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Mathematics for theoretical physicsJean Claude DutaillyTo cite this version:Jean Claude Dutailly. Mathematics for theoretical physics. 2012. hal-00735107v1 HAL Id: 0735107v1Preprint submitted on 25 Sep 2012 (v1), last revised 1 Feb 2014 (v2)HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Mathematics for theoretical physicsJean Claude.DutaillyParisSeptember 25, 2012AbstractThis book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It givesan extensive and precise coverage of the subjects which are addressed, ina consistent and intelligible manner.The first part addresses the Foundations (mathematical logic, set theory, categories), the second Algebra (algebraic strucutes, groups, vector spaces tensors, matrices, Clifford algebra). The third Analysis (general topology, measure theory, Banach Spaces, Spectral theory). The fourth Differential Geometry (derivatives, manifolds, tensorial bundle, pseudo-riemannian manifolds, symplectic manifolds). The fifth Lie Algebras, Lie Groups.and representation theory. The sixth Fiber bundles and jets. The last one Functional Analysis (differential operators, distributions, ODE, PDE, variational calculus).Several signficant new results are presented (distributions over vector bundles, functional derivative, spin bundle and manifolds with boundary).The purpose of this book is to give a comprehensive collection of precisedefinitions and results in advanced mathematics, which can be useful to workersin mathematic or physics.The specificities of this book are :- it is self contained : any definition or notation used can be found within- it is precise : any theorem lists the precise conditions which must be metfor its use- it is easy to use : the book proceeds from the simple to the most advancedtopics, but in any part the necessary definitions are reminded so that the readercan enter quickly into the subject- it is comprehensive : it addresses the basic concepts but reaches most ofthe advanced topics which are required nowodays- it is pedagogical : the key points and usual misunderstandings are underlined so that the reader can get a strong grasp of the tools which are presented.The first option is unusual for a book of this kind. Usually a book starts withthe assumption that the reader has already some background knowledge. Theproblem is that nobody has the same background. So a great deal is dedicatedto remind some basic stuff, in an abbreviated way, which does not left muchscope to their understanding, and is limited to specific cases. In fact, starting1
from the very beginning, it has been easy, step by step, to expose each conceptin the most general settings. And, by proceeding this way, to extend the scopeof many results so that they can be made available to the - unavoidable - specialcase that the reader may face. Overall it gives a fresh, unified view of the mathematics, but still affordable because it avoids as far as possible the sophisticatedlanguage which is fashionable. The goal is that the reader understands clearlyand effortlessly, not to prove the extent of the author’s knowledge.The definitions choosen here meet the ”generally accepted definitions” inmathematics. However, as they come in many flavors according to the authorsand their field of interest, we have striven to take definitions which are both themost general and the most easy to use.Of course this cannot be achieved with some drawbacks. So many demonstrations are omitted. More precisely the chosen option is the following :- whenever a demonstration is short, it is given entirely, at least as an example of ”how it works”- when a demonstation is too long and involves either technical or specificconditions, a precise reference to where the demonstation can be found is given.Anyway the theorem is written in accordance with the notations and definitions of this book, and a special attention has been given that they match thereference.- exceptionnaly, when this is a well known theorem, whose demonstrationcan be found easily in any book on the subject, there is no reference.The bibliography is short. Indeed due to the scope which is covered it couldbe enormous. So it is strictly limited to the works which are referenced in thetext, with a priority to the most easily available sources.This is not mainly a research paper, even if the unification of the concepts is,in many ways, new, but some significant results appear here for the first time,to my knowledge.- distributions over vector bundles- a rigorous definitition of functional derivatives- a manifold with boundary can be defined by a unique functionand several other results about Clifford algebras, spin bundles and differential geometry.11 j.c.dutailly@free.fr2
CONTENTSPART 1 : FOUNDATIONSLOGICPropositional logicPredicatesFormal theoriesSET THEORYAxiomaticMapsBinary 37
