From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti
aa r X i v : . [ qu a n t - ph ] F e b From Boolean Valued Analysis to Quantum Set Theory:Mathematical Worldview of Gaisi Takeuti *M ASANAO O ZAWA
College of Engineering, Chubu University, 1200 Matsumoto-cho, Kasugai 487-8501, JapanGraduate School of Informatics, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan
Abstract
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematicapplications of Boolean valued models of set theory to analysis. Later, his methods werefurther developed by his followers, leading to solving several open problems in analysisand algebra. Using the methods of Boolean valued analysis, he further stepped forward toconstruct set theory based on quantum logic, as the first step to construct “quantum mathe-matics”, a mathematics based on quantum logic. While it is known that the distributive lawdoes not apply to quantum logic, and the equality axiom turns out not to hold in quantumset theory, he showed that the real numbers in quantum set theory are in one-to-one corre-spondence with the self-adjoint operators on a Hilbert space, or equivalently the physicalquantities of the corresponding quantum system. As quantum logic is intrinsic and em-pirical, the results of the quantum set theory can be experimentally verified by quantummechanics. In this paper, we analyze Takeuti’s mathematical world view underlying hisprogram from two perspectives: set theoretical foundations of modern mathematics andextending the notion of sets to multi-valued logic. We outlook the present status of his pro-gram, and envisage the further development of the program, by which we would be able totake a huge step forward toward unraveling the mysteries of quantum mechanics that havepersisted for many years.
Key words and phrases : Takeuti, Boolean algebras, set theory, Boolean valued models,forcing, continuum hypothesis, cardinal collapsing, Hilbert spaces, von Neumann algebras,AW*-algebras, type I, orthocomplemented lattices, quantum logic, multi-valued logic, quan-tum set theory, transfer principle, quantum mathematics
In 1982, Gaisi Takeuti published a book entitled
Mathematical Worldview: Ideas and Prospectsof Modern Mathematics [1]. In this book, he states that mathematics based on classical logic isan absolute truth, and it has the characteristic that all possible statements always belong to onlyone of the two, either true or false, and yet, in the future, the progress of human culture maylead to the birth of a “new mathematics” based on a “new logic.” One well-known possibility is * This paper is an extended version of the paper published in S¯ugaku Seminar 57 (2), 28–33 (Nippon HyoronSha, Tokyo, 2018) (in Japanese) by the present author. “Quantum logic” is the logic of quantum mechanics first discovered by Birkhoff and von Neu-mann [2] in 1936. Among the laws of classical logic, it is known as the logic for which thedistributive law does not hold. Birkhoff and von Neumann hypothesized that the modular lawholds true in place of the distributive law, but now a weaker orthomodular law is hypothesized.On the other hand, in intuitionistic logic, the distributive law holds, but the law of excludedmiddle, the law of double negation, and De Morgan’s law do not hold. In this sense, quantumlogic is a logic that contrasts with intuitionistic logic, and Takeuti compared the relationshipbetween classical logic, intuitionistic logic, and quantum logic to “God’s logic,” “human logic,”and “the logic of things.”The fact that the distributive law does not hold in quantum logic is related to the existenceof physical quantities that cannot be simultaneously measured in quantum mechanics due to theuncertainty principle. Propositions considered in quantum logic are referred to as observationalpropositions. The most fundamental of these is the observational proposition that “the valueof the physical quantity A is equal to the real number a ,” which is expressed as A = a . Inquantum mechanics, a physical quantity is represented by a self-adjoint operator on a Hilbertspace, and if it has no continuous spectrum, the possible values are its eigenvalues.For example, if the eigenvalues of such a physical quantity A are a and a and the eigen-values of a physical quantity B are b and b , then A = a ∨ A = a and B = b ∨ B = b aretrue respectively. However, if the physical quantities A and B have no common eigenvectors,for any pair ( a j , b k ) of eigenvalues of A and B , the proposition A = a j ∧ B = b k is false. Thismeans that ( A = a ∨ A = a ) ∧ ( B = b ∨ B = b ) is true, but ( A = a ∧ B = b ) ∨ ( A = a ∧ B = b ) ∨ ( A = a ∧ B = b ) ∨ ( A = a ∧ B = b ) is false; therefore, the distributive law does not hold. Constructing mathematics based on the logic characteristic of these kinds of physical phenom-ena seems like an absurd undertaking at first. Where did this idea originally come from? Ac-cording to Takeuti’s
