A New Form of Physical Theory
The Structure of Reality
Reality is partially made up of facts that describe the fundamental properties of objects and their relationships Additionally, we believe that these facts, whether recorded directly or indirectly through pre-theories, take a specific form.
– between the objects a 1 , , a n and finite many real numbersα 1 , , α n , there is the relationr n (a 1 , , a n , α 1 , , α n ).
Only facts related to the “domains of physics” are taken into consideration. These facts constitute what we call thephysically recordable domain, orreality domain, and is denoted byW.
The reality domain W can be expanded through hypotheses, which are tested via experiments designed to provide a tangible reference to these hypotheses This process aims to establish a real connection between recordable facts and nonrecordable or imagined phenomena, ultimately validating the proposed relationships.
Facts, both directly and indirectly observable, serve as a collection of empirical evidence that hints at deeper, more intriguing truths Much like an iceberg, the majority of these facts lie hidden beneath the surface of immediate experience, necessitating hypotheses about the submerged elements To validate these hypotheses, it is essential to establish connections between recordable and nonrecordable facts, allowing the former to serve as evidence for or against the existence of the latter.
The Physical Reality
In our understanding of physical theories, we identify three distinct domains of physical reality that align with specific conceptual levels For any given physical theory, referred to as P T ν, we differentiate between the application domain, labeled A p ν, and the fundamental domain, which is also defined within the framework of the theory.
G ν , and the reality domain, denoted by W ν Let us briefly describe these three domains.
The application domain of a particular physical theoryP T ν , denoted byA p ν , is the restriction of the reality domainW to the facts that the theory considers a priori.
The recording of facts can be made directly or indirectly by pre-theories.
The application domain \( A_p^\nu \) is constrained by the physical concepts utilized for interpretation, highlighting the importance of this restriction, as no single physical theory encompasses all of reality Common domains of physical concepts include mechanics, optics, thermodynamics, and electrodynamics, each representing a specific contextual framework for understanding physical phenomena.
We shall see later (Sect 3.1.2) that only facts denoted by sentences using terms that designate physical concepts, belonging to the context related to the application domainA p ν , can be recorded.
From a methodological point of view, A p ν is something given a priori relative to the physical theoryP T ν Something which is given a priori is not implicitly defined, it can only be shown.
This action of demonstration aims to encompass not only directly observable facts, represented by ρ, but also indirectly observable facts as articulated by various physical theories such as P T α and P T β, while deliberately excluding those defined by the current physical theory.
In the context of physical theories, an electrical current in a conductor can be considered a fact within a specific theory P T ν, particularly when it is examined through electrodynamics However, this current is only relevant
The foundation of all observations relies on the acceptance of certain facts through immediate experience, without the need for analysis For instance, in an experiment, the status of a counter is considered a fact that requires no further scrutiny Such facts are accepted without applying scientific criteria or even physics to determine their certainty.
The justification for considering immediate experiences as real facts is not addressed by physics, which relies on excluding this question to function effectively The recognition of such facts involves complex physical, physiological, psychological, and cognitive processes that remain partially understood For instance, if a hare crossed our path during a walk, our certainty about this event cannot rely on the testimonies of absent witnesses, photographs that were not taken, or any other criteria Physics does not challenge the exploration of knowledge regarding these given facts within a realm beyond its scope.
A crucial question in fundamental physics emerges: Is the foundational basis of established facts in physics consistent with the theories and principles derived from them? This issue of consistency between the original facts and the physics developed from them is of significant importance.
The "physical representation" derived from sensory perception processes is elaborated upon in [2, Chap XVII] Up to this point, there has been no evidence suggesting a lack of consistency between physics and sensory perceptions.
To summarize, in the definition of the application domainA p ν of a physical theoryP T ν , one can already include the reality domainsW α , W β , of other physical theoriesP T α , P T β , We call theseP T α , P T β , pre-theories of
P T ν The definition of the application domainA p ν is thus not trivial It is a problem offundamental physics, which we will only be able to approach later.
This application domainA p will be further explained in Sect 3.1.2.
The fundamental domain of a particular physical theoryP T ν , denoted byG ν , is the restriction of the application domainA p ν to the facts that the theory describes.
Mathematical theories employed in physical applications are fundamentally approximations of reality, applicable to specific domains only when a certain degree of inaccuracy is permitted.
The fact of having a usable theory depends on the choice of the degree of inaccuracy allowed It is necessary to distinguish between two cases:
– We have no large inaccuracies, and we say that the theory can be applied as a “good” description of the application domainA p ν ;
– We have large inaccuracies, and we say that the theory says practically nothing about the structure of the reality in such regions.
The application of this theory to the total application domain A p ν is particularly valuable in regions where a small degree of inaccuracy is acceptable In these areas, the theory provides insights into the structure of reality, making it beneficial for technical applications We refer to such regions as the fundamental domain G ν When small inaccuracies can be tolerated across the entire application domain A p ν, it follows that G ν is equivalent to A p ν Further details on the fundamental domain G ν will be discussed in Sect 3.3.3.
The reality domain of a particular physical theory P T ν , denoted by W ν , is the extension of the fundamental domainG ν to the facts (related to the new physical concepts) that the theory describes.
Our task is not only to detect “nonmeasured” realities, but also to detect new realities.
To effectively convey new physical concepts, we must introduce specific terminology that reflects these potential or imagined realities However, the challenge remains in articulating how we can "observe" these conceivable realities.
How to introduce new physical concepts will be further explained in Sect 6.3.
To explore the fundamental properties of objects and their relationships, we can rely on direct observations and preliminary theories, utilizing clearly defined methods prior to establishing a mathematical framework This task is significantly more challenging than simply introducing new physical concepts To accurately identify the factual basis of these concepts, we must trace our steps back from the comprehensive mathematical theory to the domain of reality.
W ν ⊃A p ν This reality domainW ν will be further explained in Sect 6.6.
1.2.4 The Reality Domain of all P T s
As we have seen before, only facts belonging to physical domains are taken into consideration and constitute what we call thephysically recordable domain, orreality domain, and is denoted by W.
The reality domain W encompasses all Ws derived from various P Ts, but since not all P Ts are known, W cannot be fully defined Discovering new P Ts leads to the identification of new Ws, such as atoms and elementary particles However, the physically recordable domain W is inherently restricted by the exclusion of certain directly ascertainable facts, like the harmony of a sound or the quality of a violin's tone.
The domain of attainable facts certified for physics has not been established until this day This reality domainW will be further explained in Sect 6.6.
The techniques used to establish P Ts extend beyond physics and can be applied to diverse fields, such as understanding the structure of the human species and addressing sociological issues through Weidlich’s theory.
Figure 1.1 represents a summary of the domains of physical reality.
