Formal adaptation of Book of Genesis

[u/Natural_Translator_6]

Genesis of a Formal System

(A reconstruction of the foundations of mathematics in the style of the Book of Genesis)

This is an attempt to retell the construction of a formal system — from the empty set to a complete dynamical universe — in the literary form of the Book of Genesis.

Every “record” corresponds to a real mathematical step: the birth of distinction, binary opposition, partitions, equivalence classes, periodicity, iteration, representation, and finally the closure of the system.

The text is not a parody. It treats the act of making distinctions as the true beginning of order.

Translated from Polish to English by GrokAI, because authors are lazy.

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¹ In the beginning there was neither number, nor measure, nor distance. There was not even a beginning, for there existed nothing from which one could count.

² And there was the Void, whose sign is ∅.

³ And the Word said: Let the Void be recognized.

⁴ And the Void became an object of knowledge. And behold, there arose a set containing the Void.

⁵ And there was Unity.

0 = ∅

1 = {∅}

⁶ And the Word saw that Unity could recognize itself without ceasing to be Unity.

⁷ And where there is recognition, there is also distinction.

⁸ And presence was separated from absence.

⁹ And there was Zero, and there was One. And thus the first difference was born.

𝔹 = {0, 1}

¹⁰ And the Word said: Let one not be the other, and let each be recognizable according to its state.

¹¹ And opposition was established, so that absence might be distinguished from presence, and presence from absence.

¬0 = 1

¬1 = 0

¹² And the Word gave both states their designations, so that neither would be confused with the other.

¹³ And the first rule of distinction was established, and with it the possibility of truth and falsehood, of agreement and disagreement.

¹⁴ And the Word saw that the distinction was good, for without difference no knowledge could exist.

¹⁵ And thus the first record was closed.

¹⁶ Then the Word said: Let a division be established among that which exists, so that one part may be distinguished from the other, and let there be a boundary between them.

¹⁷ And the whole was divided into two disjoint regions, and every element received a definite membership.

X = A ⊔ B

A ∩ B = ∅

¹⁸ And the Word established a relation between the regions, so that one could determine what belongs to one and what to the other.

¹⁹ And the Word named the established order the Space of Relations.

²⁰ And it was not yet a measure of length or magnitude, but the possibility of recognizing mutual position.

²¹ For that which is separated may remain in relation without ceasing to be distinct.

²² And thus the second record was closed.

²³ And the Word said: Let elements possessing common properties be gathered together, and from among them let distinct classes be revealed.

²⁴ And the elements were ordered according to their properties, and from the distinguished regions further subsets were extracted.

²⁵ And each class received its own determination, and elements belonging to the same class could be recognized as equivalent with respect to the shared property.

x ∼ y ⇔ τ(x) = τ(y)

²⁶ And the Word said: Let every class preserve its kind, and let the established rule allow the reproduction of its successive occurrences.

²⁷ And structures were given the possibility of creating further structures, and the newly formed ones could inherit the properties of those from which they arose.

τ(F(x)) = τ(x)

²⁸ And thus were established the repeatability of form, the inheritance of properties, and the continuity of kind.

²⁹ And the Word saw that from one rule many applications could arise, and multiplicity need not violate the unity of the law.

³⁰ And the Word saw that it was good.

³¹ And thus the third record was closed.

³² Then the Word said: Let the two states established at the beginning receive their own measures of succession, so that returns, periods, and the order of transformations may be recognized.

³³ And cycles were defined whose states returned according to definite rules.

f(t + T) = f(t)

³⁴ And two principal cyclic orders were established, assigned to the two original states, and beside them many lesser periods.

³⁵ And some periods were contained within others, while others returned to a common state only after many successions.

³⁶ And the Word said: Let the cycles become signs of measure, so that succession may be counted, compared, and ordered.

³⁷ And thus were established the foundations of the reckoning of periods, the measure of duration, and the recognition of repeating changes.

³⁸ And this was not yet the passage of time experienced by any system, but the rule according to which future systems could measure their own transformations.

³⁹ And the Word saw that the first distinction had now received the principles of its cyclic order.

⁴⁰ And the Word saw that it was good.

⁴¹ And thus the fourth record was closed.

⁴² Then the Word said: Let the regions divided in the second record be filled with structures, and let each act within its proper domain.