Mathematical World View [1], there are two origins for this idea: first, theidea that all the objects of mathematics are sets, and the modern mathematics equals ZFC settheory (an axiomatic set theory with Zermelo-Fraenkel’s axioms plus the axiom of choice), and2econdly, the idea that the concept of sets, originally considered in strictly two-valued logic,can be generalized to multi-valued logic that allows intermediate truth values.The first idea, which began with Frege, went through trials such as Russell’s paradox,Hilbert’s formalist program, and G¨odel’s incompleteness theorem, but came to fruition in Bour-baki’s ´El´ements de Math´ematique [3]. It can be said that it is the fundamental idea of modernmathematics.The second idea, which is famous for Zadeh’s fuzzy set theory [4,5], is to extend the two-valued logic of the membership relation to a multi-valued logic in defining subsets of a given“classical” set. The method for expanding the universe of sets in the ordinary two-valuedlogic to multi-valued logic at once was clarified with Cohen’s [6,7] independence proof of thecontinuum hypothesis in the field of foundations of mathematics.In this independence proof, Cohen [6,7] developed a method, called forcing, of expandingthe models of set theory. Scott and Solovay [8] showed that this method can be reformulatedin a more accessible way using Boolean valued models of set theory. The Boolean valuedmodels of set theory make all the sets, as the objects of mathematics, to be ”multivalued” intoa multi-valued logic with truth values in a given Boolean algebra. This view of multi-valuedlogicalization of the concept of sets further clarified the relationship between sheaf theory,topos theory, and intuitionistic set theory.Therefore, it can be said that Takeuti’s idea of constructing mathematics based on quantumlogic is to first develop this second idea, making all the sets that are the objects of mathematicsto be multi-valued at once, and then under the ”quantum set theory” obtained in the above,to develop mathematics based on set theory along the first idea. Logic has two aspects, syn-tax and semantics. Takeuti’s program is to quantum-logicalize semantics at once, leaving themathematical syntax that is based on the formal system of set theory as it is.
The majority of mathematicians accept that ZFC set theory is the foundation on which modernmathematics is based. Takeuti wrote [9, p. 171]:ZFC is a stable and powerful axiomatic system, there is almost no concern aboutcontradictions, and all modern mathematics can be developed within it. At present,it can be said that mathematics is equal to ZFC.This is an important pillar of Takeuti’s view of the mathematical world.However, it is unfortunate that the 20th century, in which these foundations of mathematicswere established, became a little distant, and the interest in this idea and the achievements ofBourbaki, which was the basis for it, diminished. Let us add some comments.Modern mathematics has two characteristics: formal and structural [10]. On the formalside, the whole mathematics can be developed using a formal language, strict deductive rules,and a set of axioms. In other words, the whole mathematics can be formalized by a single first-order predicate theory, known as ZFC set theory, and the only undefined term is the concept ofsets. The axioms are exhausted by the axioms of ZFC set theory, and all other mathematicalconcepts are defined within ZFC set theory. For example, number systems such as naturalnumbers and real numbers, geometric spaces such as Euclidean spaces, and transfinite numbersystems of ordinal and cardinal numbers are all defined inside ZFC [3].3n the other hand, on the structural side, what mathematicians actually study is not limitedto these kinds of numbers and figures, but instead it is generally considered as abstract objectscalled “mathematical structures.” Bourbaki showed that each field of mathematics is reduced tothe study of ”mathematical structures” specific to that field. Those ”mathematical structures”are defined within ZFC set theory, and that their research is conducted based on the axioms ofZFC set theory [3].While it is difficult to explain the concept of ”mathematical structure” to the general read-ership, according to its formal definition, it is a set theoretical object that is given by a set withfunctions and relations defined on