PART 2 : ALGEBRAUSUAL ALGEBRAIC STRUCTURESFrom monoids to fieldsFrom vector spaces to algebrasGROUPSDefinitionsFinite groupsVECTOR SPACESDefinitionsLinear mapsScalar product on vector spacesSymplectic vector spaceComplex vector spaceAffine spaceTENSORSTensorial productSymmetric and antisymmetric tensorsTensor product of mapsMATRICESOperations with matricesEigen valuesMatrix calculusCLIFFORD ALGEBRAMain operations in Clifford algebrasPin and Spin groupsClassification of Clifford 149155
PART 3 : ANALYSISGENERAL TOPOLOGYTopological spacesMaps on topological spacesMetric and semi-metric spacesAlgebraic topologyMEASUREMeasurable spacesMeasured spacesIntegralProbabilityBANACH SPACESTopological vector spacesNormed vector spacesBanach spacesNormed algebrasHilbert SpacesSPECTRAL THEORYRepresentation of algebrasSpectral 77
PART 4 : DIFFERENTIAL GEOMETRYDERIVATIVESDifferentiables mapsHigher order derivativesExtremum of a functionImplicit mapsHolomorphic mapsMANIFOLDSManifoldsDifferentiable mapsTangent bundleSubmanifoldsTENSORIAL BUNDLETensor fieldsLie derivativeExterior algebraCovariant derivativeINTEGRALOrientation of a manifoldIntegralCohomologyCOMPLEX MANIFOLDSPSEUDO RIEMANNIAN MANIFOLDSGeneral propertiesLévi Civita connectionSubmanifoldsSYMPLECTIC 71373380384388394400405
PART 5 : LIE ALGEBRAS AND LIE GROUPSLIE ALGEBRASLie algebras : definitionsSum and product of Lie algebrasClassification of Lie algebrasLIE GROUPSGeneral definitions and resultsStructure of Lie groupsIntegrationCLASSICAL LINEAR GROUPS AND ALGEBRASGeneral resultsList of classical linear groups and algebrasREPRESENTATION THEORYDefinitions and general resultsRepresentation of Lie groupsRepresentation of Lie algebrasRepresentation of classical groups7413417420434450463468471481491502506
PART 6 : FIBER BUNDLESFIBER BUNDLESGeneral fiber bundlesVector bundlesPrincipal bundlesAssociated bundlesJETSCONNECTIONSGeneral connectionsConnections on vector bundlesConnections on associated bundlesBUNDLE FUNCTORS8520532545553567580587601614
PART 7 : FUNCTIONAL ANALYSISSPACES of FUNCTIONSPreliminariesSpaces of bounded or continuous mapsSpaces of integrable mapsSpaces of differentiables mapsDISTRIBUTIONSSpaces of functionalsDistributions on functionsExtension of distributionsFOURIER TRANSFORMFourier seriesFourier integralsFourier transform of distributionsDIFFERENTIAL OPERATORSLinear differential operatorsLaplacianHeat kernelPseudo-differential operatorsDIFFERENTIAL EQUATIONSOrdinary differential equationsPartial differential equationsVARIATIONAL 8681688705716719726732749
Part IPART1 : FOUNDATIONSIn this first part we start with what makes the real foundations of today mathematics : logic, set theory and categories. The two last subsections are natural inthis book, and they will be mainly dedicated to a long list of definitions, mandatory to fix the language that is used in the rest of the book. A section aboutlogic seems appropriate, even if it gives just an overview of the topic, becausethis is a subject that is rarely addressed, except in specialized publications, andshould give some matter for reflection, notably to physicists.1LOGICFor a mathematician logic can be addressed from two points of view :- the conventions and rules that any mathematical text should follow in orderto be deemed ”right”- the consistency and limitations of any formal theory using these logicalrules.It is the scope of a branch of mathematics of its own : ”mathematical logic”Indeed logic is not limited to a bylaw for mathematicians : there are alsotheorems in logic. To produce these theorems one distinguishes the object of theinvestigation (”language-object” or ”theory”) and the language used to proceedto the demonstrations in mathematical logic, which is informal (plain english).It seems strange to use a weak form of ”logic” to prove results about the moreformal theories but it is related to one of the most important feature of anyscientific discourse : that it must be perceived and accepted by other workers inthe field as ”sensible” and ”convincing”. And in fact there are several schoolsin logic : some do not accept any nonnumerable construct, or the principle ofnon contradiction, which makes logic a confusing branch of mathematics. Butwhatever the interest of exotic lines of reasoning in specific fields, for the vastmajority of mathematicians, in their daily work, there is a set of ”generallyaccepted logical principles”.On this topic we follow mainly Kleene where definitions and theorems canbe found.1.1Propositional logicLogic can be considered from two points of view : the first (”models”) which isfocused on telling what are true or false statements, and the second (”demonstration”) which strives to build demonstrations from premisses. This distinction isat the heart of many issues in mathematical logic.10
1.1.1ModelsFormulasDefinition 1 An atom2 is any given sentence accepted in the theory.The atoms are denoted as latin letters A,B,.Definition 2 The logical operators are : : equivalent : imply : and (both) : or (possibly both)q : negation(notation and list depending on the authors)Definition 3 A formula is any finite sequence of atoms linked by logical operators.One can build formulas from other formulas using these operators. A formula is ”well-built” (it is deemed acceptable in the theory) if it is constructedaccording to the previous rules. Examples : if ”3 2 x”, ” 5 3 2”, ”x2 2x 1 0” are atoms then(3 2 x) x2 2x 1 0 5 3 2 is a well built formula.In building a formula we do not question the meaning or the validity of theatoms (this the job of the theory which is scrutinized) : we only follow rules tobuild formulas from given atoms.When building formulas with the operators it is always good to use bracketsto delimite the scope of the operators. However there is a rule of precedence (bydecreasing order): qTruth-tablesThe previous rules give only the ”grammar” : how to build accepted formulas.But a formula can be well built but meaningless, or can have a meaning only ifcertain conditions are met. Logic is the way to tell if something is true or false.Definition 4 To each atom of a theory is attached a ”truth-table”, with onlytwo values : true (T) or false (F) exclusively.Definition 5 A model for a theory is the list of its atoms and their truth-table.Definition 6 A proposition is any formula issued from a model2 Thename of an object is in boldface the first time it appears (in its definition)11
The rules telling how the operators work to deduce the truth table of aformula from the tables ot its atoms are the following (A,B are any formula) : A T T FFBTFTF(A B) (A B)TTFFFTTT(A B)TFFF (A B) T A T TT FF (qA)F TThe only non obvious rule is for . It is the only one which provides afull and practical set of rules, but other possibilities are mentioned in quantumphysics.Valid formulasWith these rules the truth-table of any formula can be computed (formulashave only a finite number of atoms).The formulas which are always true (their truth-table presents only T) areof particular interest.Definition 7 A formula A of a model is said to be valid if it is always true. Itis then denoted A.Definition 8 A formula B is a valid consequence of A if (A B). Thisis denoted : A B.More generally one writes : A1 , .Am BValid formulas are crucial in logic. There are two different categories of validformulas:- formulas which are always valid, whatever the model : they provide the”model” of propositional calculs in mathematical logic, as they tell how to produce ”true” statements without any assumption about the meaning of the formulas.- formulas which are valid in some model only : they describe the propertiesassigned to some atoms in the theory which is modelled. So, from the logicalpoint of view, they define the theory itself.The following formula are always valid in any model (and most of them areof constant use in mathematics). Indeed they are just the traduction of theprevious tables.1. first set (they play a specific role in logic):(A B) A; (A B) BA (A B) ; B (A B)qqA AA (B A)(A B) (A B) ; (A B) (B A)(A B) ((A (B C)) (A C))12
A (B (A B))(A B) ((A qB) qA)(A B) ((B A) (A B))2. Others (there are infinitely many others formulas which are always valid):A A;A A; (A B) (B A) ; ((A B) (B C)) (A C)(A B) ((qA) (qB))qA (A B)qqA A; q (A (qA)) ; A (qA)q (A B) ((qA) (qB)) ; q (A B) ((qA) (qB)) ; q (A B) (A (qB))Notice that A (qA) meaning that a formula is either true or false is anobvious consequence of the rules which have been set up here.An example of formula which is valid in a specific model : in a set theory theexpressions ”a A”, ”A B” are atoms, they are true or false (but their valueis beyond pure logic). And ” ((a A) (A B)) (a B) ” is a formula. Tosay that it is always true expresses a fundamental property of set theory (butwe could also postulate that it is not always true, and we would have anotherset theory).Theorem 9 If A and (A B) then : BTheorem 10 A B iff3 A and B have same tables.Theorem 11 Duality: Let be E a formula built only with atoms A1 , .Am , theirnegation qA1 , .qAm , the operators , , and E’ the formula deduced from E bysubstituting with , with , Ai with qAi , qAi with Ai then :If E then qE ′If qE then E ′With the same procedure for another similar formula F:If E F then F ′ E ′If E F then E ′ F ′1.1.2DemonstrationUsually one does not proceed by truth tables but by demonstrations. In aformal theory, axioms, hypotheses and theorems can be written as formulas. Ademonstration is a sequence of formulas using logical rules and rules of inference,starting from axioms or hypotheses and ending by the proven result.In deductive logic a formula is always true. They are built according to thefollowing rules by linking formulas with the logical operators above :i) There is a given set of formulas (A1 , A2 , .Am , .) (possibly infinite) calledthe axioms of the theory3 Wewill use often the usual abbreviation ”iff” for ”if and only if”13