W reality domain of all P T s all directly and indirectly (via pre-theories) physically recordable facts ρ, W α , W β ,
A p ν application domain of P T ν restriction ofW to the facts that theP T ν a prioriconsiders
G ν fundamental domain of P T ν restriction ofA p ν to the facts that theP T ν describes
W ν reality domain of P T ν extension of G ν to the facts, related to new physical concepts, that theP T ν describes
W, W ν reality domain of all P T s all directly or indirectly (via pre-theories) physically recordable facts
Fig 1.1.Domains of physical reality
Fairy Tales
Our task is not only to detect nonmeasured physical realities, but also to detect new physical realities.
For a particular physical theory P T ν , we have introduced the reality domain W ν as the extension of G ν to the facts related to the new physical concepts that the physical theory describes.
A theory that is based on hypothetical concepts but does not connect to real-world facts is referred to as a "fairy tale theory" at the conceptual level, while the corresponding imagined scenario is termed "imagined reality" or "fairy tale" at the reality level In Chapter 6, we will explore the notion that such a fairy tale theory is generally regarded as merely physically possible.
In quantum mechanics, numerous myths and fairy tales persist, one of the most prevalent being the notion that every microsystem possesses a definitive state represented by a vector in a Hilbert space, such as a Schrödinger wave function However, this idea has yet to be validated as a reality.
Starting with the concept of a fairy tale can be an effective method for developing a physical theory, despite the potential risk of bias in these theories Often, there has been an attempt to establish principles that the imagined scenarios must adhere to, which are frequently grounded in philosophical reasoning.
The mathematical theory, referred to as MT, is a crucial component of a physical theory, alongside its physical reality This article will succinctly outline the essential elements required for constructing a mathematical theory, with further details available in reference [6].
Mathematics involves the study of imagined objects and their relationships, necessitating the formalization of its methods and outcomes This formalization helps clarify key concepts such as "terms," "relations," "axioms," "proofs," and "theorems," aligning closely with intuitive understanding By focusing on concrete meanings of terms like "structure," "partial structure," and "relation," we provide clarity to mathematical assertions Without this structured approach, statements like "In a partial structure of a mathematical theory, we gain insights into the real structure of reality" would remain ambiguous and lack precise significance.
In this article, we will outline the formal construction of an M T, acknowledging that a completely uniform approach in mathematics is challenging Given the complexity of various formal construction possibilities, we have selected the most suitable method for our purposes, specifically the application of an M T within a P T, as discussed in Chapter 3.
Formal Language
A mathematical theory (MT) is defined as a collection of symbols governed by specific rules The ability to formulate all mathematical expressions, known as mathematical language, relies on a limited set of straightforward rules Consequently, an MT can be formally defined through these language rules, which dictate the use of symbols.
The foundational aspect of mathematical language can be understood through its syntax, which can be likened to the rules of a game involving symbols In this article, we will explore and define these essential rules that govern the use of mathematical signs.
Mathematical text is composed of letters and recognizable symbols, such as ∨, ơ, ∈, and ⊂ These symbols are organized into sequences known as assemblies, which consist of signs written in succession To ensure clarity in mathematical language, specific rules are established to define "well-formed" assemblies These rules help differentiate between assemblies that represent "objects" and those that denote "relations."
Understanding mathematical texts is crucial for physics, as it involves translating facts from the application domain into well-formed relations using mathematical language These relations must also connect to assertions about the reality domain Therefore, establishing clear rules for formulating well-formed relations in mathematical terms is essential.
To this end, we divide the signs into three categories:
1 The logical signs: ∨, ơ, τ The sign ∨ means “or,” the sign ơ means
The logical sign "not" is represented as "¬," while the sign "τ" denotes "an object which " Further clarification on the meaning of "τ" will be provided later These logical symbols are adequate for our purposes Since an in-depth exploration of all the rules is not essential, we will proceed with a more concrete approach in our writing.
“notA;” instead of∨ABalways “AorB;” instead of∨ơAB, i.e., instead of “(notA) or B” always “A⇒B,” or in words “A impliesB;” and for
“not [(notA) or (notB)]” simply “AandB.”
2 The letters; they always represent objects Assemblies can also represent objects.
3 The specific signs of the M T considered as, e.g., the sign ∈ of the set theory.
Only the assemblies that result from the following rules are allowed in an
M T are recognized as well-formed assemblies, with each specific sign in the third category requiring a distinct characteristic These signs can be either substantiative, identifying a concrete object, or relational, asserting a relationship about one or more objects Additionally, each specific sign must be assigned a weight represented by an integer n.
In the context of theM T, "terms" refer to any assemblies that start with a τ or a substantive sign, as well as those composed of a single letter Conversely, "relations" encompass all other types of assemblies within the framework.
One designates by “formative construction” in an M T a sequence of as- semblies that have the following property, i.e., for each assembly A of the sequence, one of the following conditions is satisfied:
(b)Ais identical to “notB,”B being a relation precedingAin the sequence. (c)A is identical to “B or C,”B and C being relations preceding A in the sequence.
In the sequence, A is equivalent to τ x (B), where B is a preceding relation that includes the letter x This relationship is commonly denoted as B(x) to signify that x represents an object within an assertion.
B(x)” (e.g.,x∈M, i.e.,xis an element of the setM) The termτ x (B) is a “privileged” term which, inserted inB(x), satisfies the relationB (e.g., τ x (x∈M) is a privileged element of the setM).
A is equivalent to the sequence sA1, A2, , An, where s represents a specific sign of the third category and weight n If s is a substantive sign, then the sequence A1, A2, , An forms a new object that is defined by the objects A1, A2, , An, representing an assertion about these objects.
Axioms and Proofs
The previously outlined rules are designed to define well-formed assemblies, but now we must explore how to determine the truth of an assertion This involves the establishment of axioms and the construction of proofs In mathematics, axioms are considered true by definition However, when these axioms are applied to the reality domain, their truth takes on a different meaning compared to mathematical theory Therefore, we will refrain from discussing this further.
In mathematics, the concepts of "true" and "false" are frequently determined by established axioms rather than derived from empirical knowledge Many axioms may lack perceivable truth, as their validity often does not exist in a conventional sense.
M T it is possible to pose, instead of the axiomA, the axiom “notA” (as a
“physical” example, see the “axiom of simultaneity,” in [2, Chap VII], and its “nonvalidity” in the theory of special relativity, in [2, Chap IX]).
The establishment of axioms is crucial for both mathematics and physics, necessitating a comprehensive understanding We differentiate between explicit axioms and the underlying axiomatic rules that govern their application.
An explicit axiom is defined as a relation that adheres to the guidelines outlined in Section 2.1 Multiple explicit axioms can be formulated, incorporating certain letters that represent indefinite basic objects of the mathematical theory (MT), referred to as constants These explicit axioms effectively convey true assertions regarding these basic objects, which can also be considered implicitly defined by the axioms themselves In summary, it is common to assign names to these basic objects as a shorthand for the comprehensive set of axioms established for them.
Thus, e.g., a termx(wherexis a basic object) denotes an “ordered set” if an ordering relation is defined on this term with corresponding axioms.
