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

Full text

What the Tortoise Said to Achilles:
Lewis Carroll’s paradox in terms of Hilbert arithmetic
Vasi Penchev, vasildinev@gmail.com
Bulgarian Academy of Sciences: institute of Philosophy and Sociology:
Dept. of Philosophy of Science
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).
Keywords: equality, Lewis Carroll’s paradox, Liar’s paradox, paradox of the arrow, “Achilles and the
Turtle”, Hilbert arithmetics, qubit Hilbert space
I LEWIS CARROLL’S PARADOXES IN ORIGINAL
Lewis Carroll published in 1994 and in 1995 correspondingly in Mind two concise notices
now known (or even famous) as Carroll’s “Logical paradox” (or “Barber Shop paradox”) and
“What the Tortoise said to Achilles”. The latter is the main subject of discussion in the present
paper, however the former will be interpreted accidently following the connotations of the latter
therefore allowing for their additional elucidation (once both had been suggested by the same
author almost simultaneously).
The reason for the interest to them now consists in the realized option to be understood in
terms of Hilbert arithmetic thus eventually demonstrating that it is able to resolve alleged
contradictions between (propositional) logic, (Peano) arithmetic, and (ZFC) set theory as far as
all they are necessary for the foundations of mathematics. In turn, Hilbert arithmetic can be
considered as a reading of the separable complex Hilbert space utilized by quantum mechanics,

designated especially to use the “solution of nature” to resolve the problems in the foundations of
mathematics1
The main immediate idea of Hilbert arithmetic (which can be exemplified by “What the
Tortoise said to Achilles”, figuratively speaking, after the Tortoise had been “studied” it) consists
in the option for the contradiction of logic, arithmetic, and set theory to be reconciled relating
each of them to a certain substructure of the same structure, furthermore isomorphic to the
separable complex Hilbert space (under a few technical conditions). Then, “What the Tortoise
said to Achilles” may mean the essential properties of those three bases, “whales of
mathematics”, however in such a way that they represent alternative, but absolutely consistent
aspects of its foundation.
Lewis Carroll suggested two quite different logical paradoxes very soon one after another,
therefore eventually hinting that they share the same problem though formulated so differently at
first glance. Now, that problem can be formulated (meaning also Hilbert arithmetic as a
background) as the contradiction of propositional logic interpreted once traditionally, i.e. as the
rules of conclusion after which any logical proposition is a well-ordered sequence, on the one
hand, or interpreted as an algebraic structure such as Boolean algebra, in which the ordering is
suspended fundamentally in virtue of both double commutativity and perfect symmetry of
conjunction and disjunction, on the other hand.
One can notice that quantum mechanics would been forced about two or three decades later
to reconcile analogically (and as this will be demonstrated further: even mathematically
isomorphically) any well-ordering after measurement with its coherent counterpart before
measurement definitively deprived of any well-ordering (e.g. in virtue of the theorems about the
absence of hidden variables in quantum mechanics: Neumann 1932; Kochen, Specker 1967). Just
the separable complex Hilbert space inherently supplied by the property of unitarity is able to to
reconcile and unify Heisenberg’s (well-ordered) matrix mechanics and Schrödinger
(commutative2) ondulatory mechanics.
Projected backwards, to Carroll’s paradoxes, that property of unitarity can be interpreted as
“logical unitarity” able to resolve what the English writer, mathematician and logician Charles
Lutwidge Dodgson (known as Lewis Carroll) had observed as a contradiction. Thus, logical
unitarity is to mean the identification of the traditional propositional logic of human conclusion,
and the contemporary Boolean logical calculation (e.g. in the foundations of computer science).
2The (Hermiian) operators are not commutative in general in both quantum and ondulatory mechanics,
which true statement is often abbreviated to: “Quantum mechanics is non-commutative”, a statement
being true only after that elucidation. However, one means here the commutativity of the additive
members of any wave function unlike the well-ordering of the components of any vector therefore
representing the proper mathematical sense of unitarity (and which can be visualized by the rotation in
complex plane about its beginning and conserving the length of the vectors).
1An approach, which can be called figuratively “cognitive bionics” since bionics is defined to be a new
science predetermined to utilize relevant “solutions of nature” to resolve problems in technics, and human
cognition and can be also interpreted as a branch of techniques in a wider sense.

One can describe so, preliminarily and briefly the “spectacles”, which will be put for the
interpretation of Carroll’s paradoxes; however, they should be initially represented literally, in
original:
“A Logical Paradox” (Carroll 1894) suggests the following problem as it was formulated by
Carroll himself
“There are two Propositions, A and B. It is given that:
(1) If C is true, then, if A is true, B is not true;
(2) If A is true, B is true.
The question is, can C be true?” (Carroll 1894: 438).
Obviously, C can be true since its premise in (1) is false in virtue of (2). Thus, it is not a
paradox, but a logical mistake if it is involved in a “proof” utilizing reductio ad absurdum as in
the dialogue suggested by Lewis Carroll.
However, the contemporary “obvious” conclusion is crucially influenced by the
interpretation of the theory of syllogisms as Boolean algebra3.
If one translates the structure of the alleged paradox into syllogisms, the trouble probably
meant by Carroll would be:
(1) The negation of the proposition that A implies B implies C (the comma is interpreted as
“conjunction”);
(2) A implies B.
Then, C can be both true and false since the premise of the implication, in italic in (2), is not
stated by (2).
Also:
(1’) The proposition that B implies A implies C (comma is interpreted as “disjunction”);
(2’) A implies B.
Again, C can be both true and false since the premise of the implication, in italic in (2), is not
stated by (2).
In other words, C can never be true only if the implication in italic is interpreted
(misleadingly) as logical equivalence and an eventual proof involving it into reductio ad
absurdum would hold (as in the text of Lewis Carroll).
However, one can notice that unitarity in quantum mechanics (and eventually, the suggested
now “logical unitarity”) is forced to make just this “logical fallacy”, for example, as by
identifying the reversible coherent state before measurement and the irreversible (being
well-ordered) state after measurement.
In fact, the same logical fallacy is postulated in set theory by the axiom of choice and its
provable equivalency with the well-ordering “theorem”. Indeed, one may state that both
alternatives are equivalent as to a “choice”, but not, as to an “ordering” (however both possible
orderings are equivalent to each other). 4
4One can trace back that “logical fallacy” even to “Hume’s problem of induction” and its link to “What
the Tortoise said to Achilles” whether explicitly (e.g. Otero 2008; Howson 2000; Belkind 2018) or
implicitly and more often (e.g. Jackson 2019; Lange 2011; Weintraub 2008; Tucker 2009; Lipton 2002, a
3Gillon (1950) reflected on it from the viewpoint of “contraposition” or modus tollens.

So, a possible verdict to “A Logical Paradox” suggested by Lewis Carroll in 1994 is in turn
paradoxical: though it is not a real logical paradox5(relying on a trivial mistake), its philosophic
sense is essential as a presentiment (or “foreshock”) of the huge cognitive earthquake constituted
quantum mechanics; and this can be confirmed by the specific propositions by which the alleged
logical paradox is exemplified:
“The reader will see that if, in these two Propositions, we replace the letters A, B, C by the
names Allen, Brown, Carr, and the words ‘true’ and ‘not true’ by the words ‘out’ and ‘in’ we get:
(1) If Carr is out, then, if Allen is out, Brown is in;
(2) If Allen is out, Brown is out” (Carroll 1894: 438).
So, the corresponding exemplifying question is whether “Carr be out” is possible.
The obvious allusion should be directed to entanglement able to restrict “Carr’s degree of
freedom” having been a quantum entity (a “quantum barber”) therefore limitable to be within the
“quantum barbershop” necessarily.
However, that allusion is only a metaphor. The problem eventually underlying it is: how
unitarity (i.e. the usual unitarity means by quantum mechanics) is mathematically linked to
entanglement; how one can translate entanglement into the language of logic by means of logical
unitarity. The discussion of that problem is postponed until Section VI devoted to the logical
interpretation of the solution of quantum mechanics.
Anyway, a few preliminary notices about the relevance of Carroll’s “logical paradox” to the
tension of the traditional and algebraic interpretations of propositional logic are appropriate here.
Not paying attention to that tension is usual, referring to their equivalence in the framework of
Boolean algebra. However, the process of human conclusion meant to be modelled by traditional
logic is always well-ordered, obeying to the imperative requirement for any stage and substage to
be meaningful. Thus, it is represented as a well-ordered series of implications, since their order is
single and starting from certain premises whether preliminarily justified or granted to be obvious
as “axioms”.
On the contrary, the calculation of any Boolean expression e.g. modelled by a computer
calculation is “meaningless” as far as it need only obey the laws of Boolean algebra just an
5For example according to Burks and Copi (1950: 219), “Carroll's puzzle is not a genuine logical
paradox of the type of the Russell, the Burali-Forti, the Richard, or the Grelling, but is more akin to the
so-called paradoxes of material and strict implication”.
review of Howson’s book (2000); Lantin 1998; Weintraub 1995; Jacobson 1987); one can mean also
Hume’s problem of induction implicitly, but Carroll’s paper explicitly (e.g. Schueler 1995):
Indeed, if one paraphrases it in the present context, it refers to the relation of the axiom of induction
(stating that the possible observations are always necessarily a natural number and thus, finite), and the
axiom of infinity (stating that the conclusion of all observations relates to an infinite set and thus, the
conclusion does not follow from the premises). Even more, Hume’s problem of induction means the same
interpretation in terms of propositional logic as Carroll’s paradox rather than only the same formal
structure relatable also to the Gödel incompleteness theorems (e.g.). The present context will suggest
another link (rather implicitly) to Hume’s problem of induction mediated sequentially by Kant’s
consideration of it, his doctrine of transcendentalism, and the interpretation of the latter by “scientific
transcendentalism” as here.

