We give a simple and direct proof that super-consistency implies the cut-elimination property in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut-elimination theorems in higher-order logic that involve V-complexes.

We propose a formal representation of objects, those being mathematical or empirical objects. The powerful framework inside which we represent them in a unique and coherent way is grounded, on the formal side, in a logical approach with a direct mathematical semantics in the well-established field of constructive topology, and, on the philosophical side, in a neo-Kantian perspective emphasizing the knowing subject’s role, which is constructive for the mathematical objects and constitutive for the empirical ones.

There are many classical connections between the proof-theoretic strength of systems of arithmetic and the provable totality of recursive functions. In this paper we study the provability strength of the totality of recursive functions by investigating the degree structure induced by the relative provability order of recursive algorithms. We prove several results about this proof-theoretic degree structure using recursion-theoretic techniques such as diagonalization and the Recursion Theorem.

The implicational fragment of the logic of relevant implication, $R_{\to }$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $L{R}_{\to }$, a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $L{R}_{\to }$, but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$. This calculus, ${\mathit{LT}}_{\to }^{\text{ⓣ}}$, extends the consecution calculus $L{T}_{\to }^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment. We introduce two other new calculi as alternative formulations of $R_{\to }^{\mathbf{t}}$. For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to }^{\mathbf{t}}$. These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to }$, which we present in another paper.

We compare two different notions of generic expansions of countable saturated structures. One kind of genericity is related to existential closure, and another is defined via topological properties and Baire category theory. The second type of genericity was first formulated by Truss for automorphisms. We work with a later generalization, due to Ivanov, to finite tuples of predicates and functions.

Let $N$ be a countable saturated model of some complete theory $T$, and let $(N,\sigma )$ denote an expansion of $N$ to the signature $L_0$ which is a model of some universal theory $T_0$. We prove that when all existentially closed models of $T_0$ have the same existential theory, $(N,\sigma )$ is Truss generic if and only if $(N,\sigma )$ is an e-atomic model. When $T$ is $\omega$-categorical and $T_0$ has a model companion $T_{\mathrm{mc}}$, the e-atomic models are simply the atomic models of $T_{\mathrm{mc}}$.

The aim of the present paper is to provide a robust classification of valid sentences in set theory by means of existence and related notions and, in this way, to capture similarities and dissimilarities among the axioms of set theory. In order to achieve this, precise definitions for the notions of productive and nonproductive assertions, constructive and nonconstructive productive assertions, and conditional and unconditional productive assertions, among others, will be presented. These definitions constitute the result of a semantical analysis of the notions involved. The conceptual clarification developed here results in a classification of valid sentences of set theory that goes against the standard view that extensionality is not an existence assertion.

We show that there are denumerably many Post-complete normal modal logics in the language which includes an additional propositional constant. This contrasts with the case when there is no such constant present, for which it is well known that there are only two such logics.

A construction of the real number system based on almost homomorphisms of the integers $\mathbb{Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).

The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa. A Boffa model is obtained from a model of (a fragment of) Zermelo–Fraenkel with Choice (ZFC) with an automorphism which moves a rank: the domain of the Boffa model is a rank that is moved. “Most” elements of the domain of the Boffa model are urelements in terms of the interpreted NFU. The main result of this paper is that the restriction of the membership relation of the original model of set theory with automorphism to the domain of the Boffa model is first-order definable in the language of NFU. In particular, all information about the extensions in the original model of the urelements of the model of NFU is definable in terms of NFU. A corollary (answering a question of Thomas Forster) is that the urelements in a Boffa model are not homogeneous.

Recently, an improvement in respect of simplicity was found by Rohan French over extant translations faithfully embedding the smallest congruential modal logic (E) in the smallest normal modal logic (K). After some preliminaries, we explore the possibility of further simplifying the translation, with various negative findings (but no positive solution). This line of inquiry leads, via a consideration of one candidate simpler translation whose status was left open earlier, to isolating the concept of a minimally congruential context. This amounts, roughly speaking, to a context exhibiting no logical properties beyond those following from its being congruential (i.e., from its yielding provably equivalent results when provably equivalent formulas are inserted into the context). On investigation, it turns out that a context inducing a translation embedding E faithfully in K need not be minimally congruential in K. Several related minimality conditions are noted in passing, some of them of considerable interest in their own right (in particular, minimal normality). The paper is exploratory, raising more questions than it settles; it ends with a list of open problems.

We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu \left(\varphi \right(x,b\left)\right)=0$ for all $b$, then for some $n$, ${\mu }^{\left(n\right)}\left(\exists y\right(\varphi (x_1,y)\wedge \cdots \wedge \varphi (x_n,y)\left)\right)=0$. As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$, then $X$ is generic if and only if every translate of $X$ does not fork over $\varnothing$, precisely as in stable groups, answering positively an earlier problem posed by the first two authors.