Related papers: Notes on bounded induction for the compositional t…
We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_{1}, x_{2}) is representable in the first-order Peano Arithmetic PA by a formula [F(x_{1}, x_{2}, x_{3})] which is algorithmically verifiable,…
Classically, any structure for a signature $\Sigma$ may be completed to a model of a desired regular theory $T$ by means of the chase construction or small object argument. Moreover, this exhibits $\mathrm{Mod}(T)$ as weakly reflective in…
Let $\widetilde{\mathbb{Q}_p}$ be the field of $p$-adic numbers in the language of rings. In this paper we consider the theory of $\widetilde{\mathbb{Q}_p}$ expanded by two predicates interpreted by multiplicative subgroups…
The class forcing theorem, which asserts that every class forcing notion $\mathbb{P}$ admits a forcing relation $\Vdash_{\mathbb{P}}$, that is, a relation satisfying the forcing relation recursion -- it follows that statements true in the…
In this paper a class of languages which are formal enough for mathematical reasoning is introduced. First-order formal languages containing natural numbers and numerals belong to that class. Its languages are called mathematically…
In this paper we will see deductive systems for classical propositional and predicate logic in the calculus of structures. Like sequent systems, they have a cut rule which is admissible. In addition, they enjoy a top-down symmetry and some…
Questions concerning the proof-theoretic strength of classical versus non-classical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of…
The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional…
Dependence logic provides an elegant approach for introducing dependencies between variables into the object language of first-order logic. In [1] generalized quantifiers were introduced in this context. However, a satisfactory account was…
We construct here an iterative evaluation of all PR map codes: progress of this iteration is measured by descending complexity within "Ordinal" O := N[\omega] of polynomials in one indeterminate, ordered lexicographically. Non-infinit…
We prove that, over Kripke-Platek set theory with infinity (KP), transfinite induction along the ordinal ${\epsilon}_{\Omega+1}$ is equivalent to the schema asserting the soundness of KP, where $\Omega$ denotes the supremum of all ordinals…
We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of…
We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is…
In this paper, we give two proofs of the wellfoundedness of recursive notation systems for $\Pi_N$-reflecting ordinals. One is based on $\Pi_{N-1}^0$-inductive definitions, and the other is based on distinguished classes.
Tarski's undefinability theorem states that a formal system based on conventional predicate logic (PL) cannot talk about its own truth predicate. PL is, however, not the only formal language imaginable. In this paper, it will be shown that…
We present a propositional logic with fundamental probabilistic semantics, in which each formula is given a real measure in the interval $[0,1]$ that represents its degree of truth. This semantics replaces the binarity of classical logic,…
We formulate a framework for describing behaviour of effectful higher-order recursive programs. Examples of effects are implemented using effect operations, and include: execution cost, nondeterminism, global store and interaction with a…
For a class $\Gamma$ of formulas, $\Gamma$ local reflection principle $\mathrm{Rfn}_{\Gamma}(T)$ for a theory $T$ of arithmetic is a scheme formalizing the $\Gamma$-soundness of $T$. Beklemishev proved that for every $\Gamma \in \{\Sigma_n,…
We present two new constructions of satisfaction/truth classes over models of PA (Peano Arithmetic) that provide a foil to the fact that the existence of a disjunctively correct full truth class over a model M of PA implies that Con(PA)…
Coinduction refers to both a technique for the definition of infinite streams, so-called codata, and a technique for proving the equality of coinductively specified codata. This article first reviews coinduction in declarative programming.…