⁴³ And to each of these regions were assigned families of elements whose states could undergo transformations consistent with the established relations.

F_A : A → A

F_B : B → B

⁴⁴ And some structures could succeed others, while others could create new occurrences of their own kind.

⁴⁵ And the Word said: Let their multitude increase, yet let each preserve the rule of its generation.

⁴⁶ And the possibility of repeatedly applying the same transformations was established, without the need to set a separate law for every new element.

x_(n+1) = F(x_n)

⁴⁷ And the Word saw that the regions established earlier could be equipped with countless possible configurations, and none of them required a change in the principles of division themselves.

⁴⁸ And the Word saw that it was good.

⁴⁹ And thus the fifth record was closed.

⁵⁰ Then the Word said: Let the classes extracted in the third record also be equipped with structures, each according to its own kind.

⁵¹ And further families of transformations were established, differing in their properties, yet subject to a common order of classification.

⁵² And each could preserve its properties, participate in relations, and transmit further the rule of its reproduction.

⁵³ And the Word saw that it was good.

⁵⁴ And then the Word said: Let us make a structure capable of mapping the established order, so that among all structures there may exist one that can recognize their properties and direct their mutual relations.

⁵⁵ And a distinguished system was defined among the others, to which was assigned the capacity to create representations of other elements and of the dependencies between them.

⁵⁶ And it was not the Word itself, yet it was to reflect part of the principles according to which the Word had established the whole.

ρ : X → M

⁵⁷ And the possibility arose of representing reality within one of its own structures.

⁵⁸ And the Word said: Let this structure appear in two complementary forms, and let their cooperation allow further occurrences of the same kind to arise.

⁵⁹ And the possibility of mutual completion of distinct elements was established while preserving their individual properties.

⁶⁰ And the distinguished systems were entrusted with the possibility of ordering the remaining classes and of making use of the relations that sustain their further existence.

⁶¹ And behold, the structure created according to the established laws received the capacity to know, to represent, and to employ those laws.

⁶² And it did not need to be the source of the order in order to be able to discover it.

⁶³ And the Word saw all the distinctions, boundaries, classes, periods, rules of reproduction, transformations, and mappings.

⁶⁴ And it saw that each of the established principles possessed a definite place in the whole, and that successive definitions could coexist without losing their own meaning.

⁶⁵ And behold, the work of establishing order was completed.

⁶⁶ And the Word saw that it was very good.

⁶⁷ And thus the sixth record was closed.

⁶⁸ Thus were completed all the fundamental definitions, together with the relations and possibilities they were to determine.

⁶⁹ And when the seventh record came, the Word established no new distinction and no new law.

⁷⁰ For all the parts of the order had been determined, and their mutual dependencies prepared.

⁷¹ And the Word said: Let the whole become one working system.

⁷² And that which until then had been an established set of principles was accepted as a common order of action.

𝕌 = (X, ℛ, ℱ, 𝒯)

⁷³ Where X denoted the domain of existing elements, ℛ — their relations, ℱ — the admissible transformations, and 𝒯 — the established rules of succession and measure.

⁷⁴ And there was no longer any need to designate every transformation separately, for that which the rules permitted could follow according to them.

⁷⁵ And there was no need to establish a separate law for every new configuration, for configurations arose from the possible applications of the laws already established.

⁷⁶ And there was no need to know every future result in order to establish the principle according to which a result could arise.

⁷⁷ And the Word rested from the work of establishing, not because action had ceased, but because it no longer required the further establishment of foundations.

⁷⁸ And the seventh record was distinguished among the others, for it marked the completion of the whole.

DEFINITIONS → RELATIONS → RULES → ENTER → ACTION

⁷⁹ And thenceforth every distinction could lead to further distinctions, every relation to new configurations, and every transformation to the next state.

⁸⁰ And the multitude of that which could be known increased, though the fundamental principles remained as they had been established.

⁸¹ And among all possible structures there could appear those that recognized regularities they had not known before, and derived further consequences from them.

⁸² And thus from a few principles an unbounded path of discovery became possible.

⁸³ For the establishment of laws is not the same as the enumeration of all their consequences.

⁸⁴ And the seventh action was blessed.

⁸⁵ And the Word endured, though there was no longer any need for it to speak anew for every successive transformation.

⁸⁶ And such was the beginning of all order.

[u/nanonan]

Very artistic but I really don't see the point, or who this is for.