathsf{ZFC}$.
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。
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。
If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice.
Geometric theories for real number algebra without sign test or dependent choice axiom
本文利用几何理论的动态版本，在没有依赖选择公理的构造性数学中构建尽可能完整的实数域代数性质理论，并初步勾勒了某些 o-minimal 结构的构造性理论。
ZFC without power set II: Reflection strikes back
本文通过“大真类”概念研究 ZFC$^-$ 中相关选择原理的失效，构造模型分离不同长度的 DC 方案，并证明非平凡共尾初等自嵌入不必蕴含 von Neumann 层级片段。
Funicular preorders can be prelinearized without nonprincipal ultrafilters over $\mathbb N$
本文研究了无需非主超滤子在$\mathbb N$上即可预线性化纤维序的条件，展示了在$\mathsf{ZF+DC}$模型中此类序的预线性化方法。
On Strongly-equitable Social Welfare Orders Without the Axiom of Choice
本文研究强公平且有限匿名的社会福利序（SEA），证明其存在蕴含不可勒贝格测度的实数集存在，但不蕴含自然数上非主超滤子的存在，并运用几何集合论技术解决开放问题。
A choice-free proof of the Erdős--Dushnik--Miller theorem
本文在不含选择公理的ZF集合论中，给出Erdös–Dushnik–Miller定理的纯组合证明，避免了元数学考量。
Vopěnka's Principle without Choice: Preservation under Symmetric Extensions
本文证明Vopěnka原则在ZF下对对称扩张保持，并建立与ZFC的等一致性及Löwenheim-Skolem基数存在性，同时给出独立性结果。
Hindman and Owings-like theorems without the Axiom of Choice
本文研究了在不使用选择公理的Zermelo-Fraenkel集合论中Hindman和Owings型Ramsey理论陈述的变体，发现Hindman定理在实数加法群和不可数维的Q向量空间中不成立，而Owings型配置在假设决定公理时获得了积极结果。
For a cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $\mathfrak{a}$ and all natural numbers $n$, \[ 2^{\mathrm{fin}(\mathfrak{a})^n}=2^{[\mathrm{fin}(\mathfrak{a})]^n}. \] On the other hand, it is proved that the following statement is consistent with $\mathsf{ZF}$: there exists an infinite cardinal $\mathfrak{a}$ such that \[ 2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}(\mathfrak{a})^2}<2^{\mathrm{…
In synthetic computability, pioneered by Richman, Bridges, and Bauer, one develops computability theory without an explicit model of computation. This is enabled by assuming an axiom equivalent to postulating a function $ϕ$ to be universal for the space $\mathbb{N}\to\mathbb{N}$ ($\mathsf{CT}_ϕ$, a consequence of the constructivist axiom $\mathsf{CT}$), Markov's principle, and at least the axiom of countable choice. Assuming $\mathsf{CT}$ and countable choice invalidates the law of excluded middle, thereby also invalidating classical intuitions prevalent in textbooks on computability. On the o…
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.
For a centerless group G, we can define its automorphism tower. We define G^{alpha} : G^0=G, G^{alpha +1}=Aut(G^alpha) and for limit ordinals G^delta=bigcup_{alpha < delta}G^alpha . Let tau_G be the ordinal when the sequence stabilizes. Thomas' celebrated theorem says tau_G<2^{|G|})^{+} and more. If we consider Thomas' proof too set theoretical, we have here a shorter proof with little set theory. However, set theoretically we get a parallel theorem without the axiom of choice. We attach to every element in G^alpha, the alpha-th member of the automorphism tower of G, a unique quantifier free t…
Set-theoretic absoluteness for analysts and separable \cstar-algebras without Choice
本文在不使用选择公理的情况下，研究了可分离C*-代数的理论发展，并展示了某些结果在ZF中的可证明性。
Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice
本文在无选择公理的ZF集合论中证明若干关于无限局部有限连通图的色数、边色数、区分数与区分指数的命题等价于Kőnig引理，并给出与AC或Kőnig引理等价的覆盖与支配集存在性新条件。
Interpolating Between Choices for the Approximate Intermediate Value Theorem
本文在弱假设下构造性证明近似中间值定理，通过连续插值二分选择，无需一致连续性和可数选择公理。
On the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs without AC
本文在无选择公理的集合论框架下，考察广义 Rainbow Ramsey 定理与 Erdős–Rado 二元组 Canonical Ramsey 定理的强度，及其与多种弱选择原理的关系。
Choice revision is a sort of non-prioritized multiple revision, in which the agent partially accepts the new information represented by a set of sentences. We investigate the construction of choice revision based on a new approach to belief change called descriptor revision. We prove that each of two variants of choice revision based on such construction is axiomatically characterized with a set of plausible postulates, assuming that the object language is finite. Furthermore, we introduce an alternative modelling for choice revision, which is based on a type of relation on sets of sentences, …
Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes
该研究通过构造一个内部模型，展示了在不依赖选择公理的情况下，分区原理可以成立。
We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]^I which is a bounded subset of l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F.
On Erdos--Dushnik--Miller theorem without AC
本文在无选择公理的集合论中，研究了限制于不可数顶点集的Erdos-Dushnik-Miller定理在弱选择形式层级中的位置，并部分回答了Lajos Soukup提出的问题。
Hilbert Spaces Without Countable AC
本文研究了由与可数选择公理不相容的集合构造的 Hilbert 空间，在无选择公理假设的 ZF 集合论中探讨其性质与算子，并将其视为“量子化”集合以推导集合论结果。
Partition models, Permutations of infinite sets without fixed points, and weak forms of AC
该研究在ZF、ZFA系统中探究三类数学命题与弱选择公理的新关联，证明了图论相关原理与Kőnig引理等价，还明确了有限分划模型中弱选择公理的状态及Van Douwen选择原理在两类置换模型中的成立性。
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