arithmetical calculation obeying the corresponding laws and unlike human thinking always
admitting its realization and thus forcing a well-ordering analogical or identical to that of time.
Nonetheless, Boolean algebra guarantees a class of expressions, equivalent to any certain
chain of conclusions and sharing the same result after each corresponding calculation (including
that determined by the chain of conclusions). The difference consists in the interpretation
meaning each step of human conclusion, which allows for the check and control of the process of
thought.
As to a computer calculation, the same check and control is done preliminarily and
established as a software program is to be accomplished. So, only the ultimate result of the
calculation (whether logical or arithmetic) need be meaningful. The corresponding software
program is to mean any possible course of calculation guaranteeing for the result to be
“meaningful” therefore determining the class of those courses as “meaningful”.
Then, one can define “logical unitarity” as the mapping of that class of calculations
admissible by the corresponding software program into a class of well-ordered chains of
implications (containing at least one element necessarily) and accessible to a human mind.
Carroll’s “logical paradox” is to be related to logical unirarity as follows. It makes sense only if
one can allow that logical unitarity can be violated though itself does not manage to do this since
it is a logical fallacy in both cases: as a Boolean calculation and as a series of implications.
However, one can admit that the next attempt undertaken by Lewis Carroll, namely “What
Tortoise said to Achilles” is to be realized as a paradox able to demonstrate a violation of logical
unitarity though involving an infinite series of implications6. The ultimate result seems to be just
so unattainable as Achilles to overtake the Tortoise in the famous ancient aporia of Zeno:
Though “ACHILLES had overtaken the Tortoise” (p. 278), she or he suggested him an
equivalent (to Zeno’s “Achilles and the Tortoise”) logical problem preventing any logical
implication adding again and again, ad infinitum the same implication as a new necessary
condition for itself therefore remoting every time the conclusion to remain still and still
unattainable. Here is how Lewis Carroll himself represented the arguments of the insidious
Tortoise wanting a revenge over Achilles for his victory in the race, but in a logical way:
“(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other” (p. 278).
“And if some reader had not yet accepted A and B as true, he might still accept the sequence
as a valid one, I suppose?" (p. 278) the subtle Tortoise said, and the sincere Achilles responded:
“No doubt such a reader might exist. He might say 'I accept as true the Hypothetical
Proposition that, if A and B be true, Z must be true; but, I don't accept A and B as true’.” (p.
279).
Then, the crafty Tortoise involved again the same logical step:
" Let's call it C," said the Tortoise." - but you don't accept:
6For example, Diamond and Kaul (2010) suggest a very curious interpretation in terms of the usual
human experience (namely, to clinical trials in terms of individuals versus those of groups).

(C) If A and B are true, Z must be true" (p. 279)
“(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other" (p. 279)
Well, the perfidious Tortoise continued to knit her or his underhand logical intrigue though
one cannot accept all propositions “A”, “B”, and “C”:
“(D) If A and B and C are true, Z must be true” (p. 279).
“(E) If A and B and C and D are true, Z must be true. Until I've granted that, of course I
needn't grant Z. So it's quite a necessary step, you see?"
"I see," said Achilles; and there was a touch of sadness in his tone” (p. 280).
When the narrator would return to the same spot some months afterwards, he saw that
“Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his
note-book, which appeared to be nearly full. The Tortoise was saying ‘Have you got that last step
written down?’” (p. 280).
Obviously, the tenet can be repeated ad infinitum, so Achille should be there until now
writing and writing the next step implied necessarily by the last one.
One may formulate “What the Tortoise said to Achilles” concisely and symbolically:
(or, “ ”) and so on again
(𝐴→𝐶)→{[𝐴∧(𝐴→𝐶)]→𝐶} (𝐴→𝐶)→{[𝐴→(𝐴→𝐶)]→𝐶}
and again ad infinitum: that process of an intermediate implication again and again postpones the
conclusion C not be able to be ended ever.
One might easily, but facetiously interpret this as if “What AI said to all human beings”
would be a response to the “offensive” Turing test “whether AI can think” therefore proving that
as if the human being utilizing syllogisms successive over time cannot think in fact7.
So, if one reckons with Lewis Carroll’s intention of the two successive paradoxes to
demonstrate an eventual violation of logical unitarity, then, only the latter attempt, “What the
Tortoise said to Achilles” manages to do this correctly. The essential difference to the former one
is that it involves an infinite, though implicit and iterative series of derivative implications
therefore creating a logical equivalent of Zeno’s paradox. On the contrary, his previous “logical
paradox” not introducing infinity, though eventually meaning as an analogical idea, but without
it, i.e. the ostensible violation of the symmetry of “logical unitarity” (as it is meant above), seems
not to manage just for restricting only to a finite counterexample.
Logical unitarity can be interpreted immediately in a philosophical way, very considerable
and important nowadays, namely as the problematic fundamental mismatch of AI and human
intellect. The group of “human chauvinists” suggests that (1) they are fundamentally different
from each other, and (2) human intellect superiors AI necessarily and initially, e.g., in virtue of
the fact that the former has been created by the latter.
7For example, Fumerton (2015) suggests an “internalist” interpretation, after which he justifies equally
well two opposite conclusions without being able to choose between them; also commented by Wright
(2015).

However, Lewis Carroll’s “What the Tortoise said to Achilles” means rather the opposite
alternative: (1) they are different from each other and this can be made clear involving an infinite
series of derivative implications therefore exemplifying a violation of logical unitarity (2)
however, AI is able to surpass human intellect in that situation as far as the latter runs as a
process in time: thus needing an “infinite time” to accomplish an “infinite series of
implications”; on the contrary, AI is not bound by permanently “making sense”, so a jumplike
transition is quite natural for it.
In fact, human intellect is also able to temporarily cease that constant reflection of “making
sense” by insights as if too surprising just being “leaplike”. However, propositional logic does
not include any rule referring to insights or to any sudden jump in conclusion interpreting it as a
fallacy. Nobody knows how the “jump” due to a logical mistake should be distinguished from
that of an absolutely justified creative insight in a non-probabilistic mathematical method8,
though being especially valuable in human intellect.
Furthermore, one can notice a very instructive contradiction about the interpretation of AI
(supposedly needing the capability of insight in definition) based on the contemporary binary
computers necessarily being “Turing machines” and thus able only to finite calculations. Indeed,
they are created after human beings have been reflected on propositional logic fundamentally
incapable to represent any insight (as the cunning Tortoise forces Achilles to “be logical”) .
Nonetheless, all contemporary computers as Turing machines (and thus incapable to reflect
on their work, e.g. as in “Halteproblem”) are constructed to do Boolean operations rather than
human syllogisms (though being equivalent to the former ones). Thus, they are released from the
permanent human necessity of “making sense” by reflecting from the metalevel: a Turing
machine (i.e. a contemporary computer) is able to represent any insight as a jump-like
proposition fundamentally unrepresentable as a finite series of syllogisms remaining within the
scope of Boolean algebra but out of the traditional propositional logic of syllogisms. Indeed, it is
able to represent it, but it does not make it by itself, but only if this jump is described
exhaustively in its software program (and thus, representable by an infinite series of syllogisms,
equivalent to the logical jump by itself, but cut-eliminable to a single one following Gentzen).
However, if one involves the concept of quantum computer as a generalization of “Turing
machine”, able to make “infinite calculations” and thus, equivalent to leaplike conclusions, the
pathway to the machine modeling of human insights is open. One can notice (Penchev 2021
November 18) more or less metaphorically that the Self, or the conservation of the Self
corresponds, to the “subjective time” (but correlating directly with “objective time” of mind, and
by it, to that of the world). Any insight needs to “stop the Self” (or the “continuous flow of
8One can formulate an “insight problem” as to human intellect and analogical to the “Halting problem of
Turing machine”: one cannot know in advance whether a sudden jump in a chain of logical conclusions is
a fallacy, or on the contrary, an insight eventually justifiable further, by restoring a possible logical chain
leading to it in a necessary way. The suggested analogy to the Halting problem is not accidental, but
essential and underliable by a formal mathematical structure originating from Hilbert arithmetic, and
more precisely, from the eventual and idempotent transition from a Peano arithmetic and its dual
counterpart.

consciousness” for the logical leap to be accomplishable) to be able to appear, therefore
surprising the consciousness necessarily when the Self is “switched on” again.
II THE LOGICAL ESSENSE OF THE PARADOXES
Though the logical, mathematical, physical or at all scientific paradoxes mean an exactly
determined formal structure, it is extractable almost always from a short text in a natural
language (e.g. ancient Greek, English, etc.) and which can be enumerated among the “anecdotes”
as a literary genre since it is especially suitable to embody the logical contradiction necessary for
any logical or other paradox by the opposition of common sense’s implicit interpretation of the
described situation versus the explicit meaning, which the paradox reveals suddenly following its
logical chain.
Fortunately, Lewis Carrol is both writer and logician, so both of his paradoxes are
simultaneously interesting short stories therefore containing connotations transcending
corresponding formal structures being able to be represented exhaustively even only
symbolically. They can be interpreted many times in different ways just as any literary text, after
which new specific connotations are complemented by each interpreter being explained directly.
One can debate whether one or another connotation are really available in the interpreted piece
of literature, or it is rather created by the interpreter’s imagination. Nowadays, that discussion is
rare for any interpretation even seeming generated by a “sick mind” is equally admissible as
well.
The specific connotation added to Carroll’s two paradoxes able to unify them in virtue of his
suggestable intention is the opposition of the traditional propositional logic of syllogisms versus
the contemporary interpretation of propositional logic as Boolean algebra. Indeed, it seems
admissible at the end of the 19th century when Carroll suggested them since the interpretation by
Boolean algebra had appeared at that time generating the collision embedded as a hidden
connotation eventually revealed by the present paper,
Furthermore, that connotation is in turn interpreted by the coined term of “logical unitarity”
to be linked to the inherent unitarity of the separable complex Hilbert space, and by it to that of
quantum mechanics versus the non-unitarity of the theory of quantum information and quantum
computer, on the one hand, and to “Hilbert arithmetic” via the qubit Hilbert space, on the other
hand, following the direction, objective and intention already of the present paper.
The genre of logical paradox situated on the cross of logic and literature inherently interprets
the logical contradiction at issue as the sudden and unexpected end of an anecdote or a short
story, if one considers it as a piece of literature rather than as a “dressed in words” formal
structure. Thus, it calls for (as if) loose interpretation usual for literature, hermeneutics or
philosophy, but not for the mainstream of logic.
Meaning all those conventions very relevant to Carroll’s short stories designed as an
ornament of a corresponding logical contradiction representable thoroughly formally, one can
question about which are those purely formal nuclea of his paradoxes:
“A logical paradox” exemplifies the pair of syllogisms (as Carroll himself has written in his
text):

1. (𝐴, ¬𝐵) →𝐶
2. 𝐴 →𝐵
A, B, and C are arbitrary propositions; The problem is whether C is restricted in any way by
both conditions, and the answer is not, in virtue that a false premise of an implication does not
restrict any conclusion, which can be both false or true due to other and non-formulated premises
out of the scope of Carroll’s “logical paradox”.
Still one reflection is possible if the comma as usual is interpreted as the operation
“conjunction” in Boolean algebra. Then, ; so, only excludes for
(𝐴,¬𝐵)⇔¬(𝐵→𝐴) (𝐴,¬𝐵)
to be the particular case of logical equivalence (i.e. it implies “ ” as to
"𝐴→𝐵" ¬(𝐴↔𝐵)
). Unifying both premises under that consideration, the result is: “ ”, or just
"𝐴→𝐵" (𝐴→𝐵)→𝐶
“ ” in virtue of the associativity of implication. Thus, the opposite solution of the
𝐴→𝐵→𝐶
problem is inferred: so Carroll should mean just that a pair of opposite solutions of the same
logical problem are valid, which justified the used word “paradox”.
The interpretation in the present paper bounds the former solution with propositional logic as
Boolean algebra, and the latter one, with the traditional logic of syllogisms (though represented
by the contemporary symbols borrowed from Boolean algebra).
In fact, the two opposite solutions are due to the ambiguity of the enumeration of two
different logical conditions. Indeed, the former interpretation means: “[(𝐴, ¬𝐵) → 𝐶]∧
”, while the latter interpretation: “ ” eventually justifying
(𝐴→𝐵) [(𝐴, ¬𝐵)∧(𝐴→ 𝐵)] →𝐶
itself by the ostensible fact that only the composed propositions of “A” and “B” are relevant to
each other and thus “C” has to be out of the brackets. However, the same false substitution
presupposes the conclusion that “C” unambiguously depends on “A” and “B” (and in the final
analysis, only on “A” since “B” depends on “A” as well). In other words, the wrong substitution
means the conclusion implicitly as a premise: an obvious logical fallacy.
Additional connotations to that formal structure are complemented by the exemplification of
“A”, “B”, and “C” as spatial relations: “in - out” therefore meaning the implicit premise that “in”
is the idempotent logical negation of “out”: a statement valid in classical mechanics, but not in
quantum mechanics. Thus, the connotation of “entanglement” can be added referring to the
utilized exemplification of logical structure.
On the contrary (now meaning “What the Tortoise said to Achilles)”, Boolean algebra
naturally admits operations of an infinite set of operands, furthermore suggesting necessarily an
unambiguous ultimate result of their accomplishment (though being an infinite set)9. The formal
structure is:
1. (𝐴→𝐶)→{[𝐴∧(𝐴→𝐶)]→𝐶)}
2. “1” (just above) describes an iterative logical procedure, which can be repeated therefore
ad lib or ad infinitum.
The same formal structure is interpreted in the present paper as violating of “logical
unitarity” between the traditional propositional logic of syllogisms (as pre-supposed to be a
9Probably, Lewis Carroll had noticed the analogy to Zeno’s “Achilles and the Tortoise”, which had
perhaps inspired “What the Tortoise said to Achilles”.

thought process running in time) and the calculations in Boolean algebra (not linked to time, by
itself, and thus allowing for calculations containing an infinite set of operands)10.
“What the Tortoise said to Achilles” seems not to contain any logical fallacy as a formal
structure. Anyway, it involves implicitly infinity due to an iterative repetition ad lib and thus, ad
infinitum. Indeed, the traditional propositional logic of syllogisms does not consider that kind of
infinite syllogisms for they seem to be meaningless if one has interpreted a syllogism as a human
thought running in time (that infinite syllogism would require an infinite time to end: a condition
which is obviously meaningless). The foundation and justification is to bound unitarity in
quantum mechanics and “logical unitarity” as the relation of a well-ordering to a “coherent state”
(i.e. any well-ordering and thus, all well-orderings) in both cases.
III BOTH CONNOTATIONS OF THE PARADOX (“What the Tortoise …”)
The formal logical structure does not exhaust at all the meaning, especially philosophically
meaning of “What Tortoise said to Achilles”11. A few connotations are links to other areas, which
situate the paradox among human cognition and can be qualified as its Fregean “Sinn” (i.e.
“sense” versus “meaning”: “Bedeutung”)12. Those are a few exemplifications of the formal
structure, namely:
1. “A” is a composite proposition, consisting of the conjunction of two propositions, which
can be notated as “A1” and “A2”: “ ”. Both “A1” and “A2” refer to a relation. That
𝐴=𝐴1∧𝐴2
relation is the relation of equivalence (equality). “A1” refers to two sides of a triangle; “A2” refers
to a general property of “equivalence”. All those exemplifications are unified as a single one
because they mean each stage of the paradox internally or separately unlike the second
exemplification, which hints an analogue as a whole to the ancient aporia of Zeno:
2. That allusion is introduced borrowing both personages of Zeno’s paradox: Achilles and the
Tortoise, therefore suggesting a hidden formal structure able to underlie the ancient apporia and
12 Laraudogoitia (2014) suggests still one, but rather “hidden sense” of “What the Tortoise said to
Achilles” by a continuation of the text, which can be granted to be a “logical extrapolation” of it
demonstrating that “there are free actions (in general, contingent states of affairs) that can be predicted by
means of purely logical reasons” (p. 405). Hilbert arithmetic by its inherent link to the formalism of
quantum mechanics can elucidate the same idea additionally, but quite independently. For example,
Conway and Kochen (2006; 2009) “free will theorems” being directly inferable from that formalism
under the condition of Lorentz invariance demonstrate the “free actions” e.g. of the investigated electron
as predictable by the description of quantum electrodynamics. However, Hilbert arithmetic allows for
even more: Boolean algebra (and thus propositional logic with its strict predictions) is to relate to the
metalevel of free will (e.g. that of the electron at issue and described by a corresponding wave function
and belonging to both qubit Hilbert space and Hilbert arithmetic, in a wider sense).
11 Just many connotations rather than the proper logical structure are offered, for example, in: Moctefi
and. Abeles, eds. (2016). One can also notice Ree’s paper (1951) following Carroll’s text literally to
represent the author’s logical ideas to it.
10 According to Brown (1954: 170), the “first moral” of “What the Tortoise said to Achilles” is: “the
legitimacy of inferring that since p, q, does not require the truth of the statement that if p, then q in the
same way as it requires the truth of the statement that p”. The same suggested difference is interpreted (in
the present paper) as that between logic as syllogistic, on the one hand, and as Boolean algebra, on the
other hand.

Carroll’s contradiction. Achilles cannot overtake the Tortoise just as an elementary implication
cannot be accomplished being analogically intermediated again and again ad infinitum. The
meant contradiction refers to “common sense” (as in his previous “logical paradox”) stating the
obvious: as Achilles will overtake the Tortoise very soon really, so the implication can be ended
practically instantly.
Both paradoxes (Zeno’s and Carroll’s) do not contain any logical or mathematical fallacy,
both involve infinity implicitly by a process repeatable iteratively ad infinium, therefore
suggesting that our understanding of infinity contains some ambiguity or contradiction able to
justify the conclusions in both cases, however contradicting empirical experience.
Furthermore, the unification of the two paradoxes (the one being physical or mechanical, the
other one, logical) suggests the Hegelian idea about the philosophical category of “motion” able
to unify the two sides of the Cartesian dichotomy of “mind” (or particularly, logic meant by
Carrol’s paradox) and “body” (mechanics or physics meant by Zeno’s paradox). In fact, the
category of motion can represent ontologically Kant’s transcendental solution (being traditionally
related rather to epistemology, to “mind”, though it implies by itself to be not less relatable to
“body”) of Descartes’s problem (eventually the fundamental one of the modern Western
philosophy).
Thus, Helel exemplified “motion” historically, to emphasize his ontological interpretation of
Kant’s transcendentalism: both to history of philosophy and to philosophy of history. Even more,
he introduced “dialectic logic”, which being ontological is what should correspond to “formal
logic” (i.e. only to the traditional logic of syllogisms since in Hegel’s age, the interpretation of
propositional logic as Boolean algebra had not appeared yet). One can add that Heggel’s only
and properly philosophical approach influenced crucially the course of history in the 20th
century by means of Marx’s reinterpretation as “dialectical and historical materialism” (and
embodied as the “scientific doctrine” of the “real socialism” in the Soviet Union and all other
countries more or less controlled or impacted by it).
The present section of the paper will restrict itself only to still one exemplification of the
possible unification of logical and physical (mechanical) paradoxes, namely those of the “Liar”13
and of the “Arrow”, thus directly borrowing Carroll’s kind of allusion:
One can suggest that both paradoxes being both ancient and famous need not be represented.
Even more, the pathway of their analogy seems to be obvious, or speaking loosely or
metaphorically: the “Arrow” “lies” that “it is here”; the “Liar” moves his or her statement (about
what she or he has said) not to be identifiable with itself therefore being in a kind of “logical
motion”14.
14 That analogy moreover suggesting an underlying shared formal, logical and mathematical structure of
both paradoxes is discussed in detail in: Penchev 2009.
13 The paper of Aerts, Broekaert, and Smets (1999) interprets the “Liar” by the mathematical formalism of
quantum mechanics, the separable complex Hilbert space. Thus, it anticipates the radical and fundamental
approach of the present paper stating in the final analysis that Hilbert arithmetic (respectively, the qubit
Hilbert space) is able to resolve the real paradoxes of logic (along with those of mathematics or physics).