ii) There is an inference rule : if A is a formula, and (A B) is a formula,then (B) is a formula.iii) Any formula built from other formulas with logical operators and usingthe ”first set” of rules above is a formulaFor instance if A,B are formulas, then ((A B) A) is a formula.The formulas are listed, line by line. The last line gives a ”true” formulawhich is said to be proven.Definition 12 A demonstration is a finite sequence of formulas where thelast one B is the proven formula, and this is denoted : B. B is provable.Similarly B is deduced from A1 , A2 , . is denoted : A1 , A2 , .Am , . B : . Inthis picture there are logical rules (the ”first set” of formulas and the inferencerule) and ”non logical” formulas (the axioms). The set of logical rules can varyaccording to the authors, but is roughly always the same. The critical part isthe set of axioms which is specific to the theory which is under review.Theorem 13 A1 , A2 , .AmAp with 1 p mTheorem 14 If A1 , A2 , .AmB1 , A1 , A2 , .Amand B1 , B2 , .Bp C then A1 , A2 , .Am CTheorem 15 If(A B)1.2(A B) then AB2 , .A1 , A2 , .AmB and conversely : if ABpB thenPredicatesIn propositional logic there can be an infinite number of atoms (models) oraxioms (demonstration) but, in principle, they should be listed prior to anycomputation. This is clearly a strong limitation. So the previous picture is extended to predicates, meaning formulas including variables and functions.1.2.1Models with predicatesPredicateThe new elements are : variables, quantizers, and propositional functions.Definition 16 A variable is a symbol which takes its value in a given collectionD (the domain).They are denoted x,y,z,.It is assumed that the domain D is always thesame for all the variables and it is not empty. A variable can appears in differentplaces, with the usual meaning that in this case the same value must be assignedto these variables.14
Definition 17 A propositional function is a symbol, with definite places forone or more variables, such that when one replaces each variable by one of theirvalue in the domain, the function becomes a proposition.They are denoted : P (x, y), Q(r), .There is a truth-table assigned to thefunction for all the combinations of variables.Definition 18 A quantizer is a logical operator acting on the variables.They are : : for any value of the variable (in the domain D) : there exists a value of the variable (in the domain D)A quantizer acts, on one variable only, each time it appears : x, y, . . Thisvariable is then bound. A variable which is not bound is free. A quantizercannot act on a previously bound variable (one cannot have x, x in thesame formula). As previously it is always good to use different symbols for thevariables and brackets to precise the scope of the operators.Definition 19 A predicate is a sentence comprised of propositions, quantizerspreceding variables, and propositional functions linked by logical operators.Examples of predicates : p(( x, (x 3 z)) A) q y,y 2 1 a (z 0) n ((n N ) ( p, (p a n))) BTo evaluate a predicate one needs a truth-rule for the quantizers , :- a formula ( x, A (x)) is T if A(x) is T for all values of x- a formula ( x, A(x))) is T if A(x) has at least one value equal to TWith these rules whenever all the variables in a predicate are bound, thispredicate, for the thuth table purpose, becomes a proposition.Notice that the quantizers act only on variables, not formulas. This is specific to first order predicates. In higher orders predicates calculus there areexpressions like ” A”, and the theory has significantly different outcomes.Valid consequenceWith these rules it is possible, in principle, to compute the truth table of anypredicate.Definition 20 A predicate A is D-valid, denoted D A if it is valid whateverthe value of the free variables in D. It is valid if is D-valid whatever the domainD.The propositions listed previously in the ”first set” are valid for any D. A B iff for any domain D A and B have the same truth-table.15
