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Papers by Jean-Paul BENTZ

L'invention concerne un maitre-cylindre a reaction hydraulique pour servomoteur pneumatique d... more L'invention concerne un maitre-cylindre a reaction hydraulique pour servomoteur pneumatique d'assistance au freinage, comprenant un piston principal (12) recevant une force d'assistance, et un piston de reaction (3) monte coulissant dans une chambre de reaction (4) du piston principal et recevant une force d'actionnement. Le maitre-cylindre de l'invention comprend egalement un support de siege (7) contre lequel peut s'appliquer le piston de reaction (3) et qui est monte coulissant vers une chambre a basse pression, le piston de reaction (3) et ce support de siege constituant en combinaison un ensemble etage dans la chambre de reaction (3), propre a recevoir, de la part de la pression dans la chambre de reaction (3), une force qui s'ajoute a l'effort de freinage apres application d'un coup de frein brutal.

Every proposition P of every consistent and bivalent theory T meets, in any inclusive, consistent... more Every proposition P of every consistent and bivalent theory T meets, in any inclusive, consistent and bivalent metalanguage of T, the two logical implications :
(I) [P <-> ¬d(P)] -> P, and
(II) P ->[P <-> ¬d(¬P)],
where ¬d(P) and ¬d(¬P) respectively assert that P and ¬P cannot be demonstrated in T. While these two relations are sufficient to readily demonstrate Gödel’s first incompleteness theorem starting from the logical equivalence G <-> ¬d(G) imposed by Gödel’s formula, they correlatively raise several serious issues regarding the meaning and scope of the said theorem. In particular, Gödel's formula G, which is forced to be true by axiom G#¬d(G), by the assumed consistency of arithmetic, and by theorem
[G <-> ¬d(G)] -> G, asserts its own undemonstrability. How could it be demonstrable, then? Can G provide any information about arithmetic, while it does not even provide any information about itself other than that is expressed ab initio by axiom G#¬d(G)?

This article first describes a phenomenological approach used to develop a simple logical system ... more This article first describes a phenomenological approach used to develop a simple logical system complying with the limits and logical constraints of natural language. Then, the resulting 3-valued, modal logic, which, nevertheless, is extensional and encompasses classical binary logic, is successfully applied to the analysis of different well-known paradoxes, including the Liar Paradox and the Surprise Examination Paradox.

Drafts by Jean-Paul BENTZ

This paper proposes an informal model assimilating all axiomatic theory to a deterministic univer... more This paper proposes an informal model assimilating all axiomatic theory to a deterministic universe subject to the principle of causality, and leads to the conclusion that the incompleteness of arithmetic, as stated by Gödel's theorem, is incompatible with this model.

Cet article propose un modèle informel assimilant toute théorie axiomatique à un univers détermin... more Cet article propose un modèle informel assimilant toute théorie axiomatique à un univers déterministe soumis au principe de causalité, et conduit à la conclusion que l'incomplétude de l'arithmétique, telle que posée par le théorème de Gödel, est incompatible avec ce modèle.

This article offers a new reflection on the reasoning followed by Gödel in the proof of the incom... more This article offers a new reflection on the reasoning followed by Gödel in the proof of the incompleteness theorem(s). It takes the form of a short thought experiment using a variant of this theorem, obtained by cloning, and leading to an obviously unacceptable conclusion.

Dans la mesure où aucune des thèses fondées sur des arguments de nature sémantique et invitant à ... more Dans la mesure où aucune des thèses fondées sur des arguments de nature sémantique et invitant à reconsidérer la légitimité du cheminement logique suivi par Gödel dans la démonstration de son théorème d'incomplétude n'a trouvé écho, le présent article aborde le sujet de façon plus simple sous la forme d'une expérience de pensée utilisant une variante de ce théorème obtenue par clonage et aboutissant à une conclusion à l'évidence aussi radicale qu'inacceptable.

Jean-Paul BENTZ, 2022

The notion of instantiated infinity used in Cantor's diagonal argument appears to lead to a serio... more The notion of instantiated infinity used in Cantor's diagonal argument appears to lead to a serious paradox

As exposed in this paper, Cantor's diagonal argument seems to lead to a serious paradox.

