We prove, in ZFC, that certain Σ^1_2 functions cannot injectively embed ω _1 into a Borel class of fixed countable rank. This had been proved under determinacy or large cardinals by Harrington and Hjorth for all Σ^1_2 functions. Our contribution is to identify conditions under which the determinacy and large cardinal assumptions can be removed. These conditions are sufficient for a recent use of the non-existence of Σ^1_2 injections of ω _1 into Borel classes by Day and Marks.
We present a type-disciplined, compositional reconstruction of the diagonal strategy in Gödel’s first incompleteness theorem. The goal is not a new or “more direct” proof of incompleteness, but an explicit computational unpacking of the mechanism. The paper exhibits a concrete pipeline of code-transformers between primitive recursive syntactic domains, thereby avoiding well-definedness issues arising in Lindenbaum-quotient formulations. A central feature is the separation between a graph-mediated witness form of the Gödel sentence and a canonical normal form, connected by an internal normalization lemma formalizable in the base theory. The paper also presents the construction categorically in a category of primitive recursive code domains and morphisms, and extracts a general diagonal-flipping blueprint clarifying how type stratification distinguishes paradoxical from limitative diagonal arguments.
We study the Sigma 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma <^>0_1$$\end{document}-fragment of the Kreisel-Putnam axiom and its Sigma n0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma <^>0_n$$\end{document}-extensions in the context of intuitionistic arithmetic and analysis. Among other things, we show that the Sigma 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma <^>0_1$$\end{document}-fragment of the Kreisel-Putnam axiom is exactly the principle that fills the gap between the lesser limited principle of omniscience and the disjunctive Markov principle in constructive reverse mathematics. In addition, we introduce two variants of the linearity axiom, which are related to the Kreisel-Putnam axiom, and show that the Sigma 10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma <^>0_1$$\end{document}-fragments of these three axioms do not lie in any previously known layers in the arithmetical hierarchy of logical axioms.
We introduce a generalization of the Tukey reducibility, which we call the pre-Tukey reducibility. While the basics of the original Tukey reducibility heavily rely on the axiom of choice, the pre-Tukey reducibility works well in (without the axiom of choice) to compare cofinal types of directed sets. In , we show that two directed sets are pre-Tukey equivalent if and only if there exists a directed set into which both directed sets can be cofinally embedded. In this paper, we investigate the pre-Tukey reducibility between σ -directed sets (ω ^ω , ≤ ^*), (ℳ, ⊆ ), (𝒩, ⊆ ), ([ω ^ω ]^ω , ⊆ ) and (ω _1, ≤ ) where ≤ ^* denotes the dominating relation on ω ^ω , and ℳ and 𝒩 denote the ideal of meager and null sets of 2^ω , respectively. We show that under + and certain assumptions on sets of reals, these directed sets have pairwise distinct cofinal types. The assumptions we consider hold in the Solovay model and in L(ℝ) satisfying the axiom of determinacy.
The calculus C was introduced by H. Wansing as a constructive logic with strong negation. In addition, C validates the theses of connexive logic that are attributed to Aristotle and Boethius. A further remarkable property of C is that it is a non-trivial but negation inconsistent system: it has a formula and its negation as theorems. From a bilateralist-minded perspective, such a contradiction can be seen as the existence of both a verification and a falsification of one and the same formula. Relatedly, it has been noted by Wansing that there seems to be a kind of correspondence between these two types of derivations when it comes to a proof of contradiction. Following this observation, we attempt in this paper to introduce a precise notion for such a correspondence. We thence establish that this correspondence obtains in propositional and first-order versions of C, via formulations of suitable sequent and tableau calculi.
We study the Σ ^0_1 -fragment of the Kreisel-Putnam axiom and its Σ ^0_n -extensions in the context of intuitionistic arithmetic and analysis. Among other things, we show that the Σ ^0_1 -fragment of the Kreisel-Putnam axiom is exactly the principle that fills the gap between the lesser limited principle of omniscience and the disjunctive Markov principle in constructive reverse mathematics. In addition, we introduce two variants of the linearity axiom, which are related to the Kreisel-Putnam axiom, and show that the Σ ^0_1 -fragments of these three axioms do not lie in any previously known layers in the arithmetical hierarchy of logical axioms.
We prove that Arithmetical Comprehension is equivalent to the determinacy of all clopen integer games in which each player has at most two moves per turn.
We study the generalized dominating number 𝔡_μ at a singular cardinal μ of cofinality κ . We prove two basic lower bounds: in ZFC , cf( [μ ]^κ ,⊆) ≤𝔡_μ , and under mild cardinal-arithmetic assumptions, 2^<μ≤𝔡_μ . We also clarify when 𝔡_μ can differ from 2^μ : assuming GCH and κ = cf( μ) > ω , a finite-support iteration of Cohen forcing of length μ ^++ yields 𝔡_μ< 2^μ . On the other hand, for κ = cf( μ) = ω , natural μ -cc posets force 𝔡_μ= 2^μ .