That observation can be further represented in detail and logically consistently by the
following formal structure: if “A” is false, “A” implies the logical negation of “A”, So, let “A”
be “The arrow is here”: indeed, that “The arrow is here” is false, implies that “The arrow is not
here”. Analogically, let “A” be “The liar is lying” (therefore emphasizing that he or she lies just
saying “I lie”): then if “The liar is lying” is false, this implies that “The liar is not lying”. Thus,
there exists no formal logical contradiction in both cases for “A” can be true or false in both
cases though the paradoxes mean the option of “false”. The alleged contradiction is due to the
confusion of “ ” (which can be as true, where “A” is false, as false, where “A is true”)
𝐴→¬𝐴
with “ ” (which is identically false).
𝐴∧ ¬𝐴
One can notice that the formal structure implies that both paradoxes are also valid if as “ ”
¬𝐴
is granted as “A” in the notations above. That idempotent pair is sometimes interpreted as the
definition of the Hegelian “dialectic contradiction” or the Marxist “objective contradiction” (e.g.
Petrov 1971), namely and formally:
“ ”
𝐶𝑜𝑛𝑡𝑟𝑎𝑑𝑖𝑐𝑡𝑖𝑜𝑛 (𝑑𝑒𝑓)≡[(𝐴→¬𝐴)∧(¬𝐴→𝐴)] ⇔(𝐴↔¬𝐴)
One can easily watch that both “Liar” and “Arrow” do not satisfy that definition for the
idempotent exchange of “A” and “ A” in them means the logical operation of “disjunction”, but
¬
quite not, “conjunction”. Thus, the alleged “definition of objective contradiction” turns out to be
an identically true tautology:
“Alleged ”
𝐶𝑜𝑛𝑡𝑟𝑎𝑑𝑖𝑐𝑡𝑖𝑜𝑛 (𝑑𝑒𝑓)≡[(𝐴→¬𝐴)∨(¬𝐴→𝐴)] ⇔¬(𝐴↔¬𝐴)
Thus, the exemplified meta-allusion of “What the Tortoise said to Achilles” by the pair of the
“Liar” and “Arrow”, furthermore containing still one very essential allusion to Hegel’s doctrine
of motion as ontological, is able to suggest the following conjecture referring to both pairs: (1)
Zeno and Carroll’s paradoxes; (2) the “Arrow” and the “Liar”:
The conception of infinity can be “repaired” (meaning Gödel’s dichotomy about the relation
of (Peano) arithmetic to (ZFC) set theory: either incomplete or contradictory) investigating the
condition under which the contradiction of finiteness and infinity can be transformed into an
“alleged contradiction” (as it is defined just above rigorously). Particularly and very important as
to Gödel’s dichotomy, this means a dual counterpart of Peano arithmetic to be introduced so that
if “A” is to mean: “Peano arithmetic is available”, “ ” should be able to mean “The dual
¬𝐴
counterpart of Peano arithmetic is available” (i.e. being able to be equivalent to “Peano
arithmetic is not available”).
Meanwhile, one has to notice that Bohr’s quantum complementarity (initially applied to, or
extracted from quantum mechanics; then, generalized to many sciences and to philosophy)
satisfies just the transformation of “dialectic contradiction” into an alleged contradiction
relatively painlessly: or in other words, that of Hegel’s ontological motion into quantum motion
(however very often restricted only to area of quantum physics or sciences utilizing its results).
Then, the conjecture which means that Bohr’s quantum complementarity can be considered
as a universal solution of all paradoxes whether ancient or contemporary, whether logical,

mathematical or physical, seems to be reasonable. Furthermore, the structure of Hilbert
arithmetic can embody that universal principle of quantum complementarity ...
IV THE UNIFIED PROBLEM OF CONCLUSION, MOTION AND EQUALITY IN A
HELELIAN MANNER
Hegel’s doctrine, though originating from Kant’s transcendentalism, possesses a fundamental
specific feature: it means a unification of logic, philosophy, history and even physics by a kind of
“philosophical logic” called dialectic logic. However, that “dialectic logic” turned out to
contradict formal logic rejecting the law of (non-)contradicting therefore postulating for itself to
be a constitutionally non-falsifiable (or “metaphysical” after Popper) dogma without any
scientific value, allowing for an arbitrary statement to be inferred and thus “proved” by it in the
final analysis.
Many, many attempts have been undertaken to “repair” propositional logic modifying its
axiomatic basis in a way to make dialectic logic both formal and falsifiable15. None of them has
received recognition and distribution outside the specialized field of non-classical logic.
The idea of Hilbert arithmetic shares the same intention, but approaches radically differently:
it does not repair propositional logic in any way, i.e. conserving it thoroughly and literally. What
is “mended” is dialectic logic utilizing the “lesson” of nature, thought out by quantum mechanics
and reflected philosophically by Niels Bohr in his principle of complementarity interpreted
formally and logically. This is described above by the substitution of the definition of “dialectic
(or objective) contradiction” by that of “alleged contradiction”, furthermore interpreted to be
isomorphic to complementarity.
Then, the relation of logic, arithmetic, and set theory is to be modified in a way to be
complementary (in that sense of “alleged contradiction”) to the usual separable complex Hilbert
space therefore realizing the goal of Hegel’s dialectic logic, however inherently differently and
consistently to propositional logic: namely, the unification of logic, philosophy, and physics in
their, thus shared foundations.
This can be demonstrated particularly to Hegel’s category of “motion” being ontological, i.e.
underlying all of those: mechanical motion, physical or mental change, logical conclusion,
historical change in both history and history of philosophy. They can be merged as different
aspects of Hilbert arithmetic, sufficient for that objective. Accordingly, mechanical motion are
15 Along with the huge set of paraconsistent logics, there exist many quantum logics, which in turn modify
some axioms of propositional logic. The conclusions of quantum mechanics have seemed so
extraordinary that one can admit that it needs a new, “quantum” thinking and thus, a new, “quantum”
logic whether generalizing or modifying (in a narrow sense) traditional one. The present paper does not
share that motivation incited to creating new logics, and its relation to non-classical logic including all
quantum logics is elucidated in detail in Section X: there is no need of new logics, but one need clear up
and resolve the problem of finiteness and infinity (shared by the foundations of mathematics and thus, by
mathematics at all) as to traditional logic and Boolean algebra also in their concepts and terms. The
reason is that all non-classical logics can be situated on the side of classical logic in virtue of their models
in it versus its interpretation as a mathematical structure (such as Boolean algebra) and embodied as a
problem in the formal structure of “What the Tortoise said to Achilles” chosen to be the topic of the paper
just for that.

Hermitian operators as well as any physical change; mental change can be represented
analogically (and directly interpreting the foundation of quantum mechanics: Penchev 2021 July
26). Logical conclusion can be repeated isomorphically to the standard definition in the
framework of Hilbert arithmetic. The mediation of Husserl’s phenomenology interpreted
absolutely formally resolves the problem of how the historical change in both history and history
of philosophy can be embedded in the same framework (Penchev 2020 December 14)). So, one
can state that Hilbert arithmetic is able to embody the Hegelian (including Marxist) philosophical
and scientific ideas exhaustively, however absolutely consistently to propositional logic, and
thus, to science (including experimental science).
Meaning the consideration just above, Lewis Carroll’s “What the Tortoise said to Achilles”
can be interpreted as kind of “foreshock”16 anticipating Hilbert arithmetic, especially by its
connotations of merging mechanical motion (meant by Zeno’s paradox literally) and logical
conclusion (suggested by Carroll), already explained from the present viewpoint, on the one
hand, and logical equivalence and quantitative equality, on the other hand, therefore needing a
notice, which follows;
The category of identity is central in Hegel’s doctrine (opposing “philosophical identity”
including “change” versus the traditional formal and logical identity excluding any change). It
serves to justify “dialectic logic” and simultaneously (formally and logically, in a proper “vicious
circle”) philosophical identity to be inferred from dialectic logic.
Dialectic identity which means simultaneously “change” has been shared by many left
radical parties and movements (as their “ideologies” or political philosophies) preaching
“revolution” meaning an extremely different, new political power and social order often getting
the idea of violence for it to be able to happen. The paper, however, discusses “dialectic identity”
in an abstract, logical and properly philosophical aspect, thus absolutely independent of the
above connotations, which are social and political: i.e. they would be irrelevant to the present
context.
In fact, the use of “identity” includes “change” to some extent, fussily determined, due to
which it is a subject of a well-known logical paradox: the “sorit paradox”. Quantum mechanics
also suggest a solution of it since it was forced to resolve an analogical scientific puzzle: how to
unify consistently the description of both: (1) the leaplike, quantum change of the studied
microscopic entity; and (2) the continuous and even smooth change of the macroscopic apparatus
obeying the differential equations of classical mechanics:
Its solution suggests a wave function able to describe that unification, and thus. a
corresponding probability distribution, to which the wave function at issue is its characteristic
function: that distribution can be interpreted as a solution of the sorit paradox therefore
presupposing that “identity” is transformed into “change” both gradually and probabilistically,
16 Philosophy of science follows the natural attitude of human experience and science therefore describing
history of science as a causal process over time and thus excluding any reverse causality by the influence
of any future events on any present phenomena. However, if one has introduced a quantum worldview to
be leading as here, the corresponding philosophy of science should discuss “foreshocks” (along with the
standard causal “aftershocks”) in history of science or history at all.

after the sorit paradox has been extrapolated in advance into the terms we are properly interested
in: those of the Hegelian or marxsist “dialectic change”:
Following concepts of the dialectic law of how “quantitative changes pass into a qualitative
change”, one can distinguish the changes within the same identity as quantitative versus the
change meant in a proper sense: i.e. generating a new quality. Then, the corresponding solution
of quantum mechanics would state that both quantitative changes and qualitative change(s) share
the same probability distribution (implying a wave function as its characteristic function
unambiguously), so that it is a mapping of the set of quantitative changes into the probabilities of
the qualitative change(s) to be observed empirically or experimentally. In other words, if that
probability distribution is visualized as a two-dimensional graphic, its abscissa would correspond
to the quantitative changes conserving the same identity, and its ordinate accordingly, to the
probabilities of qualitative change(s) implying a new identity.
Meaning the concepts of Hilbert arithmetic introduced rigorously and in detail in other
papers (e.g. Penchev 2021 August 24)), that mapping means the probability distribution (or a
wave function as its corresponding characteristic function) of finite quantitative (literally,
arithmetic) changes to be considered as the new quality of infinity. In other words, Hilbert
arithmetic can be considered as a relevant tool to describe “dialectic identity” (respectively,
“dialectic change”) rigorously and consistently to propositional logic.
One meets still one trouble, rather properly logical and mathematical than philosophical,
discussing identity and referring to the problem whether logical equivalence (notated further as
) can be treated to be the same as quantitative (mathematical) equality (further, “=”);
"⇔"
symbolically and particularly as to the transitivity of both sharing the same formal structure of a
“relation of equivalence”, where they can be interpreted correspondingly as logical equivalence
and mathematical equality:
Does the proposition “ mean the same as the proposition “ ”? If
𝑎=𝑏=𝑐" 𝑎⇔𝑏⇔𝑐
“Yes”, are mutually interchangeable? For example, does make
"=", "⇔ " “𝑎=𝑏⇔𝑐"
sense? Properly, the question relates to the self-referential use of both logical equivalence and
mathematical equality as two exemplifications of the relation of equivalence. Indeed, one can ask
analogically: if (respectively, “ ”), do “ and “ ” mean the same?
"𝑎=𝑏" 𝑎⇔𝑏 𝑎" 𝑏
The problem can be generalized, e.g. so: can one exchange arbitrarily the interpretations in a
framework of the same class of equivalence? Its basis is the puzzle about the admissibility of
self-predicativity: whether or where a level and the corresponding next metalevel can be mixed
without any logical fallacy.
One can distinguish the indefinitely (infinitely) increasing hierarchy of levels, each of which
can be enumerated by a natural number, on the one hand, and an idempotent structure consisting
of only two levels so that each of them can be considered as the metalevel to the other one and
thus being mutually notatable by “logical negation”, on the other hand. The latter case is realized
in any mathematical structure able to be dual, such as the separable complex Hilbert space of
quantum mechanics and borrowed in Hilbert arithmetic.

