arXiv subjects

Marcoen J. T. F. Cabbolet

Publications and source records attributed to Marcoen J. T. F. Cabbolet.

14 recordsLinked to original sources

arXiv2026

Relative consistency of a finite nonclassical theory incorporating ZF and category theory with ZF

Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker than ZF, that is finitely axiomatized, and that does not have a countable model (if it has a model at all, that is). Here we prove that T is relatively consistent with ZF. We conclude that this is an important step towards showing that T is an advancement in the foundations of mathematics.

math.GM↗

arXiv2025

A paradox in the quantum-mechanical treatment of destructive measurements on photons

Measurements on photons are frequently cited as confirmations of predictions of quantum mechanics (QM), in particular in the context of Bell's theorem. In this paper we show, however, that we cannot ever claim to have measured a property of a photon if we treat a destructive measurement {of the value of a property of a photon prepared in a superposition of eigenstates} in the framework of orthodox QM.

quant-ph↗

arXiv2024

Addendum to `On the nonreality of the PBR theorem': disproof by generic counterexample

The PBR theorem is widely seen as one of the most important no-go theorems in the foundations of quantum mechanics. Recently, in Found. Phys. 53(3): 64 (2023), it has been argued that there is no reality to the PBR theorem using a pair of bolts as a counterexample. In this addendum we expand on the argument: we disprove the PBR theorem by a generic counterexample, and we put the finger on the precise spot where Pusey, Barrett, and Rudolph have made a tacit assumption that is false.

physics.gen-ph↗

arXiv2024

Relative consistency of Set Matrix Theory with ZF

Set Matrix Theory (SMT) has been introduced in Log. Anal. 225: 59-82 (2014) as a generalization of ZF, in which matrices constructed from sets are treated as urelements, that is, as objects that are not sets but that can be elements of sets. Here we prove that SMT is relatively consistent with ZF.

math.LO↗

arXiv2023

On the nonreality of the PBR theorem

The PBR theorem has been hailed as one of the most important theorems in the foundations of quantum mechanics (QM), , cf. E. Samuel Reich, "Quantum theorem shakes foundations", Nature (2011). Here we argue that the special measurement, used by Pusey et al. to derive the theorem, is nonexisting from the Einsteinian view on QM.

physics.gen-ph↗

arXiv2021

Hyperreal delta functions as a new general tool for modeling physical states with infinitely high densities

This paper introduces the expanded real numbers as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and defines the set of all integrable functions from the real numbers to the expanded real numbers. This allows to identify the Dirac delta with a special hyperreal-valued function of a real variable: the Dirac delta function thus defined is a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.

math.GM↗

arXiv2021

A marriage of category theory and set theory: a finitely axiomatized nonstandard first-order theory implying ZF

It is well known that ZFC, despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim-Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a nonstandard first-order language with countable many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom scheme, and it is shown that it does not have a countable model--if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

math.GM↗

arXiv2019

On the fundamentally probabilistic nature of the knowledge of a microsystem in the framework of the Elementary Process Theory

A fundamental question in the debate about the interpretations of quantum mechanics (QM) is whether the universe is fundamentally deterministic or fundamentally probabilistic. This self-contained paper shows for a microsystem made up of a single neutron that is initially at rest in a stationary force-free environment, that even if the individual processes by which the microsystem evolves are fundamentally deterministic as described by a strictly deterministic model of the Elementary Process Theory (EPT), then still our most precise knowledge of the outcome of a position measurement on the microsystem is fundamentally probabilistic. Generalizing, the conclusion is that the EPT is inconsistent with orthodox QM, but consistent with $ψ$-epistemic QM.

physics.gen-ph↗

arXiv2018

Comment to 'On the reality of the quantum state'

In Nature Phys. 8:475-478 (2012), Pusey, Barrett, and Rudolph have claimed that ψ-epistemic quantum mechanics (QM) is inconsistent with predictions of standard QM. Here we show that Pusey et al. err in the beginning of their argument by making an assumption that violates the Projection Postulate.

