A New System of Axioms Instead of ZF
(ver. 2.02)
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__Preface
__The axioms shown below have the contents that should overturn the set theory of today. If you believe that set theory is a perfect theory which includes no doubt, you should not read the followings. On the other hand, if you have some doubts about the set theory of today while you feel several disharmonies and no beauty there, you had better read the followings.
__
__Chapter 1
__
__[1] History
__Nowadays we believe that set theory is a right theory. However, set theory has a fundamental problem. It is the Russell's paradox. This problem was born of the basic ideas of set theory.
__How can we avoid this problem? Set theory of modern times took the following way.
__"To formalize the theory and adjust axioms so that they will not bring any inconsistency."
__This might be a way of certain opportunism. Anyway, this way got some fruits or success.
__However, we find that the first problem has not been solved in essence. This way can avoid the occurrence of the inconsistency but can not give the answer to the question that why the Russell's problem occurred inevitably.
__In addition, this way succeeded in avoiding the Russell's paradox but at the same time it brought other problems, which are not solved today. For Example, the problem of "V = L", the distinction of sets and classes and so on. You might expect we would be able to solve these problems someday. However, there remain other many problems that seem to be inconsistent with mathematics itself. For example, the following two problems.
__(1) ZF set theory forbids each thing in this real world to invade into ZF universe as an element. (because of the axiom of regularity). This means that ZF set theory cannot replace the naive set theory which can take up a lot of things in the real world as elements in its universe. ZF set theory and the naive set theory are incompatible. Both are inconsistent. Nevertheless, ZF is based on the naive set theory. The axioms of ZF are based on the ideas of the naive set theory that ZF itself denies. ZF was born of the naive set theory and regards the naive set theory as "a wrong theory" or "what is inconsistent with me." ZF resembles a child who betrays its mother. We cannot receive such a theory as a right theory or believe it. --Thus the ground of ZF is shaken.
__(2) In the universe of ZF, cardinal numbers grow up one by one from the beginning (null set). The axiom of the power set makes infinite cardinal numbers grow up endlessly. In this real world, however, infinite cardinal numbers are only countable and continuum and therefore the upper infinite cardinal numbers cannot exist. Nevertheless, ZF set theory insists that the upper infinite cardinal numbers more than continuum should exist. It insists not only that they can exist but also that they must exist. This is odd because it is impossible to put a large thing in a small thing.
__To sum up, we cannot recognize that the way of the current set theory succeeded. It brought some fruits but we cannot hold the conviction that it is absolutely right.
__I have told the history until now above.
__
__[2] Another way
__Now, what should we do?
__In case you lead to a dead end, you should rethink the problem from the beginning. Let's get back to the principle to consider what is a set.
__A set is, originally, regarded as the following.
__"A set is one package of plural things."
__When you get plural things and pack them into one, this one is a set. Such an operation as packing brings a set. The axioms of ZF are only the mathematical formalizations of such an operation.
__Now we get a question. What means this operation as packing plural things into one in essence? To say briefly, it means putting plural elements into a couple of brackets. For example, when you get elements such as,
____________p, q, r, s, t
__then you can put these letters into a couple of brackets and make a line of signs as the following.
__________{ p, q, r, s, t <>}
__If you define this line of signs as X, the new sign X means a certain one thing. Thus you can pack plural things into one. Such an operation is the essence of set theory.
__By the way, can we consider a group of plural things without such an operation?
__It seems hardly to be possible. It is, however, possible really. We can build a new theory that takes up a group of plural things without packing them by brackets.
__I will show it in the followings.
__
__[3] Atom
__This new theory that is shown on this homepage is named ward theory. Ward theory is the theory that takes up nude sets, which are made by stripping the brackets from usual sets.
__This theory uses no brackets when it takes up a group of plural things. Then, how does it take up plural things?
__In the universe of ward theory, plural things are not packed into one. Plural things are taken as plural things and a single thing is taken as a single thing. Plural things and a single thing must be distinguished clearly. A group of things can not be plural elements and a single set at the same time. Such a doubling does not occur.
__In this universe a single thing is named an atom. This term means that "it can not be divided nor split." (Of course this term has its origin of Greek philosophy.)
__[cf.] An atom resembles a singleton of set theory but both are different because the former has no bracket.
__If we use atoms, we can build the theory of nude sets. However, readers would not be able to believe this insistence. Then I must show this system of axioms concretely.
__This system is written in the next section.
__
__[4] Axioms of ward theory (temporary expression)
__In the followings I tell a theory named ward theory or the ward theory of temporary expression.
__Detailed explanations for beginners are not written here because this homepage is intended for experts. Read only the following axioms and understand their meanings by yourself.