At the same time, it is able to embody the indefinite (infinite) hierarchy always notatable by
natural numbers, however as if ordering it to be situated on the sublevel to that of idempotency.
In fact, this is only seeming, and can be easily fitted by introducing the qubit Hilbert space
(being equivalent to the separable complex Hilbert space under only technical conditions) as it:
as a whole is contained in any qubit; thus, restoring the perfect symmetry of hierarchy and
idempotency for which it is inherently designated.
Meaning the above observation, one can ignore the distinction of the initial sets (which are
different arguments of the mappings), of both cases of equivalence and equality: propositions in
the former case, but elements of any set in the latter. However, the much more important
statement is that Hilbert arithmetic is a relevant tool to be unified (and thus, to be ignored the
corresponding distinction) of the logical mapping into only two elements (e.g.: “false” and
“true”) and assignable to idempotency, and the mathematical mapping in an infinite set (e.g that
of all natural numbers), i.e. assignable to hierarchy.
If one takes in account, that the set of two elements is able to represent any finite set (e.g. as
the Turing machine tape is able to demonstrate), the unification of idempotency and hierarchy,
accomplishable as Hilbert arithmetic by the qubit Hilbert space, resolves simultaneously the
fundamental mathematical and philosophical problem how to be unified finiteness and infinity
(e.g. exemplified by the Gödel dichotomy about the relation of (Peano) arithmetic to (ZFC) set
theory).
One is to notice furthermore that the idea of idempotency and hierarchy to be unified (along
with its embodiment in Hilbert arithmetic) is embedded still in the axiom of choice (standardly
enumerated among the axioms of set theory though it can be interpreted as a meta-axiom
regulating the relation of arithmetic and set theory), and more precisely, in the equivalency of the
axiom of choice (for idempotency) and the well-ordering “theorem” (for hierarchy).
On the ground of those preliminary notices above, one can conjecture that the connotation
(conventionally enumerated as “second”) meant by “What the Tortoise said to Achilles” and
referring to “equality” hints at the eventual unification of idempotency and hierarchy, underlying
Hilbert arithmetic.
V THE MAIN PROBLEM OF QUANTUM MECHANICS AND THE “PROBLEM OF
THE TORTOISE AND ACHILLES” AS THE SAME
Meaning the above, logical and physical discussion of Zeno’s and Carroll’s paradoxes in a
“Hegelian manner”, one can speak of a single paradoxical structure underlying both and rather
referring to the relation of finiteness and infinity, and more precisely to common sense’s belief
that, speaking loosely, infinity is always “much more” than finiteness and thus, postponing ad
infinitum the moment when Achilles (on the side of finiteness) overtakes the Tortoise (on the
side of infinity) whether physically (after Zeno) or logically (after Carroll).
One can use the metaphor (or structure) of the Tortoise and Achilles to quantum
measurement, adding still one paradoxical aspect of the problem: for example as follows. The
moment of overtaking can be interpreted to correspond to the equality of a discrete quantum

measure (naturally meant by the Planck constant) and a continuous classical measure (embedded
in the apparatus since it is described by classical mechanics)17.
Then the additional antinomic aspect would consist in the necessary interpretation of Achilles
to be “quantum” (so, rather microscopic) versus the macroscopic “Tortoise of the apparatus”,
which is to be classical. One can describe that new, exchanged “quantum” reinterpretation so: the
huge, being macroscopic, Tortoise can never overtake the microscopic quantum Achilles though
any of his leaps must correspond exactly to the extremely small Planck constant. The reason is
that the Tortoise’s step has to be infinitely small since she or he moves continuously as classical
mechanics requires necessarily, and any finite quantity (such as the Planck constant in the case)
is much greater (even, infinitely greater) than the Tortoise’s step, which is to be infinitely small
for his or her motion to be continuous. So, the same prejudice of common sense about the
relation of infinity and finiteness takes place, though modified to the relation of an infinitely
small quantity and an extremely small, but finite constant.
As it is explained above, Hilbert arithmetic rejects common sense’s “infinity is always
greater than finiteness” (respectively, any finite quantity to be always greater than an infinitely
small one18) involving both in the same, but dual structure, furthemore isomorphic to the qubit
Hilbert space (and thus, to the usual separable complex Hilbert space of quantum mechanics
under relevant conventions).
So, the justification of the above “ridiculous”, “quantum” reinterpretation of the paradox
about quantum measurement intentionally consists in the relativity of finitiness and infinity in a
“Skolemian manner”, but established a long time ago also in the Standard model as the relativity
(respectively, identity) of the local space (e.g. the quantum Achilles’s one) and the global space
(e.g., that of the macroscopic, “classical” Tortoise). The sense of that reinterpretation is, speaking
loosely, in the logical equivalence that if Achilles cannot overtake the Tortoise, the Tortoise
cannot also overtake Achilles as well as vice versa.
The usual understanding of common sense about quantum mechanics restricts its subject only
to microscopic, quantum phenomena commensurable with the magnitude of the Planck constant:
thus, it is relevant to macroscopic phenomena very rarely, under too special conditions, but never
directly. On the contrary, other papers (e.g. Penchev 2020 July 16) explain the main problem of
18 One should mention the “Archimedes axiom” and the substitution of it by its negation in Abraham
Robinson’s (1966) “Nonstandard analysis”, even more so related to the axiom of choice by the weaker
version of the latter: the “lemma of ultrafilters”. The negation of Archimedes’s axiom supplies finiteness
and infinity to be distinguished from each other again, however in a different way in comparison with the
“standard analysis”: both finite small constant and (Leibniz) “differential” of nonstandard analysis can be
actual and thys similar to each other, but just the negation of Archimedes’s axiom allows for them not to
be compared. On the contrary, standard analysis divides them into different qualities (correspondingly,
either “finite” or “infinitesimal”) conserving to be comparable in virtue of Archimedes’s axiom, but only
by infinite quantities. So, both standard and nonstandard analyses share the same dichotomy being
supported by the corresponding alternative within the same dichotomy about finiteness and infinity: either
comparable different qualities or the same quality in two incomparable (“complementary”) aspects. The
latter alternative seems to be shared by both quantum mechanics and nonstandard analysis.
17 For example, Rees (2001) (as well as many other authors) describes quantum phenomena in terms of
Zeno’s paradoxes.

quantum mechanics to be: how the discrete, quantum entities to be consistently described by the
continuous measure of the apparatus according to classical mechanics, or in other words, how to
be described uniformly both discrete and continuous phenomena. Then, any entity to be quantum
is not more an absolute property (as the former viewpoint of common sense presupposes), but a
relation; thus, if its premises be satisfied, macroscopically observed phenomena (e.g. those of
entanglement) can obey quantum correlations directly.
The present paper continues further, extrapolating the relative interpretation of quantum
mechanics (after which “to be quantum” is not more a property, but a relation) to the problem of
finiteness and infinity following its exemplification by both versions of the paradox about
Achilles and the Tortoise. Just as Scolem (1922) has suggested following the conclusion of set
theory, and more especially, the axiom of choice, that finiteness and infinity should be considered
to be considered rather as relations than as properties of sets.
The justification of linking the relative interpretation of quantum mechanics (in the sense as
above) and Scolem’s “relativity of ‘set’” consists in the axiom of choice postulated in set theory
and inferrable in quantum mechanics19 (then embedded in the unitarity of Hilbert space). Then,
the relativity of finiteness and infinity can be realized to resolve as Zeno’s physical paradox as
Carroll’s logical paradox therefore bridging both to the main problem of quantum mechanics if it
is understand so: how to be reconciled, unified, and identified (in the final analyses) as the same
both the discrete quantum description of the investigated entity by itself and its continuous
readings by the apparatus in turn necessarily represented only by the smooth differential
equations of classical mechanics.
VI THE SOLUTION OF QUANTUM MECHANICS AS A SOLUTION OF THE CITED
ANTINOMIES
If the main problem is established to be just as it is formulated at the end of previous
paragraph, its solution consists in the introduction of the unitarity embedded in the separable
complex Hilbert space as a necessary condition to be unified the discrete viewpoint of
Heisenberg’s matrix mechanics (conventionally attachable to the measured or investigated
microscopic quantum entity “by itself”) and smooth one of Schrödinger’s ondulatory (realizable
as what the apparatus registers to that entity). In fact, the Schrödinger equation elucidates only
the way for both descriptions to be universally equated if the separable complex Hilbert space as
containing any possible solution (“wave function” as its element) is granted in advance.
Nowadays, the theory of quantum information interprets it as the qubit Hilbert space
therefore underlying the quantum generalization of information (equivalent to the information of
infinite sets or series: Penchev 2020 July 10). Thus, quantum information can be considered not
only as the universal substance of the physical world (e.g. Penchev 2020 July 24), but also and
not less, as unifying what any empirical or experimental experience referring to physical body
means with what its mathematical description by mind means (е.г Penchev 2021 February 26)
just in virtue of the completeness of quantum mechanics verified by the theoremes of the absence
19 That idea is developed in detail in other papers (e.g. Penchev 2021 April 12).

