# What the Tortoise Said to Achilles: Lewis Carroll’s Paradox in Terms of Hilbert Arithmetic

### EasyChair Preprint 7280

32 pages•Date: December 30, 2021### 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.

**Keyphrases**: Achilles and the Turtle, Equality, Hilbert arithmetics, Lewis Carroll’s paradox, paradox of the arrow, qubit Hilbert space, the Liar’s paradox