The demonstration of Gödel’s 1st Incompleteness Theorem relies on the recourse to a formula G sup... more The demonstration of Gödel’s 1st Incompleteness Theorem relies on the recourse to a formula G supposed to assert its own indemonstrable nature.
However, in order to build the formula G, Gödel had no other option than transgress the law of causality, i.e. the very law which precludes any possibility for an effect to act as its own cause.
The present draft, based on the author’s conviction that this law rules the abstract world with the same rigour as it does the material one, provides a detailed analysis of two lines of reasoning which, in the demonstration, fail to achieve such an unachievable result.

Part II.A of this draft already featured in "Gödel’s First Incompleteness Theorem, gloves off " a... more Part II.A of this draft already featured in "Gödel’s First Incompleteness Theorem, gloves off " and in "Gödel’s First Incompleteness Theorem and the humanly unreachable world". When translated into English, Parts II.B and II.C of this draft will appear in a new draft entitled "Gödel and the Law of Causality".

In the previous article, “Gödel’s First Incompleteness Theorem, gloves off”, it has been shown th... more In the previous article, “Gödel’s First Incompleteness Theorem, gloves off”, it has been shown that all proposition P in all consistent and bivalent axiomatic theory necessarily complies with the following two relations (I) : [P¬d(P)]―>P and (II) : P―>[P¬d(¬P)], where “” denotes the logical equivalence, “¬” the logical negation, “¬d(P)” the assertion that P is not demonstrable, and “¬d(¬P)” the assertion that ¬P is not demonstrable. This property leads to the conclusion that Gödel’s formula G of the 1st Incompleteness Theorem, which is built ab initio to meet the premise of relation (I), i.e. G¬d(G), could (or should?) be considered as an axiom.
The new developments here are an attempt to pursue another and completely different way to investigate the axiomatic or non-axiomatic nature of formula G.
However, they also lead to the conclusion that formula G cannot be non-axiomatic.

L'invention concerne un maitre-cylindre a reaction hydraulique pour servomoteur pneumatique d... more L'invention concerne un maitre-cylindre a reaction hydraulique pour servomoteur pneumatique d'assistance au freinage, comprenant un piston principal (12) recevant une force d'assistance, et un piston de reaction (3) monte coulissant dans une chambre de reaction (4) du piston principal et recevant une force d'actionnement. Le maitre-cylindre de l'invention comprend egalement un support de siege (7) contre lequel peut s'appliquer le piston de reaction (3) et qui est monte coulissant vers une chambre a basse pression, le piston de reaction (3) et ce support de siege constituant en combinaison un ensemble etage dans la chambre de reaction (3), propre a recevoir, de la part de la pression dans la chambre de reaction (3), une force qui s'ajoute a l'effort de freinage apres application d'un coup de frein brutal.

Every proposition P of every consistent and bivalent theory T meets, in any inclusive, consistent... more Every proposition P of every consistent and bivalent theory T meets, in any inclusive, consistent and bivalent metalanguage of T, the two logical implications :
(I) [P <-> ¬d(P)] -> P, and
(II) P ->[P <-> ¬d(¬P)],
where ¬d(P) and ¬d(¬P) respectively assert that P and ¬P cannot be demonstrated in T. While these two relations are sufficient to readily demonstrate Gödel’s first incompleteness theorem starting from the logical equivalence G <-> ¬d(G) imposed by Gödel’s formula, they correlatively raise several serious issues regarding the meaning and scope of the said theorem. In particular, Gödel's formula G, which is forced to be true by axiom G#¬d(G), by the assumed consistency of arithmetic, and by theorem
[G <-> ¬d(G)] -> G, asserts its own undemonstrability. How could it be demonstrable, then? Can G provide any information about arithmetic, while it does not even provide any information about itself other than that is expressed ab initio by axiom G#¬d(G)?