of hidden variables (Neumann 1932; Kochen, Specker 1967) as conclusion inferable only from
the properties of the separable complex Hilbert space (respectively, the qubit Hilbert space).
So the completeness of quantum mechanics can be reinterpreted as originating directly from
the use of a special mathematical structure rather than as implied by quantum mechanics as a
whole. As far as it is a mathematical structure (thus not needing any physical premises), it can be
involved for resolving the “annoying” incompleteness of the foundations of mathematics (after
the Gödel dichotomy about the relation of arithmetic to set theory as first-order theories to
propositional logic).
However, all history and successes of mathematics suggest indirectly that mathematics
should be grounded very well de facto. Nonetheless, Gödel’s direct reflection on the foundations
doubts the intuitive confidence of almost all scientists rather than only mathematicians. One can
admit that there is a problem in the reflection on its foundations rather than that in itself as a
huge cognitive building since and for millennia.
I have already argued in other paper (Penchev 2020 August 25) that the Gödel dichotomy
about the relation of arithmetic and set theory can be related as a statement to itself (i.e.
self-referentially) thus transforming into an independent additional axiom rather than a proper
theorem directly inferable from the explicit premises of arithmetic, set theory, and propositional
logic20. Its dichotomy can be traced back to the relation of the axiom of induction in arithmetic
(implying that all natural numbers are finite) and the axiom of infinity in set theory (postulating
that the set of all natural numbers is infinite).
Then, the use of both axioms into a single proof (as the Gödel paper published in 1931 does)
continues an ambiguity needing implicitly an additional axiom about how to interpret their
relation if both are simultaneously used as the formulations of theorems in the paper force.
Properly, that non-formulated additional axiom is equivalent to the central dichotomic conclusion
of the paper about the relation of arithmetic (containing the axiom of induction) and set theory
(containing the axiom of infinity); namely: either incompleteness or inconsistency.
In fact, that conclusion is a direct corollary from the analogical relation presupposed to
regulate the ambiguity of both axioms implicitly meaning that all natural numbers are
correspondingly either incomplete or inconsistent to the set of all natural numbers.It means
informally that “set” is a substantive and independent mathematical entity thus irreducible to its
elements and indirectly justifying set theory as one of the “whales” supporting mathematics.
Its basis can be revealed in the organization of contemporary cognition rather than in
mathematics itself: our “episteme” (by Michel Foucault’s term and concept) originates from
Cartesian dualism opposing “mind” to “body” and then creating a model of that opposition
within mathematics itself by arithmetic itself (for “mind”) and set theory (for “body”) therefore
in turn being dichotomic or contradictory. Indeed, the availability of such a model in
mathematics suggests for it to contain the world as a whole (just as it contains the model at
issue), on the one hand, but on the other hand, mathematics is not less contained within the world
just as “mind” (for mathematics) in “body” (for the world).
20 The idea is represented for the first time in: Penchev 2010 (in Bulgarian).

Then, the Gödel incompleteness axiom (i.e. not a “theorem” after the present consideration)
chooses only the latter aspect (mathematics within the world) ignoring absolutely the former one
(the model of mathematics and the world within mathematics itself). So, that Gödel
incompleteness axiom is rather a meta-axiom postulating how to be interpreted the relation of
mathematics and the world: namely, situating the latter within the former (which seems to be
natural for empirical experience, experimental science and the contemporary organization of
cognition dominated by them).
This fundamental branch of mathematics accepting or implicitly granting that meta-axiom
can be called Gödel mathematics. On the contrary, if mathematics shares the negation of the
Gödel incompleteness axiom (namely, the former aspect of the world within mathematics in
virtue of the availability, neglected by Gödel, of the relation of mathematics and the world within
the former), it can be called Hilbert mathematics.
All subareas of mathematics not needing the simultaneous use of arithmetic and set theory
(more precisely, the simultaneous use of the axiom of induction and axiom of infinity) can
remain the same in both Gödel and Hilbert mathematics and they comprise almost all
mathematics. Nonetheless, there exists a few fields in which the dichotomic choice between
them is unavoidable, and the foundations of mathematics (as the Gödel incompleteness
demonstrates itself) or quantum mechanics (in virtue of the theorems of the absence of hidden
variables implies) are among them.
Indeed, one can trace back the pathway in which Hilbert mathematics naturally underlain by
Hilbert arithmetic supplies an alternative relation of the axiom of induction and the axiom of
infinity and then and thus, a different solution of the Gödel problem about the relation of
arithmetic and set theory in the foundations of mathematics.
It is grounded on the interpretation of the relation of arithmetic and set theory (respectively,
that of the axioms of induction and infinity) also by means of the idea of “alleged contradiction”
implying for them to be complementary to each other: or in other words, both arithmetic and set
theory are applicable, but never simultaneously. So, the contradiction between them is never
actual, and only potential since one has to choose whether the axiom of induction (thus in the
framework of arithmetic) or the axiom of infinity (and thus, in the framework of set theory) to
use in each case21.
Hilbert arithmetic can be considered as a structure inferred from the qubit Hilbert space (in
turn originating from the separable complex Hilbert space of quantum mechanics) and
designated just for supplying the complementary use of “either arithmetic or set theory”,
furthermore each of them to be complementary to the utilization of the qubit Hilbert space thus
interpreting any entity and issue as physical rather than as mathematical:
Obviously, that formal solution based on Hilbert arithmetic can be generalized to the
fundamental problem of Western philosophy originating from Cartesian dualism: “mind” and
21 Speaking loosely, the sense of the axiom of choice can be interpreted as the necessary and dichotomous
choice to be used either the axiom of induction (and then, the axiom of choice has the form of the
well-ordering theorem) or the axiom of infinity (and thus, the proper axiom of choice).

“body” to be complementary to each other and this to be the way for regulating of both kinds of
description: either in terms of “body” (for the physical description by the qubit Hilbert space) or
in terms of “mind” (for the proper mathematical description in terms of Hilbert arithmetic in a
narrow sense).
Then, Zeno’s paradox is to be interpreted in two complementary ways, after which the
contradiction would appear only if one confuses both descriptions: on the one hand, either
mathematical or physical, or on the other hand: either arithmetic or set-theoretical if the case is
former, i.e. a mathematical description.
Indeed the physical description is what is naturally granted to be the real one in our
“episteme”, furthermore relevant to empirical human experience, and in which Achilles
overtakes the Tortoise unambiguously and after a finite interval of time. The same is valid in the
framework of one of both possible mathematical descriptions, namely that in terms of set theory.
Nonetheless, the statement is false in the other one of them, in terms of arithmetic. However, the
contradiction can be never actualized in virtue of the complementarity of both descriptions:
Speaking loosely, Achilles needs infinity weather mathematical or physical (realizable by the
concept of continuity) in order to manage to overtake the Tortoise; thus, an imaginary Achilles
competing with the Tortoise within an only arithmetic “reality” can never win, but our usual
reality granted by human experience is not arithmetic: anyway, it can be consistently granted to
be mathematical, but only set-theoretical though the completeness of mathematics needs its
complementary counterpart, which is arithmetic.
The same kind solution can be immediately interpreted in the terms of Carroll’s paradox in
virtue of the isomorphism: the implication cannot be ever ended in a proper arithmetical reality22,
but propositional logic is standardly related to physical reality and thus, to mathematical reality
meant in set theory. Indeed, this is a result indirectly reconfirmed by the Gödel (1930)
completeness paper. Thus, Hilbert arithmetic is able to describe “What the Tortoise said to
Achilles” equally well both physically and logically.
VII THE PROBLEM OF THE COMPLETENESS OF MATHEMATICS BY THAT OF
QUANTUM MECHANICS: THE IDEA OF HILBERT ARITHMETIC
An experimental science such as quantum mechanics is privileged in the episteme of
Modernity in comparison with mathematics (originating from the “mind” in the same
framework) to aspire to “completeness” as far as all mental models (among which the
mathematical ones are) ought to obey empirical and experimental experience (collected and
generalized e.g. by quantum mechanics as in the case). Independently of that “natural attitude”
confessed by the scientific “mainstream” nowadays at least implicitly, the proof of completeness
of quantum mechanics based on the theorems about the absence of hidden variables in it is
properly mathematical: not needing or even admitting referring to experiments. Particularly, the
link from them to entanglement is not logically necessary: its phenomena are only consistent
22 How arithmetic can be interpreted as first-order logic if that is the case is discussed in detail in Section
IX.

with the absence of hidden variables and can be interpreted as “confirmations” only in that
practical sense (unlike any eventual refutation).
So, the actual and thus acting episteme can be “surrounded” de facto, suggesting that the
qubit Hilbert space being an only mathematical structure is sufficient to supply the completeness
of mathematics (cherished more than a century) though expected initially for quantum mechanics
in virtue of what the general organization of cognition in Modernity can accept or “swallow”
(since the eventual completeness of mathematics implies an epistemic “revolution” and thus an
earthquake in science at all).
Nonetheless, the proof of completeness though applied to quantum mechanics (obeying the
dominating prejudice of common sense) is mathematical by itself and therefore, it is able to
resolve the crisis in the foundations of mathematics revealed still in the beginning of the 20th
century. In other words, Hibert arithmetics only develops the completeness of the separable
complex Hilbert space in the relevant mathematical concepts merely neglecting an eventual
cognitive earthquake which it can cause (and many aftershocks in all sciences).
So, the troubles (rather than problems) about Hilbert arithmetic are philosophical, needing a
new place of mathematics in cognition as well as the relevant rethinking of its relations with
physics and philosophy after needing their unification and merging the three foundations. The
corresponding shift of the worldview leads to a new, contemporary form of Pythagoreanism,
which can be called “quantum neo-Pythagoreanism”.
The essential corpus of basic mathematical ideas to underlie Hilbert arithmetic can be
reduced to a few. The arithmetical units to be defined as classes of equivalence of qubits sharing
the corresponding arithmetic unit at issue, i.e. “empty qubits” or those of a quantum Turing
machine tape before recording any value in each qubit of it. Thus, it is natural to be
complemented by the corresponding (Peano) arithmetic originating from the dual qubit Hilbert
space therefore both satisfying Peano axioms, but avoiding the direct contradiction to the axiom
of infinity in set theory (as in the Gödel incompleteness theorems) due to the set-theoretical
complement of the dual Peano arithmetic to the set of all natural numbers. After that,
propositional logic and set theory can be identified as isomorphic to each other as the same
Boolean algebra and then both, as isomorphic to a nonstandard interpretation of Peano
arithmetic (which, speaking more precisely, should be called “anti-nonstandard”, or
“anti-standard”23), in which the function successor is defined to be by adding a new member of
equality rather than an arithmetical unit (e.g. Penchev 2021 August 24; or Penchev 2020 August
25).
At last, the dual counterpart of Peano arithmetic can be naturally interpreted as a qubit
Hilbert space, therefore generating the representation of Hilbert arithmetic (in a narrow sense)
and the qubit Hilbert space as dual to each other, and thus, the duality of mathematics and
physics unifiable by a relevant Pythagorean kind of philosophy. The axiom of choice generates a
23 The addition of “anti-” is due to the anti-isometry of the dual qubit Hilbert space (originating from that
of the dual separable complex Hilbert space): to be “anti-nonstandard” (or respectively, “anti-standard”)
means the non-standardness being idempotent to be interpreted by anti-isometry also being idempotent.

mapping of any infinite set in natural numbers, but it cannot be constant, but variable (even
fundamentally randomly variable) for avoiding the contradiction of “all natural numbers” to the
“set of all natural numbers”: thus, the “backdoor” of the set-theoretical “Dedekind finiteness”
(rejecting only the constant mapping of an infinite set into it finite subset) can be utilized and
then mapped bijectively (only in virtue of the axiom of choice and the well-ordering “theorem”)
into the dual counterpart of Peano arithmetic, in which any unit can be already interpreted as a
certain value “recorded” into the -th empty qubit after mapping.
𝑛
Then, set theory being a first order logic of propositional logic (embedded in advance into the
“anti-nonstandard” interpretation of Peano arithmetic) is also doubled due to the axiom of choice
as: (1) Boolean algebra isomorphic to that of propositional logic (this is the one of both doubled
copies of set theory, namely that before applying the axiom of choice); (2) two dual to each
other, but standard Peano arithmetics and each of which satisfying the Gödel dichotomy and
thus, either incomplete or inconsistent to set theory:
Particularly and as it will be discussed in detail in Section IX, “What the Tortoise said to
Achilles” can be interpreted as a statement inverse to Gentzen’s “cut elimination rule” (Gentzen
1935), but being false unlike the latter. In other words, even an infinite series of implications can
be “cut”, but no finite series of implications can imply an infinite one, and thus, the syllogistic
scheme suggested by the Tortoise (the number of syllogisms of which is always finite in virtue of
the axiom of induction) can never end. The paradox is “true” in that sense: i.e. this is a true
statement (namely that the proposition reverse as above to the cut elimination rule is false),
which contradicts only the prejudice of common sense.
VIII HILBERT ARITHMETIC AS THE “CLASS OF ALL CATEGORIES” (in category
theory)
Category theory is a proper mathematical theory referring directly to the foundations of
mathematics. Unlike set theory (being historically first), category theory shares fundamental
philosophical ideas about the unification of physics, mathematics, and even philosophy just in
their common “root” in cognition and reality (e.g. Kuś and Skowron, eds. 2019; Krömer 2007;
Giandomenico, ed. 2006; Peruzzi 2006, etc.) That is just as the worldview of the present paper
though adhering to the more traditional approach meaning arithmetic, set theory and
propositional logic as underlying mathematics, but relevantly modified as aspects of Hilbert
arithmetic, in turn isomorphic to the formalism of quantum mechanics. Then, a natural
conjecture is the similarity of Hilbert arithmetic and category theory can be discussed even as
one or another degree of mathematical and philosophical equivalency:
An analogue of the basic ideas as they are sketched above can be discovered in category
theory attempting to suggest an alternative viewpoint to the classical foundations of mathematics
based on propositional logic, arithmetic, and set theory eventually overcoming the accompanying
antinomies. The natural conjecture is that analogue can be formulated rigorously and
mathematically: that is as an isomorphism in the final analysis.
Category theory intends to classify all mathematical structures in categories according to
their axiomatic definitions and to describe the transformations between structures both within a

category and from a category into another. Those transformations (“functors”) are interpreted as
mappings (“morphisms”) but rather of logical collections of essential features and established by
the corresponding axioms than structures (i.e. only extensionally representable as elements of
sets).
Thus, the distinction of propositional logic and both fundamental first-order logics of
arithmetic and set theory seems to be partly removed and partly substituted by the
newly-invented distinction of categories (relevant rather to first-order logics) and functors or
morphisms (describing their transformations rather logically).
The viewpoint to logic (including propositional) logic is also modified corresponding to the
contemporary approach as a specific class of structures (e.g. the category of lattices) rather than
the traditional one as series of syllogisms and relevant only to propositional logic (extremely
restricted only to Boolean lattices) in the former approach. Thus, category theory tends to resolve
logic in mathematics (into categories, on the one hand, and into functors or morphisms, on the
other hand) leading the rethinking of logic started by its interpretation as Boolean algebra to a
radical result after which the difference of logic and mathematics is rather due to the tradition
and thus, conventional.
Furthermore, physical interpretations of category theory are also possible (and even
emphasized by its creators as their motivation24): for example interpreting structures and
categories as “bodies”, and functors as “motions” or “changes” of those “bodies”.
The idea of an eventual isomorphism suggested in the present section is underlain just by the
interpretation of “morphisms” (or “functors”) in category theory as transformation of tuples of
axioms, each tuple of which is relevant to a certain category (rather than an extensional
enumeration of all structures belonging to a certain category, and their mappings as morphisms
or functors after that)25. Then, any category (i.e. equivalent to a tuple of propositions, granted to
be axioms, and thus obeying the rules of propositional logic) can be assigned unambiguously to a
relevant interpretation of the “axes” of the qubit Hilbert space so that:
25 Without any additional discussion (though necessary, but postponed for another paper), one can notice
the possible conjecture that the double exchange of finiteness and infinity, on the one hand, and the
intensional definitive property (or properties) of an infinite set and its extension consisting of an infinite
number of elements, on the other hand, can be considered to be mutually conservable passing into each
other by virtue of a certain common measure (e.g. such as probability). The former exchange is
admissible in Hilbert arithmetic as that mediated by the dual Peano arithmetic, and the latter one, too,
again mediated by the dual Peano arithmetic, but now relating to the pair of propositional logic and
arithmetic (rather that to the pair of set theory and arithmetic as in the former case).
24 For example, Eilenberg and Maclane (1945; in the introductory section of their first and fundamental
paper on category theory) elucidate their idea by two dual vector spaces such as those utilized by quantum
mechanics therefore suggesting or at least hinting at the approach in this section: the exhaustive
interpretation of category theory by the qubit Hilbert space of quantum information. Furthermore, they
emphasize expressively: “whenever new abstract objects are constructed in a specified way out of given
ones, it is advisable to regard the construction of the corresponding induced mappings on these new
objects as an integral part of their definition” (Eilenberg, Maclane 1945: 236). This declaration seems to
be a translation (into a mathematical language) of the philosophical definition of “physical entity”:
changing itself by itself.

1. An axis corresponds to an item in a certain axiomatic tuple.
2. The axiom at issue and its negation correspond to the pair of two qubits (respectively,
“axes”) idempotently dual to each other.
3. Any mathematical variable able to accept whether a finite or an infinite set of values can
be represented equivalently as an axiomatic tuple correspondingly finite or infinite.
4. Mathematical variables assigning a single value are representable as values of a relevant
qubit.
5. Logical variables being propositions assigning a binary value (i.e.: either “true” or “false”)
and can be represented by any pair of dual qubits,
6. Finally, the opposition of logical and mathematical variables is relative after the double
exchange of logical and mathematical variables, on the one hand, and the global space consisting
of bits with any local space of a single qubit.
As a conclusion, the qubit Hilbert space can be interpreted as the class of equivalence of all
categories in category theory26, and thus, as a trivial isomorphism due to the identity to a class of
equivalence in that sense is supplied.
IX LEWIS CARROLL’S PARADOX BY HILBERT ARITHMETIC
There exists the following problem about the relation of propositional logic, set theory, and
arithmetic after they have been unified within the framework of Hilbert arithmetic as above. If
set theory and propositional logic are consistent to each other (for example, after the Gödel
(1930) completeness theorems and in that sense), but arithmetic and set theory are not (for
example, after the Gödel (1931) incompleteness theorems), what is the relation of arithmetic and
propositional logic? Is arithmetic consistent as a first-order logic to propositional logic? Can the
Gödel dichotomy of arithmetic to set theory be transferred into the relation of propositional logic
and arithmetic, therefore generating an analogical “either inconsistency, or incompleteness” of
arithmetic to propositional logic, for example, as a first-order logic? Is possible an analogue (or
even an isomorphic interpretation) of the Godel incompleteness theorems, in which set theory is
substituted by propositional logic in virtue of sharing the same structure of Boolean algebra
(though it being exhaustive for propositional logic, but only necessary for set theory needing
furthermore two dual Peano arithmetics in the framework of Hilbert arithmetic)? Finally the last
question (and most relevant to “What the Tortoise said to Achilles” interpreted by Hilbert
arithmetic), does this import a kind of inconsistency in Hilbert arithmetic even if one grants that
Hilbert arithmetic is free of Gödel’s dichotomy?
Properly, “What the Tortoise said to Achilles” once interpreted in terms of Hilbert arithmetic
brings the trouble and doubt emphasized by it. As if: the Gödel incompleteness is relevantly
transposed within any implication therefore generating the ridiculous impossibility to be
completed obviously contradicting common sense and human experience,
26 This is a rather philosophical consideration which corresponds to many proper mathematical
approaches or methods for establishing “quantum category theory” (e.g. Rennela, Staton 2018;
Moskaliuk, Moskaliuk 2013; Chikhladze 2011; Davis 2006; Holdsworth 1977, etc.).