that set and several number systems (and the sets obtained byrepeating direct products and power sets of these). Examples of mathematical structures includealgebraic systems (such as groups, rings, and fields), topological space, manifolds, and measurespaces, as well as topological algebraic systems (such as Hilbert spaces and C*-algebras).In this way, the word “axiomatic system” has two different aspects corresponding to theformal and structural aspects of mathematics. In other words, these are the axioms of ZFC settheory, and the axiomatic system that characterizes each mathematical structure and is includedin its definition. The concept of a mathematical structure gradually became clearer through theapplication of the axiomatic method proposed by Hilbert to various fields of mathematics, andresulted in Bourbaki’s ´El´ements de math´ematique [3]. Each field of mathematics is relativelyconsistent from ZFC set theory if there is an example of the “mathematical structure” that thefield studies.Hilbert’s formalism program can currently be interpreted as founding all mathematics inZFC set theory and having a finistic proof for the consistency of ZFC set theory. Hilbert firstshowed that the consistency of geometry was reduced to the consistency of real number theory,and he questioned the proof of consistency of real number theory in the second problem ofhis famous 23 problems, and he called for the axiomatizations of physics theories, includingprobability theory as the 6th problem. As a result, probability theory was axiomatized byKolmogorov [11], by means of measure theory, and became the basis for giving mathematicalproofs to important hypotheses of statistical mechanics such as the ergodic hypothesis. Measuretheory is a study of the mathematical structure of measure space, and probability theory hasbeen axiomatized as one of the “mathematical structures” in ZFC set theory. In measure theory,the existence of Lebesgue non-measurable sets is a theorem of ZFC set theory, but it is knownto be independent of set theory without the axiom of choice. In addition, von Neumann’saxiomatization of quantum mechanics shows that the two formulations of quantum mechanics,Heisenberg’s matrix mechanics and Schr¨odinger’s wave mechanics, are unified by a common“mathematical structure” of the Hilbert space [12].Theory of operator algebras on Hilbert spaces plays an important role in the axiomatizationof quantum field theory [13]. Recently, however, a problem independent of ZFC set theory havebeen identified within classical problems in operator algebras [14]. In these fields, we can get aglimpse of the reality of “modern mathematics equals ZFC”. However, Bourbaki’s ´El´ements demath´ematique [3] does not include “probability theory” nor “operator algebras”. Nevertheless,it is certain that they are the most Bourbaki-like fields in mathematics.Returning to Hilbert’s program, G¨odel’s incompleteness theorem showed that Hilbert’s pro-gram is not feasible as it was. So, is Takeuti’s trust in ZFC, saying “there is almost no concernabout contradictions” unfounded? Putting aside the optimistic viewpoint that as long as con-tradictions are handled when they occur, existing mathematics will not be lost, Bourbaki states4he following regarding the proof of consistency [15, p. 44].The theorem of G¨odel does not however shut the door completely on attemptsto prove consistency, so long as one abandons (at least partially) the restrictionsof Hilbert concerning “finite procedures”. It is thus that Gentzen in 1936 [16],succeeds in proving the consistency of formalised arithmetic, by using “intuitively”transfinite induction up to the countable ordinal ε . The value of the “certainty”that one can attach to such reasoning is without doubt less convincing than for thatwhich satisfies the initial requirements of Hilbert, and is essentially a matter ofpersonal psychology for each mathematician; it remains no less true that similar“proofs” using “intuitive” transfinite induction up to a given ordinal, would beconsidered important progress if they could be applied, for example, to the theoryof real numbers or to a substantial part of the theory of sets.Actually, this achievement hoped for by Bourbaki came to pass afterward, and the results arealready in our hands. Takeuti’s fundamental conjecture [17] and its solution [18,19] gavea Gentzen-style consistency proof for the theory of real numbers, and Arai [20,21] gave aGentzen-style consistency proof for a substantial part of ZFC set theory. According to Takeuti [22], the universe of set theory is composed of the following two princi-ples.
C1.
Power set composition principle
C2.
