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.
In this paper we prove that the maximum principle in forcing is equivalent to the axiom of choice. The maximum principle is the property of forcing: p ||- exists x theta(x) iff for some name tau p ||- theta(tau). We also look at three similar partial orders in the Basic Cohen model for the failure of the axiom of choice. We show that despite their apparent similarity the maximum principle holds for only one of the three.
Bolzano-Weierstrass principle of choice extended towards ordinals
波尔札诺-维尔斯特拉斯选择原理是集合论中最古老的方法，传统上用于数学分析。我们将其扩展到超限序列的步骤序号索引。我们介绍的概念：徒步旅行者的轨道，徒步旅行者的地图和语句$P_n(X, Y, m)$ ;这是类似的有限，可数和不可数的情况下使用。作为应用，给出了Ramsey定理和Erdös-Rado定理的新证明.
We introduce a formal theory called Flow where the intended interpretation of its terms is that of function. We prove ZF, ZFC and ZFU (ZF with atoms) can be immersed within Flow as natural consequences from our framework. Our first important application is the introduction of a model of ZFU where the Partition Principle holds but the Axiom of Choice fails, if Flow is consistent. So, our framework allows us to address the oldest open problem in set theory: if the Partition Principle entails the Axiom of Choice.
We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a $\subseteq$-maximal subfamily with the finite intersection property and the principle asserting that if $P$ is a property of finite character then every set has a $\subseteq$-maximal subset of which $P$ holds. We show that these principles and their variations have a wide range of strengths in the context of second-order arithmetic, from being equivalent to $\mathsf{Z}_2$ to bei…
We introduce a general theory of functions called Flow. We prove ZF, non-well founded ZF and ZFC can be immersed within Flow as a natural consequence from our framework. The existence of strongly inaccessible cardinals is entailed from our axioms. And our first important application is the introduction of a model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds but not the Axiom of Choice (AC). So, Flow allows us to answer to the oldest open problem in set theory: if PP entails AC.
Assume ZFC. Let $κ$ be a cardinal. A ${<κ}$-ground is a transitive proper class $W$ modelling ZFC and such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<κ}$. The $κ$-mantle is the intersection of all ${<κ}$-grounds. We prove that certain partial choice principles in the $κ$-mantle are the consequence of $κ$ being inaccessible/weakly compact, and some other related facts.
We introduce a new formulation of the axiom of dependent choice that can be viewed as an abstract termination principle, which generalises the recursive path orderings used to establish termination of rewrite systems. We consider several variants of our termination principle, and relate them to general termination theorems in the literature.
We investigate computability theoretic and descriptive set theoretic contents of various kinds of analytic choice principles by performing detailed analysis of the Medvedev lattice of $Σ^1_1$-closed sets. Among others, we solve an open problem on the Weihrauch degree of the parallelization of the $Σ^1_1$-choice principle on the integers. Harrington's unpublished result on a jump hierarchy along a pseudo-well-ordering plays a key role in solving the problem.
The sequential form of a statement $\forallξ(B(ξ) \rightarrow \existsζA(ξ,ζ))$ is the statement $\forallξ(\forall n B(ξ_n) \rightarrow \existsζ\forall n A(ξ_n,ζ_n))$. There are many classically true statements of the first form whose proofs lack uniformity and therefore the corresponding sequential form is not provable in weak classical systems. The main culprit for this lack of uniformity is of course the law of excluded middle. Continuing along the lines of previous work by Hirst and Mummert, we show that if a statement of the first form satisfying certain syntactic requirements is provable …
We contribute to the knowledge of the quantifier completions and their applications by using the language of doctrines. This algebraic presentation allows us to properly analyse the behaviour of the existential and universal quantifiers. We wish to convey the following points: the first is that these completions preserve the lattice structure and the distributive lattice structure of the fibres under opportune hypotheses which turn out to be preserved as well; the second regards the applications, in particular to the dialectica construction; the third is that these free constructions carry on …
The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos as defined by van den Berg.