Axiomatic rules are not relations as defined in Section 2.1; rather, they serve as foundational principles from which new relations can be derived based on existing ones These rules are designed to yield "identically true" relations, ensuring that any relations applied in the context of an axiomatic rule will result in a logically valid outcome We will explore these axiomatic rules further in Section 2.3, where they will be presented as logical rules that combine logically identical true relations.
Axiomatic rules can be effectively expressed through abbreviations for assemblies, allowing these rules to be represented as symbolic relations known as implicit axioms In this context, the letters used as abbreviations are not directly present in the theory, as any relations derived from the theory can substitute them Consequently, a mathematical theory (MT) comprises a collection of distinct relations, which are essentially "true" or "valid" assertions, generated through three fundamental rules.
2 the implicit axioms, if they contain terms and relations built according to the rules of Sect 2.1;
3 of a relationB, in the case where the two relationsAand “A⇒B” appear previously in the text of theM T.
The relations derived from assertions (1) to (3), which consist of "true" assertions alongside the well-formed assertions outlined in Section 2.1, are classified as theorems of M T To enhance practicality, we also incorporate all explicit axioms into the theorems of M T.
If a well-formed relation cannot be established using the three previous rules, it is not considered a theorem in M T Importantly, the absence of the relation A as a theorem does not mean that "not A" is necessarily a theorem in M T This distinction will play a crucial role in the advancement of physical theories, especially in transitioning to more comprehensive theories and in understanding the "physical reality" of unobserved facts.
One already introduces here another concept which will become very significant in a P T This concerns the comparison of two M Ts A theory
M T 2 is considered "stronger" than M T 1 when all signs of M T 1 are also signs of M T 2, all explicit axioms of M T 1 are theorems in M T 2, and all implicit axioms of M T 1 are included as implicit axioms in M T 2 Consequently, this implies that every theorem derived from M T 1 is also a theorem within M T 2.
The transition from a weaker MT1 to a stronger MT2 is crucial during the construction and expansion of a PT As the MT strengthens, the PT it relies on becomes increasingly significant and impactful.
Logics
The initial implicit axioms pertain to the choice of logic, specifically opting for "normal" or "bivalent" logic over polyvalent or alternative forms This decision is crucial, as the relationships within an M T translate into assertions about reality in a P T context Consequently, we assume this logical framework for all P T, a concept that will be further elucidated in Chapters 3 and 6.
Efforts to alter logical frameworks have been observed in both mathematics and physics In particular, quantum mechanics has been cited as a compelling reason for the adoption of a polyvalent logic, which incorporates a probability logic characterized by a continuous spectrum of values.
“true” and “false” as limiting extremes The fact that we build a quantum theory with normal logic shows that such a necessity does not exist.
We introduce logic by the following axiomatic rules: IfA, B, C are rela- tions, then the relations
(A⇒B)⇒((C orA)⇒(C orB)) (2.3.4) are implicit axioms ofM T.
In a binary relation where values can be either "true" or "false," the expression "A or B" is considered true if at least one of the relations, A or B, holds true; otherwise, it is false Additionally, the relation "not A" is true if A is false, and vice versa Consequently, the implications outlined in (2.3.1) to (2.3.4) represent valid true relations, as "A ⇒ B" is defined accordingly.
The logical expression “(not A) or B,” represented as “A ⇒ B,” holds true when both A and B are true, or when A is false However, it is crucial to distinguish between the logical implicit axioms and the intuitive understanding of "true" or "false" as they relate to any given relations A and B.
In Section 2.2, we did not define the truth values of relations as true or false, but instead established rules for deriving new relations from axioms Intuitively, we can say that in model theory (M T), the phrase "A is a true relation" can be rephrased as "A is a theorem in M T." It is important to note that if A is not a theorem in M T, it does not automatically follow that "not A" is a theorem either; it is possible for neither A nor "not A" to be theorems in M T The logic introduced by the implicit axioms is considered normal logic only under these conditions.
In a mathematical theory (M T), if both a relation A and its negation "not A" are established as theorems, then every well-formed relation B is also deemed a theorem, rendering the M T contradictory and ineffective Such contradictions prevent the theory from conveying meaningful information As discussed in Chapter 3, this contradiction can lead to an entirely unusable proof theory (P T) even before empirical testing occurs Therefore, to maintain the integrity of the theory, we must exclude all contradictory M Ts, allowing only either A or "not A" to be valid theorems within the framework.
(b) The following principle ofproof by contradiction, often used during proofs, is also valid: If one adds to M T, as an additional axiom, the relation
“notA,” then one obtains a theoryM T strongerthanM T (in the sense of Sect 2.2); and ifM T is contradictory, thenAis a theorem inM T. (c) IfAis a theorem inM T, then “AorB” is a theorem inM T.
(d) IfB is a theorem inM T, then “AorB” is a theorem inM T.
(e) If “not A” and “not B” are theorems in M T, then “not (A or B)” is a theorem in M T; and also if “A or B” is a theorem in M T, then
“not [(not A) and (not B)]” is a theorem in M T, i.e., “not A” as well as “notB” cannot be theorems inM T.
The two criteria (a) and (b) fix the sense of “not,” and what we briefly indicate in thenew form by bivalent logic.
Criteria (c) to (e) establish a clear interpretation of "or," refining its meaning by incorporating the values of "true" and "false." In essence, these criteria define the conventional understanding of "or."
In this sense, on the basis of (a) to (e), we say finally that the bivalent logic is introduced by the implicit axioms (2.3.1) to (2.3.4).
While we won't elaborate on all the significant implications of the implicit axioms (2.3.1) to (2.3.4) for mathematical proof techniques, it's important to note that the deductions (a) to (e) are largely intuitive Readers are likely familiar with applying similar logic and proof methods in mathematics For a comprehensive analysis, please refer to [6].
The deductions (a) to (e) can be easily established using the references from [6, Chap I, Sect 3] Specifically, deduction (a) is validated in [6, Chap I, Sect 3.1], while (b) corresponds directly to C 15 of [6, Chap I, Sect 3.3] Deductions (c) and (d) naturally follow from the implicit axioms (2.3.2) and (2.3.3) and the proof rule (3) outlined in Sect 2.2 Lastly, deduction (e) arises from the equivalent relations detailed in C 24 of [6, Chap I].
I, Sect 3.5], “not (notA)⇔A” and “(AorB)⇔not [(notA) and (notB)].”
Due to their relevance, especially in the context of physics as discussed in Chapters 3 and 6, two additional relations derived from the implicit axioms (2.3.1) to (2.3.4) are included.
In the context of mathematical theory M T, when a relation A is added to its axioms, the resulting theory is denoted as M T If B is established as a theorem within this theory, it follows that the implication "A implies B" also qualifies as a theorem in M T For a detailed proof, refer to C 14 of [6, Chap I, Sect 3.3].
In the context of model theory, let A(x) and B represent relations within a structure M T, where x is not a constant of M T Consider a term T for which A(T) is a theorem in M T By augmenting the axioms of M T with the relation A(x), we form a new theory, denoted as M T, where x is treated as a constant.