1.2.2Demonstration with predicatesThe same new elements are added : variables, quantizers, propositional functions. Variables and quantizers are defined as above (in the model framework)with the same conditions of use.A formula is built according to the following rules by linking formulas withthe logical operators and quantizers :i) There is a given set of formulas (A1 , A2 , .Am , .) (possibly infinite) calledthe axioms of the theoryii) There are three inference rules :- if A is a formula, and (A B) is a formula, then (B) is a formula- If C is a formula where x is not present and A(x) a formula, then :if C A(x) is a formula, then C xA(x) is a formulaif A (x) C is a formula, then xA(x) C is a formulaiii) Any formula built from other formulas with logical operators and usingthe ”first set” of rules above plus : xA (x) A (r)A (r) xA(x)where r is free, is a formulaDefinition 21 B is provable if there is a finite sequence of formulas where thelast one is B, which is denoted : B.B can be deduced from A1 , A2 , .Am if B is provable starting with the formulas A1 , A2 , .Am ,and is denoted : A1 , A2 , .Am B1.31.3.1Formal theoriesDefinitionsThe previous definitions and theorems give a framework to review the logic offormal theories. A formal theory uses a symbolic language in which terms aredefined, relations between some of these terms are deemed ”true” to expresstheir characteristics, and logical rules are used to evaluate formulas or deducetheorems. There are many refinements and complications but, roughly, thelogical rules always come back to some kind of predicates logic as exposed inthe previous section. But there are two different points of view : the ”models”side and the ”demonstration” side : the same theory can be described using amodel (model type theory) or axioms and deductions (deductive type).Models are related to the ”semantic” of the theory. Indeed they are basedon the assumption that for every atom there is some truth-table that could beexhibited, meaning that there is some ”extra-logic” to compute the result. Andthe non purely logical formulas which are set to be valid (always true in themodel) characterize the properties of the objects ”modelled” by the theory.Demonstrations are related to the ”syntactic” part of the theory. They dealonly with formulas without any concern about their meaning : either they arelogical formulas (the first set) or they are axioms, and in both cases they are16
assumed to be ”true”, in the meaning that they are worth to be used in ademonstration. The axioms sum up the non logical part of the system. Theaxioms on one hand and the logical rules on the other hand are all that isnecessary to work.Both model theories and deductive theories use logical rules (either to compute truth-tables or to list formulas), so they have a common ground. And thenon-logical formulas which are valid in a model are the equivalent of the axiomsof a deductive theory. So the two points of view are not opposed, but proceedfrom the two meanings of logic.In reviewing the logic of a formal theory the main questions that arise are :- which are the axioms needed to account for the theory (as usual one wantsto have as few of them as possible) ?- can we assert that there is no formula A such that both A and its negationqA can be proven ?- can we prove any valid formula ?- is it possible to list all the valid formulas of the theory ?A formal theory of the model type is said to be ”sound” (or consistent) ifonly valid formulas can be proven. Conversely a formal theory of the deductivetype is said to be ”complete” if any valid formula can be proven.1.3.2Completness of the predicate calculusPredicate logic (first order logic) can be seen as a theory by itself. From a setof atoms, variables and propositional functions one can build formulas by usingthe logical operators for predicates. There are formulas which are always validin the propositional calculus, and there are similar formulas in the predicatescalculus, whatever the domain D. Starting with these formulas, and using theset of logical rules and the inference rules as above one can build a deductivetheory.The Gödel’s completness theorem says that any valid formula can be proven,and conversely that only valid formulas can be proven. So one can write in thefirst order logic : A iff A.It must be clear that this result, which justifies the apparatus of first orderlogic, stands only for the formulas (such as those listed above) which are validin any model : indeed they are the pure logical relations, and do not involveany ”non logical” axioms.A ”compactness” theorem by Gödel says in addition that if a formula can beproven from a set of formulas, it can also be proven by a finite set of formulas :there is always a demonstration using a finite number of steps and formulas.These results are specific to first order logic, and does not hold for higherorder of logic (when the quantizers act on formulas and not only on variables).Thus one can say that mathematical logic (at least under the form of firstorder propositional calculus) has a strong foundation.17