__
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
__
__________The Ward Theory of Temporary Expression.
__
__ (I) Meanings of symbols
__This theory is based on the predicate logic, which uses some symbols. However, Internet does not permit us to use these symbols. Therefore I will use substitutive symbols as the followings.
__
______~________ negative
______*________ universal quantifier (Instead of reversal "A")
______+________ existential quantifier (Instead of reversal "E")
______V________ "or" of propositions
______^________ "and" of propositions
______,________ "and" of propositions
______=>________If...then
______<=>______ equivalence
______:::________ definition
__
__(II) Ward Theory
__Ward theory uses two non-defined terms. One is "<" and the other is "$ "
__(i) The symbol "<" is a non-defined term. We use this symbol instead of the lying "U" or very wide "(". In set theory, the same symbol is used for the meaning of "is a subset of ".
__(ii) The symbol "$ " is a non-defined term. $ is named the whole universe.
__(iii) Italic alphabets
__We have already got two non-defined terms. Now we can use these terms and italic alphabets such as x, y, z and then make formulas such as:
____________x__<__$
__ If such a formula come to be valid (i.e. to be true or false), x is named ward.
__In the followings, italic alphabets such as p, q, r, x, y, z are all wards. This means, for example, that if you get a ward p, then p satisfies the following formula:
____________p__<__$
__And q, r, y, z etc. are as same as p.
__And also % is a ward.
__[Attention] Non-italic capitals U and V are not wards.
__
__Now I show the system of ward theory as the followings.
__
__Axiom-1
__(*p)______ p__<__p
__[This means that any p is included in itself.]
__Axiom-2
__(*p) (*q) (*r)______ ( p__<__q )__^__( q__<__r )__ =>__ ( p__<__r )
__Axiom-3
__(+%) (*p)______ %__<__p
__[This means that there is a ward that is included in any ward. This ward % is named null ward. This is almost equivalent to the null set of set theory.]
__Axiom-4
__ $__<__$
__[This means that $ is also a ward.]
__Axiom-5
__(*p) (*q) (+s) (*x)______ ( x__<__s )__ <=>__ ( x__<__p )__^__( x__<__q )
__[This means that there is the cap of p and q for any p, q. This s is written as__p & q ]
__Axiom-6
__ (*p) (*q) (+s) (*x)______ ( s__<__x )__ <=>__ ( p__<__x )__^__( q__<__x )
__[This means that there is the cup of p and q for any p, q. This s is written as__p U q.]
__Definition
____( p__=__q )__ :::__ ( p__<__q )__^__( q__<__p )
__[Here the left side is defined by the right side.]
__Axiom-7
__(*r) (*p) (+c)______ ( p__<__r )__ =>__ ( r__=__p U c )__^__( p & c__=__% )
__[This c is named the complement ward. A complement ward is almost equivalent to a complement set of set theory.]
__Definition
__t @ p__ :::__ [ t__<__p ]__^__ [ ~ ( t__=__% ) ]__^__(*x)[ ( x__<__t )__ =>
______________ ( x__=__% )__V__( x__=__t ) ]
__[This means the followings.
__ (i) t is included in p
__ (ii) t is not null ward.
__ (iii) If x is included in t, then x must be t or null ward.]
__[This t is named atom.]
__Axiom-8
__ (*p) (+t)______~ ( p__=__%)__ =>__ t @ p
__[This means that any ward except null ward includes an atom.]
__
__[5] Explanation for Ward theory
__I did not tell explanations for each axiom or definition. However, I must tell explanation about the whole system of axioms as the followings.
__(1) Ward theory has axiom-8. This axiom is named axiom of the atom. Set theory without axiom of choice does not have such an axiom. This fact gives the clear difference between ward theory and set theory.
__(2) Ward theory has introduced the axiom-8 and therefore can get a beautiful structure of the whole system. See axiom-1, axiom-2 and axiom-3.
__(3) We can introduce axiom-4 into this system. That is, we can defy whole universe of ward theory as a closed space. Consequently ward theory bear neither Russell's paradox nor Cantor's paradox. You can prove it easily and elementarily. Anyway, this fact gives a clear difference between ward theory and set theory.
__(4) Ward theory does not include an axiom equivalent to the axiom of the power set. Such an axiom can not be included in ward theory. (You could get the reason if you think a little. To say briefly, atoms of a group are plural things and they can not come to be a single thing.) Consequently, cardinal number does not grow up endlessly.
