Pub Date : 2024-06-08DOI: 10.1007/s00153-024-00927-4
Mikhail Peretyat’kin
We study the class of all strongly constructivizable models having (omega )-stable theories in a fixed finite rich signature. It is proved that the Tarski–Lindenbaum algebra of this class considered together with a Gödel numbering of the sentences is a Boolean (Sigma ^1_1)-algebra whose computable ultrafilters form a dense subset in the set of all ultrafilters; moreover, this algebra is universal with respect to the class of all Boolean (Sigma ^1_1)-algebras. This gives a characterization to the Tarski-Lindenbaum algebra of the class of all strongly constructivizable models with (omega )-stable theories.
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Pub Date : 2024-06-06DOI: 10.1007/s00153-024-00930-9
Wim Veldman
In the context of a weak formal theory called Basic Intuitionistic Mathematics (textsf{BIM}), we study Brouwer’s Fan Theorem and a strong negation of the Fan Theorem, Kleene’s Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene’s Alternative is equivalent to strong negations of these statements. We discuss finite and infinite games and introduce a constructively useful notion of determinacy. We prove that the Fan Theorem is equivalent to the Intuitionistic Determinacy Theorem. This theorem says that every subset of Cantor space (2^omega ) is, in our constructively meaningful sense, determinate. Kleene’s Alternative is equivalent to a strong negation of a special case of this theorem. We also consider a uniform intermediate value theorem and a compactness theorem for classical propositional logic. The Fan Theorem is equivalent to each of these theorems and Kleene’s Alternative is equivalent to strong negations of them. We end with a note on ‘stronger’ Fan Theorems. The paper is a sequel to Veldman (Arch Math Logic 53:621–693, 2014).
在被称为 "基本直观数学"(Basic Intuitionistic Mathematics)的弱形式理论的背景下,我们研究了布劳威尔扇形定理(Brouwer's Fan Theorem)和扇形定理的强否定--克莱因替代(扇形定理)。我们证明扇形定理等价于一些直觉上公认的可数选择公理的contrapositions,而Kleene's Alternative等价于这些陈述的强否定。我们讨论了有限博弈和无限博弈,并引入了一个建设性的有用的确定性概念。我们证明了范式定理等同于直觉确定性定理。这个定理说,康托尔空间(2^omega )的每一个子集,在我们这个有建构意义的意义上,都是确定的。克莱因替代法等同于对该定理一个特例的强否定。我们还考虑了经典命题逻辑的统一中间值定理和紧凑性定理。扇形定理等价于这些定理,而克莱因替代定理等价于它们的强否定。最后,我们对 "更强 "的范式定理做一个说明。本文是 Veldman(Arch Math Logic 53:621-693, 2014)的续篇。
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Pub Date : 2024-06-03DOI: 10.1007/s00153-024-00932-7
Karim Khanaki
We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula (phi (x,y)), the following are equivalent: