Related papers: Comparing Hume's Principle, Basic Law V and Peano …
Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law…
Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume's Principle is analytic then in the standard setting the answer appears to be yes. Hodes's work pointed to a way out by…
Akama et al. systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification…
Induction is typically formalized as a rule or axiom extension of the LK-calculus. While this extension of the sequent calculus is simple and elegant, proof transformation and analysis can be quite difficult. Theories with an induction…
Throughout the course of mathematical history, generalizations of previously understood concepts and structures have led to the fruitful development of the hierarchy of number systems, non-euclidean geometry, and many other epochal phases…
General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might…
Separation logic is successful for software verification of heap-manipulating programs. Numbers are necessary to be added to separation logic for verification of practical software where numbers are important. However, properties of the…
We study reflection principles of Peano Arithmetic PA which are based on both proof and provability. Any such reflection principle in PA is equivalent to either $\Box P\!\rightarrow\! P$ ($\Box P$ stands for `$P$ is provable') or $\Box^k…
The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic…
In \cite{MacResField} the second author gave a systematic analysis of definability and decidability for rings $\mathcal M/p\mathcal M$, where $\mathcal M$ is a model of Peano Arithmetic and $p$ is a prime in $\mathcal M$. In the present…
We present an alternative cyclic proof system for Peano arithmetic that could be simpler than the existing ones and well-adapted both for proof analysis and for automatizing inductive proof search. In addition, we will show how various…
We describe a "slow" version of the hierarchy of uniform reflection principles over Peano Arithmetic ($\mathbf{PA}$). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower…
The general completeness problem of Hoare logic relative to the standard model $N$ of Peano arithmetic has been studied by Cook, and it allows for the use of arbitrary arithmetical formulas as assertions. In practice, the assertions would…
Higher Type Arithmetic (HA$^w$) is a first-order many-sorted theory. It is a conservative extension of Heyting Arithmetic obtained by extending the syntax of terms to all of System T: the objects of interest here are the functionals of…
In introductory books about natural numbers, a common kind of assertion - often left as exercise to the reader - is that certain forms of induction on $\mathbb{N}$ (regular/ordinary, complete/strong) are equivalent one to each other and to…
Frege's definition of the real numbers, as envisaged in the second volume of \textit{Grundgesetze der Arithmetik}, is fatally flawed by the inconsistency of Frege's ill-fated \textit{Basic Law V}. We restate Frege's definition in a…
Peano Arithmetic is known to be provably equivalent to reflection over Elementary Arithmetic. We prove a characterization of Predicative Analysis in the guise of ATR0 in terms of stronger reflection principles.
In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or…
Hoare's logic is an axiomatic system of proving programs correct, which has been extended to be a separation logic to reason about mutable heap structure. We develop the most fundamental logical structure of strongest postcondition of…
In a recent paper, Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic. THEOREM: The first-order theories of Peano arithmetic and ZF with the axiom of infinity negated are…