Axiom Of Choice Research Articles

Quantum Computer: Quantum Model and Reality

Any computer can create a model of reality. The hypothesis that quantum computer can generate such a model designated as quantum, which coincides with the modeled reality, is discussed. Its reasons are the theorems about the absence of “hidden variables” in quantum mechanics. The quantum modeling requires the axiom of choice. The following conclusions are deduced from the hypothesis. A quantum model unlike a classical model can coincide with reality. Reality can be interpreted as a quantum computer. The physical processes represent computations of the quantum computer. Quantum information is the real fundament of the world. The conception of quantum computer unifies physics and mathematics and thus the material and the ideal world. Quantum computer is a non-Turing machine in principle. Any quantum computing can be interpreted as an infinite classical computational process of a Turing machine. Quantum computer introduces the notion of “actually infinite computational process”. The discussed hypothesis is consistent with all quantum mechanics. The conclusions address a form of neo-Pythagoreanism: Unifying the mathematical and physical, quantum computer is situated in an intermediate domain of their mutual transformation.

Two Strategies to Infinity: Completeness and Incompleteness. The Completeness of Quantum Mechanics

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse 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.

Parametrized Measuring and Club Guessing

We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\omega_1$ is measured by some club subset of $\omega_1$. The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles. Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $\sigma$-centered forcings, as well as from BPFA. We also consider strong versions of Measuring in the absence of the Axiom of Choice.

The Frontier of Time: The Concept of Quantum Information

The concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information.

Cyclic Mechanics: the Principle of Cyclicity

Cyclic mechanic is intended as a suitable generalization both of quantum mechanics and general relativity apt to unify them. It is founded on a few principles, which can be enumerated approximately as follows: 1. Actual infinity or the universe can be considered as a physical and experimentally verifiable entity. It allows of mechanical motion to exist. 2. A new law of conservation has to be involved to generalize and comprise the separate laws of conservation of classical and relativistic mechanics, and especially that of conservation of energy: This is the conservation of action or information. 3. Time is not a uniformly flowing time in general. It can have some speed, acceleration, more than one dimension, to be discrete. 4. The following principle of cyclicity: The universe returns in any point of it. The return can be only kinematic, i.e. per a unit of energy (or mass), and thermodynamic, i.e. considering the universe as a thermodynamic whole. 5. The kinematic return, which is per a unit of energy (or mass), is the counterpart of conservation of energy, which can be interpreted as the particular case of conservation of action “per a unit of time”. The kinematic return per a unit of energy (or mass) can be interpreted in turn as another particular law of conservation in the framework of conservation of action (or information), namely conservation of wave period (or time). These two counterpart laws of conservation correspond exactly to the particle “half” and to the wave “half” of wave-particle duality. 6. The principle of quantum invariance is introduced. It means that all physical laws have to be invariant to discrete and continuous (smooth) morphisms (motions) or mathematically, to the axiom of choice. The list is not intended to be exhausted or disjunctive, but only to give an introductory idea.

Two deductions: (1) from the totality to quantum information conservation; (2) from the latter to dark matter and dark energy

The paper discusses the origin of dark matter and dark energy from the concepts of time and the totality in the final analysis. Though both, and especially the latter, seem to be rather philosophical, nonetheless they are postulated axiomatically and interpreted physically, and the corresponding philosophical transcendentalism serves heuristically. The exposition of the article means to outline the “forest for the trees”, however, in an absolutely rigorous mathematical way, which to be explicated in detail in a future paper. The “two deductions” are two successive stage of a single conclusion mentioned above. The concept of “transcendental invariance” meaning ontologically and physically interpreting the mathematical equivalence of the axiom of choice and the well-ordering “theorem” is utilized again. Then, time arrow is a corollary from that transcendental invariance, and in turn, it implies quantum information conservation as the Noether correlate of the linear “increase of time” after time arrow. Quantum information conservation implies a few fundamental corollaries such as the “conservation of energy conservation” in quantum mechanics from reasons quite different from those in classical mechanics and physics as well as the “absence of hidden variables” (versus Einstein’s conjecture) in it. However, the paper is concentrated only into the inference of another corollary from quantum information conservation, namely, dark matter and dark energy being due to entanglement, and thus and in the final analysis, to the conservation of quantum information, however observed experimentally only on the “cognitive screen” of “Mach’s principle” in Einstein’s general relativity therefore excluding any other source of gravitational field than mass and gravity. Then, if quantum information by itself would generate a certain nonzero gravitational field, it will be depicted on the same screen as certain masses and energies distributed in space-time, and most presumably, observable as those dark energy and dark matter predominating in the universe as about 96% of its energy and matter quite unexpectedly for physics and the scientific worldview nowadays. Besides on the cognitive screen of general relativity, entanglement is available necessarily on still one “cognitive screen” (namely, that of quantum mechanics), being furthermore “flat”. Most probably, that projection is confinement, a mysterious and ad hoc added interaction along with the fundamental tree ones of the Standard model being even inconsistent to them conceptually, as far as it need differ the local space from the global space being definable only as a relation between them (similar to entanglement). So, entanglement is able to link the gravity of general relativity to the confinement of the Standard model as its projections of the “cognitive screens” of those two fundamental physical theories.

A Formal System of Axiomatic Set Theory in Coq

Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal system of axiomatic set theory based on the Coq proof assistant. The axiomatic system used in the formal system refers to Morse-Kelley set theory which is a relatively complete and concise axiomatic set theory. In this formal system, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano’s postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schroeder-Bernstein theorem. The whole formalization of the system includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “axiomatic set theory” formal system is free from the more apparent paradoxes, and a complete axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

Representation and Reality by Language: How to Make a Home Quantum Computer?