In fact, any implication in arithmetic as a first-order logic really cannot be accomplished
directly: nonetheless, it can really be done absolutely consistently, but only in virtue of the
consistency of propositional logic and set theory, on the one hand, and cut-elimination rule
allowing the transition from an infinite series (respectively, set) of implication to a finite one,
even to a singlе syllogism27. At that, cut-elimination is valid unambiguously only in that
“direction”: i.e. from an infinite series to a single syllogism, but not vice versa28: no single
syllogism implies unambiguously an infinite series of implications, due to which it needs new
implications again and again as in Carroll’s paradox29: just that falsity of the statement (reverse
to cut-elimination in a sense) is the subject of “What the Tortoise said to Achilles”.
Speaking loosely, arithmetic by itself cannot be complete, and no implication can be
accomplished being rigorously restricted to its framework just as the Tortoise stated or
suggested: this is the analogue of the Gödel dichotomy30, but rethought in the relation of
propositional logic and arithmetic (or respectively, it as a first-order logic). One needs set theory
(though as a “Wittgenstein ladder”) only after which any implication on;y within arithmetic can
be consistently accomplished. Indeed, the dual copy of Peano arithmetic embedded in Hilbert
arithmetic can be really always removed as a “Wittgenstein ladder” in virtue of the fact that the
two copies are identical (though anti-isometric, but only as a relation to each other). Speaking
again loosely, or in the Hegelian discourse intentionally emphasizing contradictions, Peano
arithmetic just as “different from itself” (in Hegel’s manner of expression) is set theory (i.e. by
its dual counterpart identical to it in a sense).
30 Horská (2014) suggests a kind of counter-analogy between the Gödel incompleteness versus the
Gentzen completeness, which is absolutely relevant also in relation of the former to “What the Tortoise
said to Achilles”, on the one hand, and also, in relation of the latter to it, on the other hand. The Gödel
incompleteness and the Gentzen completeness can be thought as a relation and its converse relation as to
finiteness and infinity: the relation from infinity to finiteness (the Gentzen completeness) exists, but its
converse, from finiteness to infinity does not exists and this is the sense of the Gödel incompleteness.
29 For example, it can be represented by the relation of sequent calculi and cut elimination, that is, in the
conceptual framework suggested and developed by Negri and Plato (2016).
28 This statement can be also interpreted by means of Gentzen’s cut elimination rule applied to Hilbert’s
ε-calculus (e.g. Wessels 1977) since the latter in turn can be reinterpreted by or as Hilbert arithmetic.
Then, the sense of “non-identity” (respectively, “without identity”) means just the “Dedekind
set-theoretical finiteness” after which the mapping of an (or any) infinite set into infinite is possible only
without identity, and that “non-identity” can be described rigorously by a probability density distribution
unambiguously assignable to the finite-set mapping of the infinite set at issue. The characteristic function
of that probability density distribution is a wave function, and thus, a transfinite natural number,
respectively being a usual natural number, if it belongs to the dual Peano arithmetic. In the final analysis,
the asymmetry of cut-elimination and “What the Tortoise said to Achilles” relies on the asymmetry of
choice in the axiom of choice (or physically, on that of time arrow). In other words, if they were
symmetric, this would imply reversible time, or set-theoretically, the idempotent exchangeability of an
element of a set and the set itself, which is false even only in virtue of the availability of many elements
belonging to the same set.
27 Buss (2015) demonstrates how cut elimination can be applied to itself in a sense (i.e. as if
self-referentially) allowing for removing all top-level cuts from proofs. Applying that method
successively, one can reach a single syllogism finally.

Describing the same, the structure of Hilbert arithmetic substitutes the mystifying Hegelian
discourse claiming the hypothetical “dialectic logic” in a way to be perfectly consistent to
propositional logic (and thus, to the rational formal logic of syllogisms). A dual Peano arithmetic
originating from the anti-isometric qubit Hilbert space is what one means as Peano arithmetic
“different from itself” (said in the confusing Hegelian manner). Thus, that complementary
counterpart supplies what one is to understand as the “non-arithmetic, but set-theoreic” in an
explicit and constructive way. Then, just that “non-arithmetic, but set-theoretic” part is
comprised by propositional logic to be consistent to set theory so that arithmetic (as set theory) is
either incomplete or inconsistent to it, representing the essential “plot” of “What the Tortoise
said to Achilles”:
In other words or speaking loosely, Achilles can overtake the Tortoise both logically and
physically only by means of the dual counterpart though the contest runs only in the initial
Peano arithmetic and the dual “twin” can be even absolutely hidden in the ultimate result when
Achilles has overtaken the Tortoise by any finite distance.31 The “Wittgenstein ladder” of the
dual Peano arithmetic is necessary only in the infinitely small interval when Achilles is
overtaking the Tortoise actually, but not after that, due to which it is removed.
The absolutely necessary mediation of the dual counterpart can be demonstrated negatively
by the “Halting problem” of the Turing machine where the moment of overtaking is substituted
by stopping the calculation, being mathematically isomorphic. Any Turing machine is
definitively restricted to compute only in the framework of a single Peano arithmetic
(conventionally called “initial” above): thus, it cannot calculate whether it will stop or not since
it would need the mediation of a complementary Peano arithmetic (or Turing machine) rejected
in default. Speaking figuratively, the insidious Tortoise forces Achilles to think as a Turing
machine (therefore depriving him of human intellect).
The advantage of Hilbert arithmetic after the description of “What the Tortoise said to
Achilles” consists just in the explicit referring to the dual Peano arithmetic, and then describing it
in detail though it is not necessary within the ultimate result and can be removed in it as a
“Wittgenstein ladder”.
X INSTEAD OF CONCLUSION: THE UNITY OF LOGIC, PHYSICS AND
MATHEMATICS BEING COMPLETE
The necessary unification of physics, mathematics, and philosophy once Hilbert arithmetic
has been introduced is discussed in a few papers (e.g.: Penchev 2021 July 26; 2021 August 24;
2021 November 18). Now, the specific accents are two: (1) logic by means of propositional logic
in the framework of that unity; (2) completeness as the sufficient condition for the unity.
The contemporary realization of logic seems to be “forked” into specific mathematical
structures of categories (such as lattices), on the one hand, or non-classical logics, on the other
hand, modifying in a way or another the traditional formal logic of syllogisms. That bifurcation
31 The same can be represented rather intuitively following the ideas of White (1999) about the relation of
simulation of a physical object and a relevant deduction in the framework of its mathematical model
especially to cut-elimination.

can be noticed still when Boolean algebra appeared being both a mathematical structure (an
algebra) and simultaneously, granted to be equivalent to the logic of syllogisms. (Carroll’s
paradoxes are interpreted to be devoted to the troubles of that eventual equivalence). If logic is
understood as a share of mathematics (i.e. in the former branch being historically second), one
does not need to debate separately its belonging to the unity in question. However, logic as a
collection of non-classical logics changing ones or other axioms of classical logic needs a special
consideration to be added in the scope of the unity.
Just “What the Tortoise said to Achilles” centered into the problems of distinguishing
Boolean algebra and propositional logic by syllogisms is quite relevant to the discussion. The
Hegelian opposition of formal and “dialectic logic”, the latter in turn granted to be the proper
“ontological logic” including that of the physical world and mechanical motion, is a natural
complementing background of the same problem; even more so that many “paraconsistent
logics” claiming to formalize, and thus, to originate directly from dialectic logic, have been
created and belong to the scope of “non-classical logic”.
As far as any non-classical logic proves its consistency by a model in classical logic, one can
restrict the discussion only to the traditional syllogistic logic . If it is granted to be equivalent to
Boolean algebra, the problem is resolved. So, the question asked by “What the Tortoise said to
Achilles” is essential also about eventual obstacles in the pathway for logic to be thoroughly
integrated with physics (after mathematics by the category of lattice) and thus with philosophy
following Hegel’s objective, but by formal and logical tools.
Properly, the problem of the paradox (especially emphasized by Carroll by means of the
allusion to Zeno’s one) is about whether or how infinity is admissible in logic and whether that
problem (about the applicability of infinity in logic) does not distinguish Boolean algebra (where
infinity seems to be relevant as in any mathematical structure) from classical logic referring to
successive syllogisms: where an infinite series of syllogisms seems to be inaccessible whether to
the finite human mind or as a process that takes place over time32 (and thus any infinite syllogism
turns out to be never finished, moreover, each single implication entails an infinite syllogism).
Furthermore, the same opposition of infinity to finiteness is subject of the Gödel
incompleteness theorems implying his dichotomy. In fact, Lewis Carroll’s antinomy anticipates
them, since it shares the same dichotomy however exemplified by a special set of syllogisms
(that of an iterative series of syllogisms). Classical logic calls the natural numbers of arithmetic
to enumerate those syllogisms as successive to each other therefore implying for their number to
be always finite in virtue of the axiom of induction.
On the contrary, Boolean algebra being a mathematical structure is situated on the side of set
theory and infinity, naturally admitting an infinite set of propositions thus enumerable by the set
of all natural numbers. Then, one can repeat literally the Gödel dichotomy, this time in relation to
syllogistic logic and Boolean algebra:
32 Valaris (2017) justifies “that there are diachronic norms of epistemic rationality – namely, norms of
good reasoning” (the author’s italic) meaning a necessary mental jump from the premises to the
conclusion corresponding to different (“diachronic”) moments of time.

Syllogistic logic is either incomplete or inconsistent to Boolean algebra: and this statement
can be immediately visualized by “What the Tortoise said to Achilles”, after which even any
implication cannot be accomplished since it is equivalent to an infinite set of syllogisms, which
contains at least one neither true nor false. The paradox is “true”, and this means that the
completeness of syllogistic logic to Boolean algebra (though both are granted by common sense
to be equivalent to each other) is just so unprovable as arithmetic to set theory under the crucial
condition of “Gödel mathematics”. That is: it is provable, but in “Hilbert mathematics”:
Hilbert arithmetic in turn reforms and unifies consistently Peano arithmetic, set theory, and
propositional logic in a way they to be the reliable foundations of Hilbert mathematics, only after
which Carroll’s paradox about Achilles and the Tortoise can be resolved rigorously and logically
for it has been rooted in the Gödel dichotomy, which is to be removed before that.
Once Hilbert mathematics is established, all non-classical logics grounded in relevant
modified tuples of the axioms of classical logic can be reconciled consistently with their
interpretations as mathematical structure (for example belonging to the category of lattices).
What remains to be justified is “(2)” above: how the concept of completeness assists or
necessary implies (from a proper logical and mathematical viewpoint) the unity of logic,
mathematics, physics, and philosophy:
As to philosophy, the idea of Kant’s transcendentalism embodies completeness (rather
postulated by the concept of the totality) as a necessary basis for Cartesian dichotomy and
dualism to be resolved monistically: anyway it is a metaphysical (in a Popper sense) and thus, a
fundamentally unfalsifiable statement as a philosophical doctrine therefore needing a relevant
scientific reformulation to be reusable in exact and deductive sciences such as mathematics and
logic or even in an experimental science such as physics. This is its counterpart of “scientific
transcendentalism” discussed in detail in other papers (e.g. Penchev 2020 October 20).
As to physics (ibid.), completeness can be revealed in both concepts of the universe and the
proved completeness of quantum mechanics once it has been grounded into the separable
complex Hilbert space. Just the latter is the relevant bridge to completeness in mathematics,
however fundamentally impossible in the framework of “Gödel mathematics”.
Then only Hilbert mathematics based on Hilbert arithmetic is able to embody consistently
completeness: at that, following the analogical or isomorphic approach and formal structure
established by quantum mechanics. The last touch of completeness, as to logic, can be painted
investigating Lewis Carroll’s “What the Tortoise said to Achilles”, by which one can ascertain
that the completeness of logic though often granted to be self-evident, in fact is so problematic
as that of mathematics, but fortunately even isomorphic. Thus, once a solution is revealed in
mathematics, it can be merely repeated in logic in virtue of that isomorphism:
Hilbert arithmetic underlies rigorously and reliably the unity and completeness of logic,
mathematics, physics, and philosophy.

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