The homeomorphism of Minkowski space and the separable complex Hilbert space: the physical, mathematical and philosophical interpretations

A homeomorphism is built between the separable complex Hilbert space and Minkowski space by meditation of quantum information. That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering. Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy.

Quantum information as the information of infinite collections or series

The quantum information introduced by quantum mechanics is equivalent to a certain generalization of classical information: from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The “qubit”, can be interpreted as that generalization of “bit”, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the transfinite ordinal number corresponding to the infinity series in question. The transfinite ordinal numbers can be defined as ambiguously corresponding “transfinite natural numbers” generalizing the natural numbers of Peano arithmetic to “Hilbert arithmetic” allowing for the unification of the foundations of mathematics and quantum mechanics.

An Algorithm for Linearizing the Collatz Convergence

The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.

Several results on compact metrizable spaces in $$\mathbf {ZF}$$

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

The Brain and the New Foundations of Mathematics

Many concepts in mathematics are not fully defined, and their properties are implicit, which leads to paradoxes. New foundations of mathematics were formulated based on the concept of innate programs of behavior and thinking. The basic axiom of mathematics is proposed, according to which any mathematical object has a physical carrier. This carrier can store and process only a finite amount of information. As a result of the D-procedure (encoding of any mathematical objects and operations on them in the form of qubits), a mathematical object is digitized. As a consequence, the basis of mathematics is the interaction of brain qubits, which can only implement arithmetic operations on numbers. A proof in mathematics is an algorithm for finding the correct statement from a list of already-existing statements. Some mathematical paradoxes (e.g., Banach–Tarski and Russell) and Smale’s 18th problem are solved by means of the D-procedure. The axiom of choice is a consequence of the equivalence of physical states, the choice among which can be made randomly. The proposed mathematics is constructive in the sense that any mathematical object exists if it is physically realized. The consistency of mathematics is due to directed evolution, which results in effective structures. Computing with qubits is based on the nontrivial quantum effects of biologically important molecules in neurons and the brain.

Set Theory

The chapter is a concise, practical presentation of the basics of set theory. The topics include set equivalence, countability, partially ordered, linearly ordered, and well-ordered sets, the axiom of choice, and Zorn’s lemma, as well as cardinal numbers and cardinal arithmetic. The first two sections are essential for a proper understanding of the rest of the book. In particular, a thorough understanding of countability and Zorn’s lemma are indispensable. Parts of the section on cardinal numbers may be included, but only an intuitive understanding of cardinal numbers is sufficient to follow such topics as the discussion on the existence of a vector space of arbitrary (infinite) dimension, and the existence of inseparable Hilbert spaces. Cardinal arithmetic can be omitted since its applications later in the book are limited. Ordinal numbers have been carefully avoided.

The Axiom of Choice: The Last Great Controversy in Mathematics

This paper explores multiple perspectives and proofs regarding the validity of what is considered to be one of the – if not THE - most controversial proofs in the field of mathematics, historical and contemporary applications alike. We consider the various explanations and equivalents of the axiom along with the more widely receptive alternatives. Furthermore, we review the resistance the axiom has encountered in various fields and its potential use in the research and expansion of said fields.