physics.gen-ph↗

arXiv2018

A category mistake in observational claims regarding ultrashort-lived unstable particles

The physics literature contains many claims that ultrashort-lived unstable particles, such as a Higgs boson, have been observed. These claims are a matter of applying the $5σ$-convention in particle physics. This paper, however, shows that by applying this $5σ$-convention a category mistake is made, by which a pure reasoning is passed off as an observation. Not only are these two fundamentally different primitive notions at the very basis of science, but the pure reasoning in question is also weaker than an observation: what we have in each case is that the existence of the ultrashort-lived unstable particle is inferred to the best explanation, but that does absolutely not merit the stronger claim that the particle in question has been "observed". Consequently, the observational claims in question will thus have to be dismissed as overstatements. On a general note, this demonstrates that the empirical support for the Standard Model of particle physics is significantly less than hitherto thought.

physics.gen-ph↗

arXiv2015

The Importance of Developing a Foundation for Naive Category Theory

Recently Feferman (Rev. Symb. Logic 6: 6-15, 2013) has outlined a program for the development of a foundation for naive category theory. While Ernst (ibid. 8: 306-327, 2015) has shown that the resulting axiomatic system is still inconsistent, the purpose of this note is to show that nevertheless some foundation has to be developed before naive category theory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive category theory currently a 'cookbook recipe' is used for constructing categories, and it is explicitly shown with a formalized argument that this 'foundationless' naive category theory therefore contains a paradox similar to the Russell paradox of naive set theory.

math.GM↗

arXiv2015

Berkelian Idealism Regarding Properties in Orthodox Quantum Mechanics, and Implications for Quantum Gravity

Referring to the 18th century idealism of George Berkeley in which an object exists if and only if it is observed, this note shows that orthodox quantum mechanics (OQM) entails a Berkelian idealism regarding properties (BIRP): a quantum `has' a property X with quantitative value x if and only if the property X has just been measured with outcome x. It is then impossible to recontextualize GR's principle of curvature in any quantum framework that implies this BIRP, for a quantum cannot curve space-time if it doesn't have a definite energy - which is supposed to be the cause of curvature - in absence of observation to begin with. Concluding, it is ruled out that a quantum theory of gravity, in which GR's principle of curvature is built in as a fundamental physical principle, can be developed in any framework implying this BIRP.

physics.gen-ph↗

arXiv2013

Set Matrix Theory as a Physically Motivated Generalization of Zermelo-Fraenkel Set Theory

Recently, the Elementary Process Theory (EPT) has been developed as a set of fundamental principles that might underlie a gravitational repulsion of matter and antimatter. This paper presents set matrix theory (SMT) as the foundation of the mathematical-logical framework in which the EPT has been formalized: Zermelo-Fraenkel set theory (ZF), namely, cannot be used as such. SMT is a generalization of ZF: whereas ZF uses only sets as primitive objects, in the framework of SMT finite matrices with set-valued entries are objects sui generis, with a one-by-one set matrix [x] being identical to the set x. It is proved that every set that can be constructed in ZF can also be constructed in SMT: as a mathematical foundation, SMT is thus not weaker than ZF. In addition, it is shown that SMT is more suitable han ZF for the intended application to physics. The conclusion is that SMT, contrary to ZF, is acceptable as the mathematical-logical foundation of the framework for physics that is determined by the EPT.

math.LO↗

arXiv2011

Comment to a paper [arXiv:1103.4937] of M. Villata on antigravity

In a recent paper of M. Villata [arXiv:1103.4937], it is claimed that "antigravity appears as a prediction of general relativity when CPT is applied." However, the present paper argues that Villata puts the cart before the horse qua methodology, and that the resulting theory cannot be reconciled with the ontological presuppositions of general relativity. The conclusion is that Villata's suggestion for the physics that might underlie a gravitational repulsion of matter and antimatter is not acceptable in its current state of development.

physics.gen-ph↗