1.3.3Incompletness theoremsAt the beginning of the XX century mathematicians were looking forward toa set of axioms and logical rules which could give solid foundations to mathematics (the ”Hilbert’s program”). Two theories are crucial for this purpose :set theory and natural number (arithmetics). Indeed set theory is the languageof modern mathematics, and natural numbers are a prerequisite for the rule ofinference, and even to define infinity (through cardinality). Such formal theories use the rules of first order logic, but require also additional ”non logical”axioms. The axioms required in a formal set theory (such as Zermelo-Frankel’s)or in arithmetics (such as Peano’s) are well known. There are several systems,more or less equivalent.A formal theory is said to be effectively generated if its set of axiomsis a recursively enumerable set. This means that there is a computer programthat, in principle, could enumerate all the axioms of the theory. Gödel’s firstincompleteness theorem states that any effectively generated theory capable ofexpressing elementary arithmetic cannot be both consistent and complete. Inparticular, for any consistent, effectively generated formal theory that provescertain basic arithmetic truths, there is an arithmetical statement that is truebut not provable in the theory (Kleene p. 250). In fact the ”truth” of thestatement must be understood as : neither the statement or its negation canbe proven. As the statement is true or false the statement itself or its converseis true. All usual theories of arithmetics fall under the scope of this theorem.So one can say that in mathematics the previous result ( A iff A) does notstand.This result is not really a surprise : in any formal theory we can buildinfinitely many predicates, which are ”grammatically” correct. To say thatthere is always a way to prove any such predicate (or its converse) is certainlya crude assumption. It is linked to the possibility to write computer programsto automatically check demonstrations.1.3.4Decidable and computable theoriesThe incompletness theorems are closely related to the concepts of ”decidable”and ”computable”.In a formal deductive theory computer programs can be written to ”formalize” demonstrations (an exemple is ”Isabelle” see the Internet), so that theycan be made safer.One can go further and ask if it is possible to design a program such thatit could, for any statement of the theory, check if it is valid (model side) orprovable (deducible side). If so the theory is said decidable.The answer is yes for the propositional calculus (without predicate), becauseit is always possible to compute the truth table, but it is no in general forpredicates calculus. And it is no for theories modelling arthmetics.Decidability is an aspect of computability : one looks for a program whichcould, starting from a large class of inputs, compute an answer which is yes or18
no.Computability is studied through ”Türing machines”, which are schematiccomputers. A Türing machine is comprised of an input system (a flow of binarydata read bit by bit), a program (the computer has p ”states”, including an”end”, and it goes from one state to another according to its present stateand the bit that has been read), and an output system (the computer writesa bit). A Türing machine can compute integer functions (the input, outputand parameters are integers). One demonstration of the Gödel incompletnesstheorem shows that there are functions that cannot be computed : notably thefunction telling, for any given input, in how many steps the computer wouldstop.If we look for a program that can give more than a ”Yes/No” answer onehas the so-called ”function problems”, which study not only the possibility butthe efficiency (in terms of ressources used) of algorithms. The complexity ofa given problem is measured by the ratio of the number of steps required by aTüring machine to compute the function, to the size in bits of the input (theproblem).19
2SET THEORY2.1AxiomaticSet theory was founded by Cantor and Dedekind in early XX century. The initial set theory was impaired by paradoxes, which are usually the consequences ofan inadequate definition of a ”set of sets”. Several improved versions were proposed, and its most common , formalized by Zermello-Fraenkel, is denoted ZFCwhen it includes the axiom of choice. For the details see Wikipedia ”Zermelo–Fraenkel set theory”.2.1.1Axioms of ZFCSome of the axioms listed below are redundant, as they can be deduced fromothers, depending of the presentation. But it can be useful to know their names:Axiom 22 Axiom of extensionality : Two sets are equal (are the same set) ifthey have the same elements.Equality is defined as : (A B) (( x (x A x B)) ( x (A x B x)))Axiom 23 Axiom of regularity (also called the Axiom of foundation) : Everynon-empty set A contains a member B such that A and B are disjoint sets.Axiom 24 Axiom schema of spec
- the conventions and rules that any mathematical text should foo w in order to be deemed "right" - the consistency and mitations of any formal theory using these lo gical rules. It is the scope of a branch of mathematics of its own : "mathematical logic" Indeed logic is not mited to a bylaw for mathematicians : there are also theorems .
