The usual understanding of Cantor’s Theorem is that it constitutes a proof that the continuum is an uncountable set. A challenge for this position comes from the observation that using forcing one can make any given cardinal countable and make the continuum arbitrarily large. In this paper, we present a different take on the relationship between Cantor’s Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory.
In this work a general way is given to construct some examples of NSOP1 theories as limits of some Fraïssé class satisfying strong conditions. These limits will satisfy existence, that Kim-independence coincides with algebraic independence, and that forking independence is obtained by forcing base monotonicity on Kim-forking. These theories also come with a stationary independence relation. This study is based on results of Baudisch, Chernikov, Kruckman, and Ramsey.
I show that the axioms and rules of inference about the horizontal in the subsystem of Grundgesetze without value-ranges fail to capture its intended interpretation. That is, I build a nonstandard model in which-a is the true if and only if a is not the false, thereby confirming a conjecture made by Landini that a & Aogon; (-a) = a is not provable in this subsystem. This motivates our main result, namely, soundness and completeness proofs for the first-order fragment without value-ranges with respect to a new nonclassical semantics of boolean truncation. Simply put, the widespread assumption that Grundgesetze's first-order logic is classical first-order logic is mistaken.
This article investigates the proof theory of contra-classical logics, with a focus on Heinrich Wansing’s constructive connexive logic C. Drawing on Wansing’s semantic bilateral framework—which characterizes connectives in terms of support of truth and support of falsity—we introduce a bilateral labeled sequent calculus incorporating specific verification and falsification rules. The resulting calculus is shown to exhibit key structural properties: All logical rules are height-preserving invertible, all structural rules are height-preserving admissible, and the cut rule is admissible. Moreover, the system, proven to be both sound and complete, allows for terminating proof search and the extraction of countermodels from failed root-first proof search. Finally, the computational complexity of the resulting decision procedure is established. The article concludes with some observations on systems related to C, such as C3 and N4, and outlines potential directions for future work.
In 2018, Fern & aacute;ndez-Bret & oacute;n proved that Hindman's theorem is only a countable phenomenon. In 2023, Carlucci and Fern & aacute;ndez-Bret & oacute;n established a positive example of an uncountable Hindman-type theorem called the Adjacent Hindman Theorem for uncountable groups. In this article, we investigate a kind of products called syndetic finite products, and we obtain an uncountable Hindman-type theorem in semigroups for these products. Moreover, we establish a result for adjacent finite products of a fixed number of elements under large cardinal assumptions, which can be viewed as the generalization of Rado's path decomposition theorem.