M T ) IfB is a theorem inM T , thenB is a theorem inM T (Proof: see
In the context of the theories M T, if neither A nor "not A" are established as theorems, one can incorporate both A and "not A" as axioms, leading to the development of two enhanced, non-contradictory theories, M T 1 and M T 2 This scenario holds particular importance in the field of physics, especially when examining the relationship between Galileo–Newton space-time theory and special relativity theory.
In this discussion, we will not delve into the challenges surrounding the proof of non-contradiction in mathematical theories (MT) We assume that the mathematical theories in use are consistent unless a contradiction is identified If a contradiction does arise within a mathematical theory, it becomes necessary to revise the axioms to resolve the inconsistency.
If Aand B are relations, we briefly write for the relation “(A⇒B) and
(B ⇒A)”:“A⇔B” and we say that “AisequivalenttoB.” For any relations, because of the axioms introduced above, the following equivalences (C) are valid (as theorems inM T, see [6, Chap I, Sect 3.3]):
(Aand (B orC))⇔((AandB) or (AandC)),
(Aor (B andC))⇔((AorB) and (AorC)),
(not (Aand B))⇔((notA) or (notB)), (C)
(not (Aor B))⇔((notA) and (notB)),
When we treat the symbol ⇔ as an equality sign and replace "and" with "∧" and "or" with "∨," the logical theorems discussed align with the calculation rules of a complemented distributive lattice.
Set Theory
In exploring the role of axioms within set theory, we focus on key elements that are crucial for their application in a physical theory (P T) While the implications of set theory in the mathematical representation of a physical theory (M T) will be addressed in future discussions, we will not delve into the physical interpretations at this stage.
In the set theory there appears as a new relational sign: “z∈y” (concretely,
In set theory, the notation "x ⊂ y" signifies that all elements z of set x are also elements of set y, which can be expressed as "(∀z)((z ∈ x) ⇒ (z ∈ y))." This indicates that x is a subset of y, meaning y contains all elements of x Conversely, when stating "not (z ∈ y)" or "not (x ⊂ y)," it implies that there are elements in x that are not found in y.
The relational sign∈will become of decisive importance for the use of an
M T in aP T Concretely, aP T includes assertions about the facts of reality as elements of a set (see Chap 3).
In set theory, a set is understood as the complete collection of its elements; however, this concept becomes problematic in physics The idea of a "whole of all" elements is questioned, particularly in the context of statements like the "set of all electrons," which lacks meaning in the construction of a physical theory This raises doubts about the existence of such a totality of all electrons.
In mathematics, “to collect in a set” is a significant notion of the set theory.
If the set is a collection of its elements, then two sets must be identical if they have the same elements; for this reason, one requires as a first explicit axiom
But it is precisely this “collection in a set,” intuitively so obvious, which leads to contradictions in mathematics when one neglects to take certain precau- tionary measures.
To formally group all elements of a specific type into a set, we define a relation R(x) This is represented by the notation "Coll x R," which signifies that there exists a set y such that for every element x, x is a member of y if and only if R(x) holds true If "Coll x R" is established as a theorem, it provides a foundational principle for constructing such sets.
In M T, a relation R is described as collectivizing if it determines a set y, defined as the "set of all x that satisfy R(x)." This leads to the conclusion that if (∀x)((x∈y)⇔R) and (∀x)((x∈z)⇔R), then z equals y.
In this article, we define the set E x (R) as the collection of all x such that R(x) holds true, expressed as "E x (R) is the set of x such that R(x)." The relationship between E x (R) and R is captured by the equivalence (∀x)((x ∈ E x (R)) ⇔ R), which aligns with the concept of Coll x R It is important to note that the notation E x (R) will often be simplified to the standard form "{x | R(x)}." However, the existence of the set E x (R) is contingent upon Coll x R being a theorem within the framework of M T Interestingly, Coll x R is not universally a theorem for all R(x), raising questions about the expected existence of such sets for every R(x).
R(x).” Wouldn’t it be easy to conceive of Coll x R as an axiom for allR(x)?
All those who have dealt with problems of an “intuitive” set theory know that such a general condition contains problems For this reason we will proceed in a more careful way.
When considering the elements \( x \) for which the relation \( R(x) \) holds true, these elements form a subset of the set \( z \), which may also include elements that do not satisfy \( R \) Consequently, the collection of \( x \) satisfying \( R \) can be established as a theorem in model theory (M T) If the relation \( R \) is dependent on an object \( y \), and all \( x \) that meet the criteria of \( R \) are part of a set \( z \) that may rely on \( y \), then for at least one element \( y \) from a set \( u \), all \( x \) that satisfy \( R \) must also constitute a valid set, as stipulated by the implicit axiom.
This will make it possible to obtain, starting from sets, new sets using rela- tions But to be able to produce sets we pose the following axioms:
This means that ifxandyare objects, then there is a set whose only elements arexandy We indicate this by{x, y} This axiom is very easy to interpret in a
In set theory, a finite set is defined as a collection of a limited number of elements, such as x1, x2, , xn However, infinite sets, which will be elaborated on later, play a crucial role in the development of set theory and necessitate a rigorous axiomatization It is important to note that while these infinite sets are significant in theory, they cannot be physically interpreted, a topic that will be explored in greater detail in Section 3.2.4.
To advance set theory, it is essential to introduce a new term represented by a pair (x, y), which combines two individual objects into a single entity.
The pair (x, y) is different from the set{x, y}! In the pair, according to (2.3.4), the componentsxandy are ordered.
(∀x)Coll y (y⊂x) (2.4.5) means that “the set of all the subsets of a setxexists.” The last axiom postulates the existence of an infinite set (2.4.6)
An infinite set is precisely a set not finite A finite set is defined by the fact that the cardinality changes if one adds one element to the set.
EachM T used in aP T is stronger than the set theory, i.e., all the axioms indicated until now are valid in M T In what follows, we suppose that each
M T is stronger than the set theory.
In the realm of physics, as discussed in Section 3.2.4, a potential seventh axiom could be introduced to set theory, asserting that no set exists with a size strictly between that of the integers and the continuum This axiom can be demonstrated to be independent of previous axioms Alternatively, it may be necessary to stipulate that every set within an M T is either countably infinite or a subset of a collection of sets that are at most countable.
In an M T (stronger than the set theory), starting from n sets (terms)
In the context of set theory, one can construct new sets step by step from given sets E1, , En The notation P(E) represents the collection of all subsets of E, while E1 × E2 denotes the set of all ordered pairs (x, y) where x belongs to E1 and y belongs to E2 By repeatedly applying the operations P and × to the sets E1, , En, one can generate new sets through a finite process known as echelon construction The resulting set from this process is referred to as an echelon, denoted by S(E1, , En), where S signifies the scheme used for the construction If E1, , En are distinct sets, S(E1, , En) will also represent an echelon based on the same construction scheme but with the specified base sets.