Transfinite generation principlePrinciple C1 states that for an arbitrary set, the totality of its subsets becomes a set again.Principle C2 states that when a method for creating a new set is provided, the method canbe repeated a transfinite number of times as much as possible. The most typical case of thisprinciple is the generation of ordinal numbers, which starts from the empty set ∅ and endlesslyrepeats the operation of collecting the ordinal numbers constructed so far, and the whole ordinalnumber is generated, as follows. ∅ , { } , { , } , . . . , ω = { , , , . . . } , ω +1 = { , , , . . . , ω } , . . . . Sets like ω = { , , , , . . . } which do not contain the final ordinalnumber are called limit ordinal numbers. Then the order relation for two ordinals α, β is definedas α < β iff α ∈ β . Using these two principles, the universe V of sets can be constructed asfollows. F1.
When ordinal number 0 arises, V is composed as ∅ . F2.
When creating ordinal number α + 1 from ordinal number α , V α +1 is composed as P ( V α ) . F3.
When creating limit ordinal number α , V α is composed of S β<α V β . F4.
As long as the generation of ordinal numbers is continued, the composition of V α willbe continued endlessly. We call the union S α V α of V α over all the ordinal numbers α , as V .To consider a general method for constructing a set theory based on a multi-valued logic,consider the following general structure of semantics of a logic. Let L be a partially ordered setsuch that every subset S has the supremum W S and the infimum V S , and that for the arbitraryelement x ∈ L , its complement x ⊥ is defined with the following properties: x ∧ x ⊥ = 0 , x ∨ x ⊥ = 1 , x ⊥⊥ = x , and x ≤ y ⇔ y ⊥ ≤ x ⊥ . Here, we write x ∧ y = V { x, y } , x ∨ y = { x, y } . The element W L is the maximum element of L representing “true”, the element V L is the minimum element of L representing “false”, and the other elements of the set L represent intermediate truth values. Such a structure L is called a complete orthocomplementedlattice.In the process of constructing the universe of sets, the concept of a power set appears in F2as a sole procedure for creating a new set, so that to construct the universe of L -valued sets, itsuffices to extend this part to the L -valued logic. Incidentally, using the idea of fuzzy sets, forany set X , any function A : D → L from any subset D of X to L can be considered as an L -valued subset of X . Therefore, the universe V ( L ) of sets based on L -valued logic is definedas follows.(1) V ( L )0 = ∅ .(2) V ( L ) α +1 = { u | u : dom( u ) → L , dom( u ) ⊆ V ( L ) α } .(3) V ( L ) α = S β<α V ( L ) β if α is a limit ordinal.(4) V ( L ) = S α ∈ On V ( L ) α .Here, On stands for the set of ordinals.Define the L -valued truth values of any closed formulas of the language of set theory aug-mented by the names of elements of V ( L ) as follows [23,24,26].(1) [[ ¬ φ ]] = [[ φ ]] ⊥ .(2) [[ φ ∧ ψ ]] = [[ φ ]] ∧ [[ ψ ]] .(3) [[( ∀ x ∈ u ) φ ( x )]] = V x ∈ dom( u ) ( u ( x ) → [[ φ ( x )]]) . (4) [[( ∀ x ) φ ( x )]] = V x ∈ V ( L ) [[ φ ( x )]] .Here, the operation → on L is defined by P → Q = P ⊥ ∨ ( P ∧ Q ) . We consider the logicalsymbols ∨ , ( ∃ x ∈ y ) , and ( ∃ x ) to be defined by(5) φ ∨ ψ := ¬ ( ¬ φ ∧ ¬ ψ ) .(6) ( ∃ x ∈ u ) φ ( x ) := ¬ (( ∀ x ∈ u ) ¬ φ ( x )) .(7) ( ∃ x ) φ ( x ) := ¬ (( ∀ x ) ¬ φ ( x )) .The L -valued truth values of the membership and the equality relations between two elements u, v of V ( L ) are recursively defined as follows.(8) [[ u ∈ v ]] = [[ ∃ x ∈ v ( x = u )]] .