相关论文: On Cantor's Theorem

Proofs of Tychonoff's theorem often seem to require a bit of magic. Machinery such as ultrafilters, nets or maximal families with the finite intersection property are employed to give proofs that can be very neat, but not the kind of thing…

Emergence is a pregnant property in various fields. It is the fact for a phenomenon to appear surprisingly and to be such that it seems at first sight that it is not possible to predict its apparition. That is the reason why it has often…

Quantum mechanics marks a radical departure from the classical understanding of Nature, fostering an inherent randomness which forbids a deterministic description; yet the most fundamental departure arises from something different. As shown…

In the several contexts such as combinatorial number theory, families of sets of positive integers closed under taking subsets have been investigated. Then it is sometimes useful to give bijections between the set of the one-sided infinite…

Human agents happen to judge that a conjunction of two terms is more probable than one of the terms, in contradiction with the rules of classical probabilities---this is the conjunction fallacy. One of the most discussed accounts of this…

In a recent paper, Enayat and Le lyk [2024] show that second order arithmetic and countable set theory are not definitionally equivalent. It is well known that these theories are biinterpretable. Thus, we have a pair of natural theories…

In this paper, we introduce a notion of quantum discrepancy, a non-commutative version of combinatorial discrepancy which is defined for projection systems, i.e. finite sets of orthogonal projections, as non-commutative counterparts of set…

If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible…

Optics, aka functional references, are classes of tools that allow composable access into compound data structures. Usually defined as programming language libraries, they provide combinators to manipulate different shapes of data such as…

Agents' judgment depends on perception and previous knowledge. Assuming that previous knowledge depends on perception, we can say that judgment depends on perception. So, if judgment depends on perception, can agents judge that they have…

Quantum theory demands that, in contrast to classical physics, not all properties can be simultaneously well defined. The Heisenberg Uncertainty Principle is a manifestation of this fact. Another important corollary arises that there can be…

Generalized uncertainty relations may depend not only on the commutator relation of two observables considered, but also on mutual correlations, in particular, on entanglement. The equivalence between the uncertainty relation and Bohr's…

Librationist set theory \pounds ${}$ is developed. It descends from semantics for truth, initiated by Kripke, and others. # extends \pounds, of Librationist closures of the paradoxes in Logic and Logical Philosophy 21(4), 323-361, 2012.…

The independence of the continuum hypothesis is a result of broad impact: it settles a basic question regarding the nature of N and R, two of the most familiar mathematical structures; it introduces the method of forcing that has become the…

We discuss the idea that superpositions in quantum mechanics may involve contradictions or contradictory properties. A state of superposition such as the one comprised in the famous Schr\"odinger's cat, for instance, is sometimes said to…

There has been a considerable amount of work on uncertainty in knowledge-based systems. This work has generally been concerned with uncertainty arising from the strength of inferences and the weight of evidence. In this paper we discuss…

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions…

We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability…

We construct a direct natural bijection between descending plane partitions without any special part and permutations. The directness is in the sense that the bijection avoids any reference to nonintersecting lattice paths. The advantage of…

Recently, it has been emphasized that the possibility theory framework allows us to distinguish between i) what is possible because it is not ruled out by the available knowledge, and ii) what is possible for sure. This distinction may be…