Related papers: Unprovability in Mathematics: A First Course on Or…
This paper examines the application of Tarski's Undefinability Theorem to first-order arithmetic. The generally accepted view is that for this case the Theorem establishes that arithmetic truth is not arithmetic. A careful examination of…
The arithmetic of the natural numbers can be extended to arithmetic operations on planar binary trees. This gives rise to a non-commutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and…
Harvey Friedman's gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the…
To investigate objects without a describable notion of distance, one can gather ordinal information by asking triplet comparisons of the form "Is object $x$ closer to $y$ or is $x$ closer to $z$?" In order to learn from such data, the…
`What more than its truth do we know if we have a proof of a theorem in a given formal system?' We examine Kreisel's question in the particular context of program termination proofs, with an eye to deriving complexity bounds on program…
We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A…
Many data management applications must deal with data which is uncertain, incomplete, or noisy. However, on existing uncertain data representations, we cannot tractably perform the important query evaluation tasks of determining query…
The notion of clause set cycle abstracts a family of methods for automated inductive theorem proving based on the detection of cyclic dependencies between clause sets. By discerning the underlying logical features of clause set cycles, we…
This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the L\"owenheim-Skolem theorem, Craig interpolation, Beth's…
This work addresses the problem of computing measures of recognisable sets of infinite trees. An algorithm is provided to compute the probability measure of a tree language recognisable by a weak alternating automaton, or equivalently…
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is…
We introduce ordinal collapsing principles that are inspired by proof theory but have a set theoretic flavor. These principles are shown to be equivalent to iterated $\Pi^1_1$-comprehension and the existence of admissible sets, over weak…
An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…
We propose an extension of Poole's independent choice logic based on a relaxation of the underlying independence assumptions. A credal semantics involving multiple joint probability mass functions over the possible worlds is adopted. This…
We present a version of G\"odel's Second Incompleteness Theorem for recursively enumerable consistent extensions of a fixed axiomatizable theory, by incorporating some bi-theoretic version of the derivability conditions. We also argue that…
This paper presents a complete axiomatization of Monadic Second-Order Logic (MSO) over infinite trees. MSO on infinite trees is a rich system, and its decidability ("Rabin's Tree Theorem") is one of the most powerful known results…
We introduce the split principles and show that they bear tight connections to large cardinal properties such as inaccessibility, weak compactness, subtlety, almost ineffability and ineffability, as well as classical combinatorial objects…
The minimal pairs paradigm of comparing model probabilities for contrasting completions has proven useful for evaluating linguistic knowledge in language models, yet its application has largely been confined to binary grammaticality…
We consider the satisfiability problem for the two-variable fragment of first-order logic over finite unranked trees. We work with signatures consisting of some unary predicates and the binary navigational predicates child, right sibling,…
We establish an explicit lower bound for Kruskal's weak tree function at n=3, proving that tree(3) >= 844,424,930,131,960 = 3 * 2^48 - 8. This is achieved by constructing an explicit sequence of unlabeled rooted trees satisfying the…