(9) [[ u = v ]] = [[ ∀ x ∈ u ( x ∈ v ) ∧ ∀ y ∈ v ( y ∈ u )]] .As a result of the above, the L -valued truth value [[ φ ]] ∈ L is defined for any closed formula φ in set theory with constant symbols naming elements of V ( L ) .Let V be the universe of the standard sets based on the 2-valued logic. Then there exists an L -valued set ˇ a corresponding to each a ∈ V . In fact, ˇ a is determined as an element of V ( L ) such that dom(ˇ a ) = { ˇ x | x ∈ a } and that ˇ a (ˇ x ) = 1 if x ∈ a . Then the relationship between thestandard sets a and b is isomorphic to the relationship between L -valued sets ˇ a and ˇ b .6hen L satisfies the distributive law, P ∧ ( Q ∨ R ) = ( P ∧ Q ) ∨ ( P ∧ R ) , P ∨ ( Q ∧ R ) = ( P ∨ Q ) ∧ ( P ∨ R ) , L is called a complete Boolean algebra. If B is a complete Boolean algebra, V ( B ) is called aBoolean value model of set theory. In this case, the following theorem holds [23]. Theorem 1 (Scott-Solovay) . If φ ( x , . . . , x n ) is provable in ZFC set theory, [[ φ ( u , . . . , u n )]] = 1 holds for any u , . . . , n n ∈ V ( B ) . Let us show what kind of B can be used to prove the independence of the continuum hy-pothesis (CH) (i.e., CH cannot be proved in ZFC set theory); note that the consistency of CH(i.e., the negation of CH cannot be proved in ZFC set theory) was proved by G¨odel [25]. Let I be an index set of the cardinality larger than ℵ . Let X = 2 ℵ × I be the generalized Cantorspace, a product topological space of the direct product of ℵ × I copies of { , } . Let B ( X ) be the Borel σ -field of subsets of X . Let m be a product measure on B ( X ) defined asfollows. m ( { p ∈ X | p ( j ) = a , . . . , p ( j n ) = a n } ) = (cid:18) (cid:19) n , where a j ∈ { , } ( j = 1 , . . . , n ) . The existence of this kind of measure is based on Kol-mogorov’s extension theorem. Let N be the collection of measure zero subsets. Then thequotient Boolean algebra B = B ( X ) / N is a complete Boolean algebra, called the measurealgebra of m . For this B , the value of [[CH]] can be calculated, and the value [[CH]] = 0 isobtained [23, p. 173]. If CH can be proved from ZFC, then from Theorem 1, [[CH]] = 1 for any B , so that the independence of the continuum hypothesis is proved in this way. Let φ be a logical formula in ZFC set theory representing a mathematical theorem. Accordingto theorem 1, the proposition “ [[ φ ]] = 1 ” is a new, different theorem. Because the proposition [[ φ ]] = 1 is constructed by recursive rules, we can analyse its meaning, so that we will be ableto restate [[ φ ]] = 1 using familiar concepts. If ψ is a mathematical proposition written with ourfamiliar concepts, and if [[ φ ]] = 1 and ψ can be proved equivalent, ψ will also become a newmathematical theorem. In other words, if we prove that [[ φ ]] = 1 and ψ are equivalent, thenrather than proving ψ directly, we can prove ψ by proving φ instead. In this case, φ is a farsimpler proposition than ψ , and proving the equivalence of [[ φ ]] = 1 and ψ is similar to languagetranslation in many ways. As a result, this approach is known to often lead to very promisingprospects.To apply this method to analysis, in Two Applications of Logic to Mathematics [27] Takeutistudied the structure of the real numbers in V ( B ) for a complete Boolean algebra B of pro-jections on a Hilbert space H and showed that the real numbers in V ( B ) are in one-to-onecorrespondence with the self-adjoint operators on H such that their spectral projections belongto B . Here, the real numbers in V ( B ) is defined as the set of elements u of V ( B ) satisfying [[R(u)]] = 1 , where R(x) is a logical formula in ZFC meaning “ x is a real number”. This7hows that from theorems φ related to real numbers, the theorem [[ φ ]] = 1 for these kinds ofself-adjoint operators can be systematically obtained. For example, from the theorem stating“Every upper bounded set of real numbers has its supremum”, we obtain the theorem stating“Every upper bounded set of mutually commuting self-adjoint operators has its supremum.”This is a very powerful method, which Takeuti referred to as Boolean valued analysis. Inhis paper Von Neumann algebras and Boolean valued analysis [28], Takeuti took this a stepfurther, showing that the class of all Hilbert spaces in V ( B ) are in one-to-one correspondencewith the class of all normal *-representations of the commutative von Neumann algebra A generated by B ; see also [29]. In addition, he further showed that the class of all von Neumannfactors (von Neumann algebras with trivial centers) in V ( B ) are in one-to-one correspondencewith the class of von Neumann algebras such that their centers are isomorphic to A .Based on this finding, theorems for general von Neumann algebras are systematically de-rived from theorems on von Neumann factors. In