This article first describes a phenomenological approach used to develop a simple logical system ... more This article first describes a phenomenological approach used to develop a simple logical system complying with the limits and logical constraints of natural language. Then, the resulting 3-valued, modal logic, which, nevertheless, is extensional and encompasses classical binary logic, is successfully applied to the analysis of different well-known paradoxes, including the Liar Paradox and the Surprise Examination Paradox.

This paper proposes an informal model assimilating all axiomatic theory to a deterministic univer... more This paper proposes an informal model assimilating all axiomatic theory to a deterministic universe subject to the principle of causality, and leads to the conclusion that the incompleteness of arithmetic, as stated by Gödel's theorem, is incompatible with this model.

Cet article propose un modèle informel assimilant toute théorie axiomatique à un univers détermin... more Cet article propose un modèle informel assimilant toute théorie axiomatique à un univers déterministe soumis au principe de causalité, et conduit à la conclusion que l'incomplétude de l'arithmétique, telle que posée par le théorème de Gödel, est incompatible avec ce modèle.

This article offers a new reflection on the reasoning followed by Gödel in the proof of the incom... more This article offers a new reflection on the reasoning followed by Gödel in the proof of the incompleteness theorem(s). It takes the form of a short thought experiment using a variant of this theorem, obtained by cloning, and leading to an obviously unacceptable conclusion.

Dans la mesure où aucune des thèses fondées sur des arguments de nature sémantique et invitant à ... more Dans la mesure où aucune des thèses fondées sur des arguments de nature sémantique et invitant à reconsidérer la légitimité du cheminement logique suivi par Gödel dans la démonstration de son théorème d'incomplétude n'a trouvé écho, le présent article aborde le sujet de façon plus simple sous la forme d'une expérience de pensée utilisant une variante de ce théorème obtenue par clonage et aboutissant à une conclusion à l'évidence aussi radicale qu'inacceptable.

Jean-Paul BENTZ, 2022

The notion of instantiated infinity used in Cantor's diagonal argument appears to lead to a serio... more The notion of instantiated infinity used in Cantor's diagonal argument appears to lead to a serious paradox

As exposed in this paper, Cantor's diagonal argument seems to lead to a serious paradox.

The demonstration of Gödel’s 1st Incompleteness Theorem relies on the recourse to a formula G sup... more The demonstration of Gödel’s 1st Incompleteness Theorem relies on the recourse to a formula G supposed to assert its own indemonstrable nature.
However, in order to build the formula G, Gödel had no other option than transgress the law of causality, i.e. the very law which precludes any possibility for an effect to act as its own cause.
The present draft, based on the author’s conviction that this law rules the abstract world with the same rigour as it does the material one, provides a detailed analysis of two lines of reasoning which, in the demonstration, fail to achieve such an unachievable result.

Part II.A of this draft already featured in "Gödel’s First Incompleteness Theorem, gloves off " a... more Part II.A of this draft already featured in "Gödel’s First Incompleteness Theorem, gloves off " and in "Gödel’s First Incompleteness Theorem and the humanly unreachable world". When translated into English, Parts II.B and II.C of this draft will appear in a new draft entitled "Gödel and the Law of Causality".

In the previous article, “Gödel’s First Incompleteness Theorem, gloves off”, it has been shown th... more In the previous article, “Gödel’s First Incompleteness Theorem, gloves off”, it has been shown that all proposition P in all consistent and bivalent axiomatic theory necessarily complies with the following two relations (I) : [P¬d(P)]―>P and (II) : P―>[P¬d(¬P)], where “” denotes the logical equivalence, “¬” the logical negation, “¬d(P)” the assertion that P is not demonstrable, and “¬d(¬P)” the assertion that ¬P is not demonstrable. This property leads to the conclusion that Gödel’s formula G of the 1st Incompleteness Theorem, which is built ab initio to meet the premise of relation (I), i.e. G¬d(G), could (or should?) be considered as an axiom.
The new developments here are an attempt to pursue another and completely different way to investigate the axiomatic or non-axiomatic nature of formula G.
However, they also lead to the conclusion that formula G cannot be non-axiomatic.