In the context of M T, let f i represent mappings of the sets E i onto themselves, ensuring that for every x in E i, the mapping f i (x) also belongs to E i By utilizing these mappings, one can straightforwardly construct canonical extension mappings from E 1, , E n onto E 1, , E n through a systematic step-by-step process.
1 By defining a mappingg of P(E) ontoP(E ) starting from a mapping f of E ontoE such that, for a subset e⊂E, g(e) is defined as the subset of allf(x) such thatx∈e.
2 By defining a mappinggofE 1 ×E 2 ontoE 1 ×E 2 , starting from the map- pingsf 1 ofE 1 ontoE 1 andf 2 ofE 2 ontoE 2 , byg(x, y) = (f 1 (x), f 2 (y)).
The application ofS(E 1 , , E n ) onto S(E 1 , , E n ) thus obtained is de- noted byf 1 , , f n S
If all functions \( f_i \) are injective (or surjective), then the composite function \( f_1, \ldots, f_n S \) is also injective (or surjective), as this property holds for each step in the echelon construction scheme \( S \) When \( f_i \) are mappings from \( E_i \) onto \( E_i \) and \( g_i \) are also mappings from \( E_i \) onto \( E_i \), the composition of these mappings is denoted as \( g_i f_i \) Consequently, it follows that \( g_1 f_1, \ldots, g_n f_n S = g_1, \ldots, g_n S f_1, \ldots, f_n S \).
If all f i are bijective (i.e., injective and surjective), with g i = f i − 1 , then f 1 , , f n S is also bijective and
(f 1 , , f n S ) −1 =f 1 −1 , , f n −1 S , wheref −1 is the inverse bijection off.
In the context of echelons G1, G2, , Gp, if multiple elements s1, s2, , sp are present, one can construct an element s = (s1, s2, , sp) that belongs to the set G1 × G2 × × Gp, which also qualifies as an echelon Additionally, when a relation R(x1, , xp) exists, it is possible to examine the implications of this relation within the defined structure.
Otherwise, one can also takeRas a relation of only onexofG=G 1 × .×G p There is the theorem: Coll x
R(x) andx∈G , i.e.,R(x) determines inG a subsetH ⊂Gsuch that{x∈H⊂R(x) andx∈G}.
The set H, previously referred to as E x (R(x) and x ∈ G), will now be denoted as {x | x ∈ G and R(x)} This set H remains an element of P(G), indicating that a relation R can be represented by a subset or as an element of an echelon Similarly, functions and applications can also be characterized as elements of an echelon.
Recording Process
The initial stage in transitioning from reality to mathematics involves a recording process of facts, represented by ↔ This process captures facts pertinent to the application domain \( A_p \) of the theory, expressed in simple, natural language sentences known as the basic language of the theory.
The existence of an intuitive language guarantees a nonproblematic semantic framework, characterized by a complete and exclusively empirical interpretation This language relies solely on concepts derived from direct observations or indirect observations facilitated by pre-theoretical insights.
Natural language, despite its inherent syntactic and semantic ambiguities, remains the cornerstone of formal languages In this context, we introduce a simplified natural language known as basic language (B l), which expresses facts about reality through natural sentences that describe the properties of objects and their relationships.
(r) between the objectsa 1 , , a n and finite many real numbersα 1 , , α n , there is the relationr n (a 1 , , a n , α 1 , , α n ).
To formulate sentences of the forms (p) and (r), it is essential to first establish the objects, their properties, and their relationships Developing a physical theory requires the ability to determine these realities independently of the new theory being proposed However, this does not imply the absence of any theoretical framework; rather, we often rely on pre-theories, which are established theories that inform our understanding Later, we will explore how existing theories can help us identify new realities.
To establish a foundational understanding of physics, we must start with theories that rely solely on direct sensory observations, avoiding any reliance on pre-existing theories This approach emphasizes the importance of tangible realities in the study of physics, rather than delving into the complexities of constructing an all-encompassing framework for the discipline.
1 Later, in Chap 6, we will extend the basic languageB l to anextended basic lan- guage, denoted byB l ex , by introducing ‘new words’ designating “new concepts.”
By these “new concepts” we detect also - new facts -.
In this book, we assert that the field of physics is defined by observable realities and the methodologies used to develop physical theories While we acknowledge the existence of realities beyond direct physical observation, we do not accept them as part of the observable realm of physics Throughout the text, we will briefly reference these undetectable realities.
The methodology of physical theories presented in this book extends beyond the realm of physics, allowing application to various domains by considering realities not directly observable by physics For instance, instead of focusing on the live sound of a symphony, we can analyze a CD recording, which captures the vibrations produced by the orchestra To appreciate the essential value of the symphony as conveyed through the CD, one must create an effective hi-fi sound system, necessitating a solid understanding of both physical and psychological principles.
The basic formulations of sentences in the language are inadequate, as they do not account for the role of real numbers in expressions Real numbers cannot be directly observed in nature, as we only encounter whole numbers, such as counting sheep in a herd Consequently, real numbers emerge through the use of pre-theories, which define the objects that these numbers represent.
Using Newton's theory of space-time as a foundational concept, we can represent time with a single real number \( t \) indicating a specific time point, while a set of three real numbers \( (x_1, x_2, x_3) \) denotes a location in space To accurately define these values, it is essential to establish a fixed space-time reference system.
Real numbers in physics (r) require clarification regarding their pre-theoretical origins and definitions, such as using 'timet' for time or 'positionx1, x2, x3' for position, to convey their specific meanings Often, terms like 'time' and 'position' are abbreviated to letters like 't' and 'x', with their significance also influenced by their context within (r) While the use of real numbers simplifies mathematical descriptions in physics, leading to a perception that the discipline is primarily quantitative, it is essential to recognize that physics encompasses more than just quantitative analysis The methods for defining real numbers within physical theories will be explored further in Section 3.1.1.2.
To formulate sentences of the forms (p) and (r), it is essential to designate an object using a symbol, such as the letter 'a i' or any other sign For instance, if we have two glass balls, we can identify one by marking it appropriately.
In quantum mechanics, distinguishing between individual electrons is fundamentally impossible For instance, in a helium atom with two electrons, labeling one as 'a' and the other as 'b' contradicts the principles of quantum mechanics Similarly, during a collision experiment between an electron and a hydrogen atom, it is incorrect to assert that the electron exiting the collision is the same as the one that entered, as the nature of quantum particles defies such identification.
In scientific terms, objects can be defined by specific letters, while electrons do not fit this classification However, a "physical system" comprising multiple electrons and a nucleus can be considered an object if it can be distinctly identified through the comprehensive experiments outlined in the relevant theory.
Restricting the fundamental language of our proposed physics theory to clear sentence forms (p) and (r) is crucial for effective communication However, physicists often utilize complex and condensed language that can lead to misunderstandings among non-experts To clarify this issue, we will provide examples of language forms that fall outside the basic framework and fail to accurately represent the proper method of physics.
