逻辑
This note complements our paper "Categoricity-like properties in the first order realm" (Journal for the Philosophy of Mathematics, 2024).
We isolate the limit-stage filter construction needed for countable-support symmetric iterations built from standard successor-step symmetric systems. At successor stages we take the $\omega_1$-completion of the usual successor-stage…
We construct, from a ground model of $ZFC$, a transitive symmetric model $M$ satisfying $ZF + DC + PP + AC_{wo} + \neg AC$. The construction starts with a Cohen symmetric seed model $N$ over $Add(\omega,\omega_1)$ and performs an Ord-length…
Fix a field $K$. We show that $K$ is large if and only if some elementary extension of $K$ is the fraction field of a henselian local domain which is not a field. The proof uses a new result about the \'etale-open topology over $K$: if $K$…
We characterize Martin-L\"of randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem. After setting up a general framework for defining notions of merging randomness, we focus…
We prove that each finite chain in the two-branching countable ultrahomogeneous pseudotree has finite big Ramsey degrees. This is in contrast to the recent result of Chodounsk\'{y}, Eskew, and Weinert that antichains of size two have…
The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_\sigma$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof…
We investigate iterating the construction of $C(\mathtt{aa})$, the $L$-like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of $L$ to obtain a model of $V=C(\mathtt{aa})$, and to…
We confirm Martin's conjecture for a broad subclass of weakly quasi-o-minimal theories.
Given a strongly inaccessible cardinal $\lambda$, we study the Fra\"iss\'e class of all Boolean algebras of size $<\lambda$, together with regular embeddings. We prove that this is indeed a Fra\"iss\'eclass, and its limit has the same…
In recent work, the notion of $m$-rigidity was introduced as a sufficient condition for the existence of infinite antichains of $1$-degrees inside many-one degrees. Motivated by a recent preprint of Richter, Stephan, and Zhang on finite-one…
A finite relational structure A is called compact if for any infinite relational structure B of the same type, the existence of a homomorphism from B to A is equivalent to the existence of homomorphisms from all finite substructures of B to…
Sandqvis's semantics for classical logic without bivalence resolves the question of an anti-realist account of classical reasoning after Dummett. This paper applies the framework to the essential questions of metamathematics. The system…
Joel Hamkins asks whether there is a $\Pi^0_1$-formula $\rho(x)$ such that $\rho({\ulcorner \phi \urcorner})$ is independent over ${\sf PA}+\phi$, if this theory is consistent, where this construction is extensional in $\phi$ with respect…
The aim of this paper is to give natural examples of $\mathbf{\Sigma}_1^1$-complete and $\mathbf{\Pi}_1^1$-complete sets. In the first part, we consider ideals on $\omega$. In particular, we show that the Hindman ideal $\mathcal{H}$ is…
We prove some results on NIP integral domains, especially those that are Noetherian or have finite dp-rank. If $R$ is an NIP Noetherian domain that is not a field, then $R$ is a semilocal ring of Krull dimension 1, and the fraction field of…
We review some independence results in a finite axiom-schematization of classical first-order logic introduced by Norman Megill. We also prove that a certain axiom scheme of this system is independent although all of its instances are…
Joel Hamkins asks whether there is a $\Pi^0_1$-formula $\rho(x)$ such that $\rho(\phi)$ is independent over ${\sf PA}+\phi$, if this theory is consistent, where this construction is extensional in $\phi$ with respect to ${\sf PA}$-provable…
We introduce propositional team-based logics expressively complete for (quasi) downward and (quasi) upward closed properties in a syntactically dual way, by using variants of the inclusion atom. In particular, the variants of the primitive…
In this paper we study the conjugacy relation on one-sided subshifts in the viewpoint of descriptive set theory. We show the conjugacy relation on one sided subshifts with the alphabet set $\{0,1\}$ is non-treeable and non-amenable.