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced.

God, Logic, and Quantum Information

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latters. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis.

Some remarks about nonstandard methods in analysis. I

This and forthcoming articles discuss two of the most known nonstandard methods of analysis---the Robinsons infinitesimal analysis and the Boolean valued analysis, the history of their origination, common features, differences, applications and prospects. This article contains a review of infinitesimal analysis and the original method of forcing. The presentation is intended for a reader who is familiar only with the most basic concepts of mathematical logic---the language of first-order predicate logic and its interpretations. It is also desirable to have some idea of the formal proofs and the Zermelo--Fraenkel axiomatics of the set theory. In presenting the infinitesimal analysis, special attention is paid to formalizing the sentences of ordinary mathematics in a first-order language for a superstructure. The presentation of the forcing method is preceded by a brief review of C.Godels result on the compatibility of the Axiom of Choice and the Continuum Hypothesis with Zermelo--Fraenkels axiomatics. The forthcoming article is devoted to Boolean valued models and to the Boolean valued analysis, with particular attention to the history of its origination.

Infinite Hausdorff spaces may lack cellular families or discrete subsets of cardinality ℵ0

In set theory without the Axiom of Choice (AC), we investigate the deductive strength of the statements “every infinite Hausdorff space has a countably infinite cellular family” and “every infinite Hausdorff space has a countably infinite relatively discrete subspace”, and of variants of the above statements for certain classes of Hausdorff spaces, their mutual relationship, as well as their relationship with various weak choice principles.Among several results, we construct a model N of ZF (i.e., of Zermelo–Fraenkel set theory without AC) in which there exists a dense-in-itself, zero-dimensional, Hausdorff topology on ω such that c(ω)=s(ω)=ω (where c(ω) and s(ω) are the cellularity and the spread of ω, respectively), but ω has no infinite discrete subsets in N, and thus neither has infinite cellular families in N.

REALIZING REALIZABILITY RESULTS WITH CLASSICAL CONSTRUCTIONS

AbstractJ. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.

Homotopy type-theoretic interpretations of constructive set theories

AbstractWe introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

Juhász’s topological generalization of Neumer’s theorem may fail in 𝖹𝖥

In set theory without the Axiom of Choice ( A C \mathsf {AC} ), we investigate the open problem of the deductive strength of Juhász’s topological generalization of Neumer’s Theorem from his paper On Neumer’s Theorem [Proc. Amer. Math. Soc. 54 (1976), 453–454]. Among other results, we show that Juhász’s Theorem is deducible from the Principle of Dependent Choices and (when restricted to the class of T 1 T_{1} spaces) implies the Axiom of Countable Multiple Choice, and hence implies van Douwen’s Countable Choice Principle, but does not imply either the full van Douwen’s Choice Principle or the axiom of choice for linearly ordered families of nonempty finite sets. Furthermore, we prove that Juhász’s Theorem (for T 1 T_{1} spaces) implies each of the following principles: “ ℵ 1 \aleph _{1} is a regular cardinal”, “every infinite set is weakly Dedekind-infinite”, and “every infinite linearly ordered set is Dedekind-infinite”. We also establish that Juhász’s Theorem for T 2 T_{2} spaces is not provable in Z F \mathsf {ZF} . In contrast to the above results, we show that Neumer’s Theorem and Juhász’s Theorem for compact T 1 T_{1} spaces are both provable in Z F \mathsf {ZF} .

Uncountable dichromatic number without short directed cycles

AbstractHajnal and Erdős proved that a graph with uncountable chromatic number cannot avoid short cycles, it must contain, for example, (among other obligatory subgraphs). It was shown recently by Soukup that, in contrast of the undirected case, it is consistent that for any there exists an uncountably dichromatic digraph without directed cycles shorter than . He asked if it is provable already in ZFC (i.e., Zermelo ‐Fraenkel set theory with the Axiom of choice). We answer his question positively by constructing for every infinite cardinal and a digraph of size with dichromatic number at least without directed cycles of length less than .

Determinate logic and the Axiom of Choice

Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice.We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite arity from the language of Takeuti's proof system, then cut elimination is equivalent to a determinacy hypothesis provable, e.g., in ZFC + “there are infinitely many Woodin cardinals.” A slight extension of the proof system admits cut elimination for countable sequents under the same assumptions.A simple modification of the proof system yields analogs of the results above for the Axiom of Real Determinacy and uncountably many Woodin cardinals.

Spectra of uniformity

We study some limitations and possible occurrences of uniform ultrafilters on ordinals without the axiom of choice. We prove an Easton-like theorem about the possible spectrum of successors of regular cardinals which carry uniform ultrafilters; we also show that this spectrum is not necessarily closed.

The axiom of determinacy implies dependent choice in mice

AbstractWe show that the Axiom of Dependent Choice, , holds in countably iterable, passive premice constructed over their reals which satisfy the Axiom of Determinacy, , in a background universe. This generalizes an argument of Kechris for using Steel's analysis of scales in mice. In particular, we show that for any and any countable set of reals A so that and , we have that .

Deeply concatenable subgroups might never be free

We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer–Specker group $\mathbb{Z}^{\omega}$ and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented.

Łoś's theorem and the axiom of choice

AbstractIn set theory without the Axiom of Choice (), we investigate the problem of the placement of Łoś's Theorem () in the hierarchy of weak choice principles, and answer several open questions from the book Consequences of the Axiom of Choice by Howard and Rubin, as well as an open question by Brunner. We prove a number of results summarised in § 3.