An example of such a sentence that could be misunderstood is the following:
‘A ruler in motion is shorter than a ruler at rest’.
Such a sentence makes no sense If one analyzes it, it is syntactically correct. Nevertheless, it is semantically incorrect This sentence is in contradiction with itself.
The concept of "in motion" is relative, meaning that when we say the first ruler is in motion, it implies movement in relation to the second ruler Consequently, this also indicates that the second ruler is moving relative to the first This relativity leads to a contradiction, as it suggests that the second ruler is shorter than the first ruler.
The above sentence is an incorrect formulation of the following reality:
Mathematization Process
The second step in transitioning from reality to mathematics involves the amathematization process, referred to as (cor) This process converts natural sentences expressed in the basic language B1, which pertains to the application domain Ap, into formal sentences articulated in a mathematical language that aligns with the theory.
This paragraph introduces a formal mathematical language designed to accurately transcribe natural sentences expressed in the basic language B l, which represent facts within the application domain A p.
We begin with the mathematical language M T, characterized by "well-formed" combinations of signs, logic, and set theory, as outlined in Chapter 2 This language may include certain constants and their associated axioms, although this is not a requirement To develop a more comprehensive mathematical language that meets our specific needs, we will take additional steps.
In the framework of model theory, we introduce two types of relations to the model MT: for each property p of the set Bl, we establish a formal relation p(x), and for each relation r within Bl, we create a formal relation r(x1, , xn, α) These relations, p and r, are newly defined constants within the model MT Notably, if a relation r includes real numbers sourced from pre-theories, the corresponding formal relation will retain those same real numbers.
– the natural sentence of the basic languageB l ‘the objectahas the property p’ corresponds to the formal sentence of the enrichedM T ‘p(a)’;
– the natural sentence of the basic languageB l ‘between the objectsa 1 , a 2 , , and the numberα, there is the relationr(a 1 , a 2 , , α)’ corresponds to the formal sentence of the enrichedM T ‘r(a 1 , a 2 , , α)’.
In a similar way, the natural sentences compounded by the connective words ‘and’ and ‘not’ inB l correspond to the formal sentences compounded by the logical signs ‘∧’ and ‘ơ’ in the enrichedM T.
The natural sentences in B l correspond directly to the formal sentences in the enriched M T Additionally, the constants p, r, and the logical symbols ‘∧’ and ‘ơ’ are defined through the process of mathematization.
(ii) We add toM Tnot only constants but also two axioms for these constants.
InB l some properties are characterized as basic properties Let p 1 ,p 2 ,p 3 be the basic properties in B l , and let p 1 , p 2 , p 3 be the corresponding relations added by (i) toM T We then define the relationp 0 =p 1 ∨p 2 ∨p 3
The first axiom is thecollectivizing axiom
∃y(p 0 (x)⇔x∈y) and, briefly, p 0 (x)⇔x∈y are equivalent The set y is then uniquely determined by p 0 (x); we write y={x|p 0 (x)} Instead ofp 1 (x)∨p 2 (x)∨p 3 (x) we can write x∈y.
The axiom (3.2.2) asserts that we can only document a finite number of facts regarding the properties of objects and their relationships, meaning there can only be a limited number of objects in the textA This can be demonstrated through experiments, such as counting the elements in a set representing a herd of sheep However, in physics, there exist sets with such vast quantities of elements that it becomes impossible to count or record every single object.
In conclusion, we take together the constants p, r and the axioms (3.2.1) and (3.2.2) under the signΘ For the theory enriched in this way we write
In our model M, we introduce two new constants, p and r, where p represents a relation of weight 1 denoted as p(x), and r signifies a relation of weight 3 expressed as r(x1, x2, α), with α being a real number from a finite set of real numbers IR We consider p as a fundamental property Consequently, based on axiom (3.2.1), we establish that there exists a set M such that x is an element of M if and only if p(x) holds true.
On the basis of the axiom (3.2.2),M is a finite set.
The introduction of M T Θ serves to highlight that the "collectivizing" axiom underpins the standard form M T Θ, which will be utilized in all subsequent descriptions of physical theories.
To enhance mathematical clarity, we propose a slight modification to the definition of Θ, replacing the constants p and r with M0 and several others, along with updated axioms.
M 0 is a finite set; (3.2.3) and for the variouss, s⊂M 0 (3.2.4) or s⊂M 0 ìM 0 ì ã ã ã ìIRì ã ã ã , (3.2.5) where IR is afiniteset of real numbers.
We define the following correspondences betweenM T Θ andM T Θ :
{x|p 0 (x)}corresponds to M 0 , (3.2.6) p(a) (for a propertyp)corresponds toa∈s
(for the pcorresponding tos), (3.2.7) r(a 1 , a 2 , , α) (for a relationr)corresponds to (a 1 , a 2 , , α)∈s
Since we have postulated that all objects aof the application domainA p have the propertyp 0 , we can add this propertyp 0 to everyp(x) and to every r(a 1 , a 2 , , α):
The above correspondences (3.2.6) to (3.2.8) follow then from the theorems (see Sect 2.3)
The relationship between the natural sentences of B l and the formal sentences of M T Θ establishes a significant correspondence, denoted as B l (cor) M T Θ This connection highlights the alignment between these two systems, reinforcing their interrelatedness.
It is often more convenient not to introduce p 0 (x) and the corresponding
To establish a comprehensive framework, we define a base set M_i for each fundamental property while retaining all essential characteristics This approach creates a correspondence between the natural sentences of B_l and the formal sentences of M_T Θ, facilitating a clearer understanding of their interrelations.
– ‘the objectahas the basic propertyp i ’corresponds to‘a∈M i ’;
In the context of object properties, if an object \( a \) possesses a fundamental property \( p_i \), it also exhibits the property \( p \) This relationship can be expressed as \( a \in s \subset M_i \), where \( s \) represents both properties \( p_i \) and \( p \) Furthermore, for two objects \( a_1 \) and \( a_2 \), with basic properties \( p_1 \) and \( p_2 \) respectively, the existence of a relation \( r_n(a_1, a_2) \) can be denoted as \( (a_1, a_2, \ldots) \in s \subset M_{i_1} \cap M_{i_2} \).
Thus the mathematization process B l (cor)M T Θ is well defined also for this general case where we take all basic properties separately.
As an axiom we only postulate that allM i are finite sets.
We add toM T two new constantsM ands On the basis of the axiom (3.2.3),
M is a finite set On the basis of the axiom (3.2.5), we have s⊂M×M ×IR.
The mathematization processB l (cor)M T Θ , i.e., the transcription of natural sentences formulated in the basic language into formal sentences formulated in the formal language, is given by
‘the measured distance between aandbisα±ε’ (cor) ‘(a, b, J)∩s=∅’. HereJ is the intervalα−εto α+ε.
Through the mathematization process, we have created a correspondence between a finite collection of natural sentences, referred to as A, expressed in the basic language B l, and a finite collection of formal sentences, also denoted by A, formulated in mathematical language.
