The Completeness: From Henkin's Proposition to Quantum Computer

The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all infinite sets. However, only Henkin's proposition equivalent to an internal position to infinity is consistent . This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or " observer'' , or "user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint

Paradoxes of Demonstrability

In this paper I consider two paradoxes that arise in connection with the concept of demonstrability, or absolute provability. I assume—for the sake of the argument—that there is an intuitive notion of demonstrability, which should not be conflated with the concept of formal deducibility in a (formal) system or the relativized concept of provability from certain axioms. Demonstrability is an epistemic concept: the rough idea is that a sentence is demonstrable if it is provable from knowable basic (“self-evident”) premises by means of simple logical steps. A statement that is demonstrable is also knowable and a statement that is actually demonstrated is known to be true. By casting doubt upon apparently central principles governing the concept of demonstrability, the paradoxes of demonstrability presented here tend to undermine the concept itself—or at least our understanding of it. As long as we cannot find a diagnosis and a cure for the paradoxes, it seems that the coherence of the concepts of demonstrability and demonstrable knowledge are put in question. There are of course ways of putting the paradoxes in quarantine, for example by imposing a hierarchy of languages a` la Tarski, or a ramified hierarchy of propositions and propositional functions a` la Russell. These measures, however, helpful as they may be in avoiding contradictions, do not seem to solve the underlying conceptual problems. Although structurally similar to the semantic paradoxes, the paradoxes discussed in this paper involve epistemic notions: “demonstrability”, “knowability”, “knowledge”... These notions are “factive” (e.g., if A is demonstrable, then A is true), but similar paradoxes arise in connection with “nonfactive” notions like “believes”, “says”, “asserts”.3 There is no consensus in the literature concerning the analysis of the notions involved—often referred to as “propositional attitudes”—or concerning the treatment of the paradoxes they give rise to

Reflection using the derivability conditions

Reflection principles are a way to build non-conservative true extensions of a theory. However the application of a reflection principle needs a proof predicate, and the effort needed to provide this is so great as to be not really practical. We look at a possible way to avoid this effort by using, instead of a proof predicate, a predicate defined using only necessary `modal' properties. Surprisingly, we can produce powerful non-conservative extensions this way. But a reflection principle based on such a predicate is essentially weaker, and we also consider its limitations

The Completeness: From Henkin’s Proposition to Quantum Computer

The Completeness: From Henkin’s Proposition to Quantum Compute

Title redacted for blind review

This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the axioms of a dynamic provability logic, which embeds GL within the modal $\mu$-calculus. Via correspondence results between modal logic and the bisimulation-invariant fragment of second-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the notion of 'intuition-that'. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions and hyperintensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages. Hyperintensionality is countenanced via four models, without a decision as to which model is to be preferred. The first model makes intuition sensitive to hyperintensional topics, i.e. subject matters. The second model is a hyperintensional truthmaker semantics, in particular a novel epistemic two-dimensional truthmaker semantics. The third model is a topic-sensitive non-truthmaker epistemic two-dimensional semantics. The fourth model is a topic-sensitive epistemic two-dimensional truthmaker semantics

Topic-Sensitive Epistemic 2D Truthmaker ZFC and Absolute Decidability

This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance an epistemic two-dimensional truthmaker semantics, if hyperintenisonal approaches are to be preferred to possible worlds semantics. I examine the relation between epistemic truthmakers and epistemic set theory

Modal Logics that Bound the Circumference of Transitive Frames

For each natural number $n$ we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than $n$ and no strictly ascending chains. The case $n=0$ is the G\"odel-L\"ob provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When $n=1$, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to intuitionistic propositional logic. A topological semantic analysis shows that the $n$-th member of the sequence of extensions of S4 is the logic of hereditarily $n+1$-irresolvable spaces when the modality $\Diamond$ is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of $\Diamond$ as the derived set (of limit points) operation. The variety of modal algebras validating the $n$-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by $n$. Moreover each algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them