We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded $T$-convex theory.

The goal of this article is to take a step toward the resolution of the problem of finding an analytic sequent calculus for the logic of proofs. For this, we focus on the system Ilp, the intuitionistic version of the logic of proofs. First we present the sequent calculus Gilp that is sound and complete with respect to the system Ilp; we prove that Gilp is cut-free and contraction-free, but it still does not enjoy the subformula property. Then, we enrich the language of the logic of proofs and we formulate in this language a second Gentzen calculus ${\mathbf{Gilp}}^{\ast }$. We show that ${\mathbf{Gilp}}^{\ast }$ is a conservative extension of Gilp, and that ${\mathbf{Gilp}}^{\ast }$ satisfies the subformula property.

The algebra of embeddings at the I3 level has been deeply analyzed, but nothing is known algebra-wise for embeddings above I3. In this article, we introduce an operation for embeddings at the level of I0 and above, and prove that they generate an LD-algebra that can be quite different from the one implied by I3.

Operators can be compared in at least two respects: expressive strength and deductive strength. Inspired by Hintikka’s treatment of question embedding verbs, the variations of noncontingency operator, and also the various combinations of modal operators and Boolean connectives, we propose a logic with (deductively) strong noncontingency operator as the only primitive modality. The novel operator is deductively but not expressively stronger than both noncontingency operator and essence operator, and expressively but not deductively weaker than the necessity operator. The frame-definability power of this new logic is in between standard modal logic and noncontingency logic. A notion of bisimulation is proposed to characterize this logic within standard modal logic and first-order logic. Axiomatizations over various frame classes are presented, among which the minimal logic is related to the treatment of an alternative semantics of the agreement operator proposed by Lloyd Humberstone.

An ${\aleph }_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ${\aleph }_1$-Souslin tree.

In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a ${\lambda }^{+}$-Souslin tree. This class includes Prikry, Magidor, and Radin forcing.

In line with Ken Manders’s seminal account of Euclid’s diagrammatic method in the “The Euclidean Diagram,” two proof systems with a diagrammatic syntax have been advanced as formalizations of the method FG and Eu. In a paper examining Eu, Nathaniel Miller, the creator of FG, has identified a variety of technical problems with the formal details of Eu. This response shows how the problems are remedied.

We give a counterexample to the claim that every provably total function of Basic Arithmetic is a polynomially bounded primitive recursive function.

This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space $\{0,1{\}}^{\mathbb{N}}$ with the uniform measure and the usual shift (erasing the first bit). We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets $\mathcal{P}$ of positive measure holds for iterations starting at the point. This means that for each $k\in \mathbb{N}$ there is an $n$ such that $n,2n,\dots ,kn$ shifts of the point all end up in $\mathcal{P}$. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, ${\Pi }_1^0$ with computable measure, and ${\Pi }_1^0$. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the $k$ commuting shift operators on $\{0,1{\}}^{{\mathbb{N}}^k}$.

For $n<\omega$, we say that the${\Pi }_n^1$-reflection principle holds at $\kappa$ and write ${Refl}_n\left(\kappa \right)$ if and only if $\kappa$ is a ${\Pi }_n^1$-indescribable cardinal and every ${\Pi }_n^1$-indescribable subset of $\kappa$ has a ${\Pi }_n^1$-indescribable proper initial segment. The ${\Pi }_n^1$-reflection principle ${Refl}_n\left(\kappa \right)$ generalizes a certain stationary reflection principle and implies that $\kappa$ is ${\Pi }_n^1$-indescribable of order $\omega$. We define a forcing which shows that the converse of this implication can be false in the case $n=1$; that is, we show that $\kappa$ being ${\Pi }_1^1$-indescribable of order $\omega$ need not imply ${Refl}_1\left(\kappa \right)$. Moreover, we prove that if $\kappa$ is $(\alpha +1)$-weakly compact where $\alpha <{\kappa }^{+}$, then there is a forcing extension in which there is a weakly compact set $W\subseteq \kappa$ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and $\kappa$ remains $(\alpha +1)$-weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down.

Goodman’s theorem states that ${\mathsf{HA}}^{\omega }+\mathsf{AC}+\mathsf{RDC}$ is conservative over $\mathsf{HA}$. The same result applies to the extensional case, that is, $\mathsf{E}\text{-}{\mathsf{HA}}^{\omega }+\mathsf{AC}+\mathsf{RDC}$ is also conservative over $\mathsf{HA}$. This is due to Beeson. In this article, we modify Goodman realizability and provide a new proof of the extensional case.

The Adams Thesis holds for a conditional $\to$ and a probability assignment $P$ if and only if $P(A\to B)=P(B\mid A)$ whenever $P\left(A\right)>0$. The restriction ensures that $P(B\mid A)$ is well defined by the classical formula $P(B\mid A)=P(B\cap A)/P\left(A\right)$. Drawing on deep results of Maharam on measure algebras, it is shown that, notwithstanding well-known triviality results (Lewis, etc.), any probability space can be extended to a probability space with a new conditional satisfying the Adams Thesis and satisfying a number of axioms for conditionals. This puts significant limits on how far triviality results can go.