In this article, we differentiate between recorded facts as either "stated," where the factual truth is explicitly mentioned, and "not stated," where there is uncertainty about whether they were acknowledged In Chapter 6, we will explore how to articulate real facts using aP T, even when they haven't been directly observed or previously theorized The text primarily includes sentences that reflect facts established through direct observation or pre-theoretical frameworks, but it may not encompass every instance of "stated facts," focusing instead on a selection of them.
Mathematics is utilized in physics because it allows us to express fundamental concepts in a clear and straightforward manner.
If we add toM T Θ the results of a mathematization process, i.e., the text
Idealization Process
Mathematical theories in physical applications are often mere approximations of reality, functioning effectively within specific domains only when a certain degree of inaccuracy is accepted This section will enhance the understanding of mathematical theories by incorporating the concept of idealization.
The system Θ, defined by constants added to M T to form M T Θ, is characterized by base sets M i and subsets s ⊂ M i Similarly, the system ∆ is characterized by the same constants as Θ, along with a specific procedural framework.
– For every base setM i is defined in M T a setQ i ,
– For everys⊂M i ì ã ã ãis defined inM T a subsets⊂Q i 1 ì ã ã ã ìIRì ã ã ã, – For everyQ i and everyQ i ì ã ã ã ìIRì ã ã ãwiths⊂Q i ì ã ã ã ìIRì ã ã ãare defined inM T “inaccuracy sets”U i respectively U s
As “inaccuracy setU for a setX” is denoted a setU ⊂X×Xwhich comprises the diagonal set D of all pairs (x, x), for which the relation (x, y) ∈ U ⇒
(y, x)∈U is a theorem, and for which the later relation (3.3.4) is valid.
We now start to get the axiom for∆ By this axiom we want to postulate that theQ i , s are “idealized” picture sets of the “physical” setsM i , sof Θ.
All mappings of product spacesM i ìã ã ãìIRìã ã ã onto Q i ìã ã ãìIRìã ã ãwhich are generated canonically by the φ i and by the identical mapping IR → IR will be denoted byφ.
If the φ i were bijective and if for U the diagonal sets are chosen, then
In many physical theories, the bijective mappings φ i and diagonal sets D can lead to contradictions in the context of M T ∆ A Consequently, we view the Q i and s as "idealized" representations of the physical sets M i and s.
The physicist often employs a specialized language that may lead to misunderstandings among those outside the field Regarding the correlation of M T Θ with reality, we will also assert that the components of M i and s are significant.
In Chapter 6, we will explore the challenge of determining whether the elements of Mi and the relations s can be considered "real." We will demonstrate that, under specific conditions, this is indeed possible However, it is important to note that physicists frequently refer to the elements of Qi and s as physical entities, even though they merely represent "imprecise" interpretations of reality.
How can we describe that theφ i are only “imprecise” mappingsM i →Q i ?
At first we postulate that theφ i are injective (3.3.2)
The postulate (3.3.2) says that the elements of theQ i can distinguish between different elements of the M i , i.e., that two elements of M i will also have different pictures.
Since the M i are finite sets the relation (3.3.2) says practically nothing about theφ i , since if (3.3.2) was not valid, then we could replaceQ i by a set
The equation Q i × R i indicates that the R i do not affect the s, suggesting that the expression (3.3.2) merely serves as a refinement of the mathematical representation rather than conveying any insights about reality Consequently, we propose a less stringent relation instead of focusing on surjectivity.
Postulate (3.3.3) indicates that while there are sufficient elements in M i from a mathematical perspective, it does not guarantee that all these elements represent "real" objects We assume, however, that M i exclusively includes elements that pertain to "real" entities This topic will be explored further in Chapter 6, where we will examine the criteria for distinguishing what is considered "real" versus fictional.
We want to express (independently of the φ i ) that a set (also infinite)X
(theQ i or the s) is anidealization of a finite set Therefore, we introduce as an additional condition for the definition of an inaccuracy setU forX
Now we can formulate the axiomP ∆ :
Heres is the complement ofsinM i 1 ìã ã ãìIRìã ã ãands is the complement ofsinQ i 1 ì ã ã ã ìIRì ã ã ã, and
. For such a test∆, let us look at the following.
We formulate by∆thatM , sis “approximately” a two-dimensional Euclidean geometry:
InM T Θ we define the following sets:
Q= IR×IR andsas the set s⊂Q×Q×IR of all (q 1 , q 2 , d) with d(α 1 , α 2;β 1 , β 2) (α 1 −β 1) 2 + (α 2 −β 2) 2 , whereq 1= (α 1 , β 1) andq 2= (α 2 , β 2) with real numbersα i , β i
We define the following “inaccuracy” set forQ:
; 0 is the point (0,0) in IR×IR.For IR+ we introduce the inaccuracy set
∪IR ρ + ×IR ρ + , (3.3.7) where IR ρ + ={α|α∈IR + ∧ α > ρ}.We takeδ < ε and >2rπ.
These inaccuracy sets generate in a canonical way also an inaccuracy set
U s in Q×Q×IR+ On the basis of the axiom (3.3.5) we have
(∃φ) φ:M →Q is an injective mapping with (φM) U =Q
If we want to give particular values for theε, r we must take different appli- cation domains, one for the round table and one for the surface of the earth:
– For the round table we can choose, e.g., ε = 0.1 mm and r essentially greater than the radius of the round table, e.g., ten times the radius of the table.
– For the surface of the earth we can choose, e.g.,ε= 10 cm andr= 10 km. But we can also choose other values If we choose a biggerr, we must also choose a biggerε.
The introduction ofAtoM T ∆ is thesameas the introduction ofA toM T Θ
(see Sect 3.2.3) The only difference is the axiom (3.3.5) and that therefore the addition ofA can lead to contradictions.
But there arises a major difficulty if we have in some regions of the Q i
Inaccuracies, such as Q r × Q r in equation (3.3.6), can significantly impact our analysis We have not imposed any restrictions on the "real" part represented by A, which may include elements located in the regions of "large" inaccuracies within Q i Consequently, it is essential to differentiate between two cases regarding the inaccuracy sets U.
In the initial scenario, we observe the absence of significant inaccuracies across regions Consequently, we must incorporate A into M T ∆, as outlined in Section 3.2.3, and evaluate for any contradictions with axiom (3.2.5) A positive outcome for the theory test is confirmed if no contradictions arise with the axiom However, it is essential to determine the requirements for conducting this test.
The textAis given by relations of the form (3.2.9), (3.2.10), and (3.2.11).
If we writeφa=a, these relations take the form of the following relations: a∈Q,
We have no contradiction to the axiom (3.3.5) if the relations of the form a∈Q,
In the mathematical theory \( M_T \), the equation \( (a_1, a_2, \ldots, J) \cap (s) \cup s = \emptyset \) (3.3.10) indicates that the elements \( (a_1, a_2, \ldots, J) \) are subsets of \( (s) \cup s \) without causing contradictions in \( M_T \) when treated as new constants We denote the theory with the addition of this text as \( M_T A \) A positive test outcome occurs when \( M_T A \) remains free of contradictions.