We generalize the main result of [13] and show the consistency of the statement “There are exactly n Q-points up to isomorphism" for any finite n. Furthermore, we show that the above statement for n=2 can alternatively be obtained by a length- ω _2 countable support iteration of Matet-Mathias forcing restricted to a Matet-adequate family.
We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if F ⊆ℝ^n is a closed set represented so that the distance function x ↦ d(x,F) can be computed, and (f^(k̅))_|k̅| ≤ m is a Whitney jet of order m on F, then we can compute g ∈ C^m(ℝ^n) such that g and its partial derivatives coincide on F with the corresponding functions of (f^(k̅))_|k̅| ≤ m.
We study structural consequences of the existence of a Reinhardt embedding (an embedding of the set-theoretic universe into itself). First, we prove that if delta is a regular cardinal greater than the critical point of the embedding and is fixed by it, then no function from delta to delta in the range of the embedding eventually dominates the embedding restricted to delta. Assuming the existence of such an embedding, we use this to obtain elementary embeddings of initial segments of the universe into themselves exhibiting a local form of extendibility. We then refine results of Gabriel Goldberg on almost supercompact cardinals by proving that certain such cardinals cannot be regular. Finally, we use these ideas to give an alternative proof of Kunen’s Inconsistency Theorem.
We investigate whether the ultrafilter number function κ↦𝔲(κ) on the cardinals is monotone, that is, whether 𝔲(λ) ≤𝔲(κ) holds for all cardinals λ< κ or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if κ is a singular cardinal with countable cofinality or a strong limit singular cardinal, then 𝔲(κ) ≤𝔲(κ^+) holds.
We consider combining the definition of a cardinal invariant and the notion of an infinite game. We focus on the splitting number 𝔰 since the corresponding cardinal invariants behave in an interesting way. We introduce three kinds of games as reasonable realizations of the combination of the notions of splitting and infinite games. Then, we consider two cardinal invariants for each game, so we define six numbers. We prove that three of them are equal to the size of the continuum 𝔠 and one of them is equal to the σ -splitting number 𝔰_σ , which is defined as the minimum size of a σ -splitting family. On the other hand, we show that the remaining two numbers are consistently different from 𝔠 , 𝔰 and 𝔰_σ . Moreover, though the two numbers share almost the same rule of the game, we prove that they can take distinct values from each other, and hence the slight difference of the rule is actually crucial in this sense.
We consider the restriction of Ramsey's theorem that arises from considering only translation-invariant colourings of pairs, and show that this has the same strength (both from the viewpoint of Reverse Mathematics and from the viewpoint of Computability Theory) as the Adjacent Hindman's Theorem, proposed by L. Carlucci (Arch. Math. Log. 57 (2018), 381–359). We also investigate some higher dimensional versions of both of these statements.
We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an ω -Erdős cardinal, we determine which of these theories are Borel complete. We develop machinery, including forbidding nested sequences which implies a tight upper bound on Borel complexity, and admitting cross-cutting absolutely indiscernible sets which in our context implies Borel completeness.
We investigate the model completeness of the theory of a mixed characteristic henselian valued field with finite ramification relative to the model completeness of the residue field and value group. We address the case in which the valued field has a value group with finite spines, and the case in which the value group is elementarily equivalent to the infinite lexicographic sum of ℤ with a minimal positive element. In both cases, we find a one-sorted language in which the theory of the valued field is model complete, if the theory of the residue field is model complete in the language of rings.
In (Intuitionistic propositional logic with galois negations. Stud Logica 111:21–56, 2023) , Ma and Li established Intuitionistic Propositional Logic with Galois negations (IGN). These logics can be viewed as the ordered dualization of Ewald’s intuitionistic tense logic IKt, where Galois negations function as the ordered duals of residuated pairs of tense operators. In Almiñana et al. (On heyting algebras with negative tense operators. Stud Logica 111:1015–1036, 2023), we introduced the concept of negative tense operators on Heyting algebras and defined the variety of tense H-algebras. This new variety of algebras serves as the algebraic semantics for IGN. In this paper, we extend our investigation into tense H-algebras, focusing specifically on their topological properties. Particularly, we provide a topological characterization of subdirectly irreducible tense H-algebras.
We continue the work from [8] and make a small—but significant—improvement to the definition of j-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this allows us to more easily lift weakly compact embeddings and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has Mitchell order of at least 2.
We show that the forcing axiom for countably compact, ω _2 -Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah [9] and sets a new limit to the generalization of Martin’s Axiom to the stage of ω _2 .
