What the Tortoise Said to Achilles: Lewis Carroll’s paradox in terms of Hilbert arithmetic

Abstract

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll's paradox) and that of the arrow (for "Achilles and the Turtle"), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called "ontological", on which basis "motion" studied by physics and "conclusion" studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll's paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)'s completeness theorems).

Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic

“…The observation at issue is an exemplification of the statement justified in another paper (Penchev 2021 November 18) and stating figuratively that "Achilles will never overtake the Tortoise" in an only "arithmetic universe" once it is interpreted in logical terms as Lewis Carroll (1895) did. Thus, the proof of FLT, though formulated only arithmetically, needs set theory just like Achilles if he "wishes" to overtake the Tortoise (which our usual experience reckons to be obvious).…”

Fermat’s Last Theorem Proved in Hilbert Arithmetic. Iii. the Quantum-Information Unification of Fermat’s Last Theorem and Gleason’s Theorem

“…In other words, the Gödel incompleteness or dichotomy is only a mathematical paraphrase of the modern episteme relying rather on the conviction of its validity than on logical and mathematical arguments 22 . The Gödel incompleteness can be also revealed in any physical motion or in the unfoundedness of any logical conclusion as, for example, the ancient aporia about Achilles and the Tortoise or its interpretation by Lewis Carroll can demonstrate (Penchev 2021 November 18).…”