(The relationAwas introduced in [1] Chap 6 by “the principles of impre- cise mapping” formulated intuitively.)
We conclude that in the first case, it suffices for a test to see whetherM T A is without contradiction.
In certain regions characterized by significant inaccuracies, it is possible to establish a mapping φ that satisfies axiom (3.3.5), aligning all elements of set A within these areas of large inaccuracies without creating contradictions However, this approach does not claim that the theory accurately represents reality on a broader scale; rather, it is valid only within specific subregions of the application domain, where small inaccuracy sets can be effectively utilized How can we express this concept using the language of M T ∆?
In our analysis, we select one or more tuples (a₁, a₂, ) from the set M₁, M₂, etc These tuples serve not only as mathematical symbols but also as representations of established real-world objects We modify axiom (3.3.5) by incorporating a condition in brackets that specifies φ(a₁, a₂, ) ∈ Q ⊂ Q₁, Q₂, , where Q denotes a subset characterized by minimal inaccuracies Occasionally, we may designate a specific element q ∈ Q, allowing us to express (3.3.11) as φ(a₁, a₂, ) = q.
If no contradictions are found in the enrichments of A through the addition of new experiments with respect to equations (3.3.5) and (3.3.11) or (3.3.12), we conclude that M T ∆ accurately represents a surrounding of Q or q, albeit with minor inaccuracies.
It can be that we can use the theory for more than oneQ (orq).
We will demonstrate this below.
In this case we record - marked spots - and the results of measurements of the - distances between these marked spots -.
Understanding how to measure distance requires a foundational knowledge of measurement principles, even before delving into the geometry of surfaces It is essential to recognize that all measurements inherently carry an "error of measurement," which signifies their imprecision When measuring the distance between two points, labeled a and b, the result is represented by a real number α However, this measurement should be interpreted as lying within an interval, specifically between α−ε and α+ε, rather than being an exact value.
Now we want to describe the use of our theoryM T ∆ for the description of the real application domain of marked spots on the earth, and their measured distances.
In Section 3.3.6, we observe an instance of the "second case" outlined in Section 3.3.2, where the area of significant inaccuracies is represented by Q r The point (0,0) is identified as the "center" of the Q r region characterized by minor inaccuracies We then select one of the designated points within A, which may include the point a 0.
(a 0 can be, e.g., a particular marked spot at Greenwich in London) We add in (3.3.8) the condition φa 0= (0,0).
Thus we get the axiom
(∃φ) φ:M →Q is an injective mapping with (φM) U =Q
Mathematical Structures
At first we recall the well-known definition of a species of structures:
Let M T represent a theory that surpasses standard set theory In this context, let x₁, x₂, , xₙ, and s denote distinct letters that are not constants in M T, while A₁, A₂, , Aₘ are terms within M T that do not include the letters x₁, x₂, , xₙ, and s Additionally, let S be defined as an echelon construction scheme applied to n + m terms.
T(x 1 , , x n , s) : s∈S(x 1 , , x n , A 1 , , A m ) is called a typificationof the letter s The term s is called a structure term.
A relation P(x 1 , , x n , s) is said to be transportable with respect to the typificationT(x 1 , , x n , s) if bijective mappings lead to equivalent relations, in other words ifM T contains the following theorem:
“T(x 1 , , x n , s)∧(f 1 is a bijective mapping ofx 1 ontoy 1)
∧(f n is a bijective mapping ofx n ontoy n )” follows the relation
The expression P(x₁, , xₙ, s) is equivalent to P(y₁, , yₙ, s), where s is defined as the function f₁, , fₙ, along with identity mappings Id₁, , Idₘ of sets Aᵢ onto themselves This relation, referred to as an axiomatic relation, emphasizes that it will be established as an axiom in subsequent discussions.
We take as textΣthe following groups of symbols: the lettersx 1 , , x n , s, the typificationT(x 1 , , x n , s), and the axiomatic relation P(x 1 , , x n , s).
This textΣ is called aspecies of structures The lettersx 1 , , x n are called theprincipal base sets, the lettersA 1 , , A m theauxiliary base sets, and the lettersthestructure term of the speciesΣ.
By incorporating the relation "T∧P" as an axiom into the theory M T, a more robust theory known as M T Σ is established This enhanced theory retains the constants from M T, along with the variables from T and P, specifically x1, , xn, and s M T Σ is referred to as the theory of the species of structures Σ, signifying its expanded framework.
In physical theories, we can define specific structural forms for various species Each mathematical theory, as a component of a physical theory, inherently includes set theory, which encompasses both real and complex number theories Additionally, these theories incorporate a structural element represented by the text Σ.
1 A certain number of principal base sets and auxiliary base sets x : x 1 , , x n , A:=A 1 , , A m
For this species of structuresΣ we write Σ(x, s) ≡ Σ (T(x, s) ∧ P(x, s) ), and for the mathematical theoryM T endowed with this species of structures Σ(x, s) we write
The expression Σ(x, s) can be represented in a general mathematical format by substituting the ν elements with the typification s ∈ P(PS 1(x, A) ∪ PS 2(x, A) ∪ ∪ PS p(x, A)) This leads to the formulation s ⊂ PS 1(x, A) ∪ PS 2(x, A) ∪ ∪ PS p(x, A), where s is defined as s = s1, , sp, with each sν belonging to PSν(x, A), indicating that sν is a subset of Sν(x, A).
The introduction of the concept of species of structures is evident, as illustrated by Θ from Section 3.2.2 and ∆ from Section 3.3.1, which serve as examples of these species.
1 the principal base setsM i and the auxiliary base set IR,
3 with the typificationss⊂M i or s⊂M i 1 ìM i 2 ì ã ã ã ìIRì ã ã ã,
1 the principal base setsM i as for Θ and the auxiliary base sets IR, IR,
3 with the typificationss⊂M i or s⊂M i 1 ìM i 2 ì ã ã ã ìIRì ã ã ãas forΘ,
4 withthe axiomatic relationP ∆ given in (3.3.5).
Now we want to introduce a relation between two species of structures which is often used, especially in the mathematical parts of a physical theory.
In the study of structures, we examine two species, Σ 1 and Σ 2, which share identical principal and auxiliary base sets as well as typification, yet differ in their axiomatic relations, P 1 and P 2 When P 1 is established as a theorem within the framework of M T Σ 2, we conclude that the species of structures Σ 2 is richer or finer compared to Σ 1, indicating that Σ 1 is poorer in its axiomatic capacity.
(or coarser) than Σ 2 If P 1 is a theorem in M T Σ 2 and P 2 is a theorem in
M T Σ 1 , then we say that Σ 1 and Σ 2 are equally rich, and sometimes also simply “equal.”
This article explores the concept of "lattice" structures within the species of structures Σ, focusing on a principal base set x without any auxiliary base sets The order relation "