operator algebras, there has been a well-known method, called the reduction theory, for deriving theorems on general von Neumannalgebras from theorems on von Neumann factors using the direct integral decompositions ofvon Neumann algebras into von Neumann factors, but there are restrictions such as separabil-ity. The Boolean valued analysis method is a powerful approach which does not have theserestrictions.To show that Boolean valued analysis is a truly powerful method, the author explored thevon Neumann factors in V ( B ) for the completely general complete Boolean algebra B , andshowed that the type I von Neumann factors in V ( B ) are in one-to-one correspondence with thetype I AW*-algebras whose central projections are isomorphic to B [30,31]; see also [29,32].There is a theorem that “every type I von Neumann factor is isomorphic to the algebra of allbounded operators on a Hilbert space, and the cardinal number representing the dimension ofthat Hilbert space is a complete invariant.” Considering [[ φ ]] = 1 for φ to be the above theorem,we obtain the theorem stating “the isomorphic invariants of type I AW*-algebras correspondsto the cardinal numbers in V ( B ) ”.By the way, when A is a type I von Neumann algebra, B satisfies the countable chaincondition locally, so that the cardinal numbers in V ( B ) can be represented by the step functionsof the standard cardinal numbers; this fact is known as the absoluteness of the cardinality in V ( B ) for complete Boolean algebras B satisfying the countable chain condition [23, p. 162].In this case, it was already known in the theory of operator algebras that “the step functionsof the standard cardinal numbers forms an isomorphic invariant of the type I von Neumannalgebras”. However, it had been an open problem whether the isomorphic invariants of the typeI AW*-algebras can be represented by the step functions of the standard cardinal numbers, sinceKaplansky [33] made a negative conjecture in 1952. This conjecture was eventually settled in1983 by the method of Boolean valued analysis [30].In fact, the case where the cardinals in V ( B ) cannot be represented by the step functionsof the standard cardinal numbers is known by the forcing method as “cardinal collapsing” [34,Ch. 5]: for any two infinite cardinals α and β in V , we can construct a complete Boolean algebra B such that ˇ α and ˇ β have the same cardinality in V ( B ) . Therefore, when the central projectionsforms such a complete Boolean algebra, it is derived that the isomorphic invariants of those typeI AW*-algebras cannot be represented by the step functions of the standard cardinal numbers.Thus, Kaplansky’s conjecture is settled [30,31]. For example, for arbitrary infinite cardinalnumbers α and β , we can construct a commutative AW*-algebra Z such that the two type I8W*-algebras, the α × α matrix algebra over Z and β × β matrix algebra over Z , are isomorphic[32].In this way, Takeuti’s Boolean valued analysis makes it possible to systematically apply theforcing method to analysis, and it plays a role of a bridge between the two fields by applyingthe results in foundations of mathematics to analysis. For further developments of Booleanvalued analysis in operator algebras, we refer the reader to [24,35–38]. If L is a lattice consisting of the projections on a Hilbert space H , then L is called the stan-dard quantum logic. In general, L is called orthomodular iff P ≤ Q implies that there existsa Boolean subalgebra of L including P and Q . The standard quantum logic is a complete or-thomodular lattice, and a complete orthomodular lattice is considered to be a general model ofquantum logic.Takeuti introduced the universe V ( Q ) of sets based on the standard quantum logic Q on aHilbert space H in his seminal paper Quantum Set Theory [24] published in 1981, and startedhis research of quantum set theory.A remarkable fact, pointed out by Takeuti, about quantum set theory is that the reals de-fined in V ( Q ) corresponds bijectively to the self-adjoint operators on H , as a straight forwardconsequence of Boolean valued analysis based on complete Boolean algebras of projections,developed by Takeuti in Two