Related papers: Debunking Cantor: New Set-Theoretical and Logical …
We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings…
In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald's constant term identities admit an extra set of free parameters, thereby…
We discuss two main ways in comparing and evaluating the size of sets: the "Cantorian" way, grounded on the so called Hume principle (two sets have equal size if they are equipotent), and the "Euclidean" way, maintaining Euclid's principle…
We prove an analogue of Morley's categoricity theorem where cardinality is replaced by the recursion-theoretic notion of arithmetic degree. We say that a complete arithmetically definable theory $T$ is $D$-categorical if any two…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
Constructor theory seeks to express all fundamental scientific theories in terms of a dichotomy between possible and impossible physical transformations - those that can be caused to happen and those that cannot. This is a departure from…
This paper examines a denumerable version of the nested-set theorem and derives from it a contradiction involving the formal consistency of the actual infinity assumed by the Axiom of Infinity.
All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…
The formulation of a new analysis on a zero measure Cantor set $C (\subset I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$ exploiting the concept of {\em relative} infinitesimals and a scale invariant…
We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the…
We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections).…
The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the…
A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a…
Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…
Following K. Mahler's suggestion for further research on intrinsic approximation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets of rational iterated function systems. We further investigate the behavior of…
We specialise a recently introduced notion of generalised dinaturality for functors $T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$ to the case where the domain (resp., codomain) is constant, obtaining notions of ends…
This text gives a rough, but linear summary covering some key definitions, notations, and propositions from Lambda Calculus: Its Syntax and Semantics, the classical monograph by Barendregt. First, we define a theory of untyped extensional…
The article is devoted to modeling of the nega-$\tilde Q$-representation of real numbers. The representation is a generalization of representation by alternating Cantor series and positive $\tilde Q$-representation is a generalization of…
We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with $\Delta_0$-induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano…
Let $T$ be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that $T$ has continuum-many countable models. The proof is purely first-order, but raises the question of Borel completeness of…