applications of logic to mathematics [27]. We refer to the abovecorrespondence between the self-adjoit operators and reals in V ( Q ) as the Takeuti correspon-dence.What formulas hold in V ( Q ) ? To answer this question, Takeuti introduced the commutator | =( S ) for any subset S of Q in [24], and used it to define the commutator ∨ ( u , . . . , u n ) of anyelements u , . . . , u n in V ( Q ) . In addition, he showed that, roughly speaking, if one rewrites anaxiom of ZFC by replacing ∀ xφ ( x ) by ∀ x ( ∨ ( x ) → φ ( x )) and replacing ∃ x by ∃ x ( ∨ ( x ) ∧ φ ( x )) ,the modified axiom holds in V ( Q ) . Then how theorems of ZFC hold true in V ( Q ) ? UsingTakeuti’s technique, the present author obtained the following theorem [26,39,40]. Theorem 2 (Quantum Transfer Principle) . If a theorem φ ( x , ..., x n ) of ZFC contains onlybounded quantifiers, then the relation [[ φ ( u , . . . , u n )]] ≥ [[ ∨ ( u , . . . , u n )]] holds for any u , . . . , u n ∈ V ( Q ) . This theorem implies that any commuting family of quantum sets u , . . . , u n satisfies ZFCtheorems, since in this case [[ ∨ ( u , . . . , u n )]] = 1 . The above theorem more generally states thatany family of quantum physical quantities u , . . . , u n satisfies ZFC theorems at least the truthvalue [[ ∨ ( u , . . . , u n )]] .In quantum mechanics, to every quantum system S there corresponds a Hilbert space H ,and the physical quantities of S are represented by the self-adjoint operators of H . Accord-ingly, Q represents the logic of quantum system S , and the real numbers in V ( Q ) represent thephysical quantities of S . Based on this, all the observational propositions on a quantum system S can be translated into logical formulas in ZFC referring to real numbers in V ( Q ) using the9akeuti correspondence [41]. Therefore, mathematics based on quantum logic is nothing but amathematics in which the reals are quantum physical quantities.A method for calculating the probability of observational proposition φ in quantum mechan-ics has been known as the Born rule. By translating an observational proposition φ into the cor-responding ZFC logical formula ˆ φ using the Takeuti correspondence between physical quanti-ties A and reals ˆ A in V ( Q ) , the probability calculation can be expressed as Pr { φ k ψ } = k [[ ˆ φ ]] ψ k [41]. However, because the equality relation A = B for two physical quantities A and B is notincluded in the observational propositions under the known quantum rule, the probability of A and B having the same value was not defined in quantum mechanics until recently. However,for the corresponding real numbers ˆ A and ˆ B in V ( Q ) , the truth value [[ ˆ A = ˆ B ]] is defined in V ( Q ) , so that by Pr { A = B k ψ } = k [[ ˆ A = ˆ B ]] ψ k , the probability of two physical quantities A and B having the same value has been newly defined [41]. This theory was used to intro-duce the most basic condition in quantum measurement theory requiring that “the measuredquantity A and the meter quantity B in the measuring instrument match”. It has contributed agreat deal to reforming the uncertainty principle and other progress in this field [41]. In thisway, quantum set theory is a mathematical theory that has the power to extend conventionalquantum mechanics, allowing it to be applied to new phenomena.We have finally reached the starting point for constructing mathematics based on quantumlogic, we were able to clarify the equality relation between real numbers, and the study of theorder relation between real numbers has just begun [42,43]. Since these relations show observ-able relationships that hold between the physical quantities of the quantum system, the resultsof quantum set theory can be immediately experimentally verified. By inheriting and develop-ing such a wonderful heritage of Takeuti’s mathematical achievements, it must be possible togreatly advance the elucidation of the interpretational problems of quantum mechanics, whichhas been regarded as a mystery for many years. Acknowledgments
This research was supported by JSPS KAKENHI, No. 17K19970, and the IRI-NU collabora-tion.
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