__(5) Ward theory itself includes no axiom about cardinal numbers. Such an axiom, however, is not forbidden to be taken. It should be added to ward theory afterwards as an external. Both the axioms of ward theory and the axiom about cardinal numbers coexist and constitute the foundation of mathematics. (This way is different from that of ZF.) Well, you must pay a special attention when you consider the axiom of infinity. This axiom is not included in ward theory and therefore we can constitute a universe of ward theory only in finite universe. In such a universe without the axiom of infinity, we can prove completeness and consistency of ward theory. (Of course we can not get the same way for ZF, which has the axiom of regularity.)
__
__[6] Difference of conclusions
__Ward theory is different from ZF as I have told above. Consequently both give different conclusions as the followings.
__(1) Ward theory doesn't have the axiom of the power set and therefore cardinal numbers do not grow up endlessly. That is, you can get the universe of finiteness, that of countable and also that of continuum. In any case you can get a closed universe, in which infinite cardinal numbers need not grow up to higher powers. Such higher powers can exist and can also not-exist. Ward theory has more freedom than ZF.
__ZF does not have such freedom. ZF bears higher infinities necessarily in its universe. For example, it cannot stop the bearing of infinities below countable or continuum.
__We get ward theory and ZF. Now, which is more suitable for this cosmos which we live in? Of course, ward theory. The cosmos which we live in has the cardinal number of continuum and cannot have higher cardinal numbers. It is impossible to put a large thing in a small thing. Meanwhile, ZF must have higher cardinal numbers than continuum.
__In other words, ZF is not a theory for the cosmos which we live in. It is a theory for another cosmos like the theory of four-dimensional cosmos. Such a theory is not suitable for three-dimensional cosmos which we live in.
__(2) Ward theory does not include the axiom of infinity or natural numbers. It also does not include the axiom of real numbers. This axiom is added to this system afterwards as an external, just like the axiom of infinity.
__In addition, the axiom of real numbers that are to be added to the ward theory must be completely different from the usual axiom of real numbers that is proposed by Dedekind. If you use this new axiom of real numbers, you are naturally led to non-standard analysis that takes up infinitesimal. (You are naturally led to it but you cannot be easily to led to it. The ward theory in chapter 1 is insufficient. You need to wait the new theory written in chapter 2.)
__
__[7] Faults
__The ward theory of temporary expression above has some small faults. It does not include inconsistency formally but some problems philosophically or meaningly.
__Remember the quantifiers (existential quantifiers and universal quantifiers). They are defined in the predicate logic as the symbols that take up individuals.
__Such a symbol qualifies grammatically not only a sign that means an individual but also a sign that means a group of individuals (plural things).
__This is proper in set theory because a group of individuals is also an individual when they are packed into one. A set is one thing that includes many things. A set has the status of a single thing and plural things. However, ward theory should not or could not get the same way. In the ward theory plural things and a single thing must be distinguished clearly. Atoms of a group are plural things and they can not come to be one. Therefore, for example, "p" of axiom-8 may be plural individuals (atoms). Nevertheless, the predicate logic demands that a sign after a quantifier must mean a single individual.
__Ward theory seems to be incompatible with the predicate logic if you consider its meanings, though it is compatible as far as you consider it only formally.
__Now, let's come back to the beginning and rethink.
__What is an individual in the predicate logic? It is, really, not defined enough. We use the word "individual" while we do not know its meaning exactly. For example, is a woman an individual if she is pregnant? Is a cup of water an individual after it is drunken? Are colors such as blue or red individuals? What gives the difference between individuals and non-individuals?
__When we use English, we believe we know what are individuals very well. However, such individuals are determined linguistically instead of logically or physically. English has many nouns that means plural things and is treated grammatically as a single.
__The predicate logic takes up individuals while it leaves them unclear. Moreover, the meanings of quantifiers are also unclear. Some people insist that if we use these symbols and get no inconsistency formally, then it is the right way. This standpoint is, however, inaccurate or irresponsible. These symbols (quantifiers) should be defined exactly or led as a theorem. A theory that persists in seeking for consistency instead of seeking for truth may be somewhat unbelievable. If you feel so, you will also feel that the system of the axioms of the predicate logic does not have beauty. It does not satisfy the following two conditions for being a good system of axioms.
__The conditions are:
__(i) Simple propositions bring complex propositions.
__(ii) Axioms are not many.
__Any theory that is recognized to be right does satisfy these two conditions. Only the predicate logic, however, does not. It can satisfy one but cannot both. Then, we cannot convince ourselves that such a theory is right. If we believe that truth and beauty are similar, we should rather abandon the predicate logic.
__
__[8] A new theory
__However, can we get a new theory instead of the current theories (the predicate logic and ZF)? Can we get such a theory that can solve all the problems above?
__Yes. I can answer so. You might, however, hardly believe it. Then, I should tell this theory in the followings.
__In chapter 2, I will show a new theory, which is named the ward theory of regular expression.
__The ward theory of temporary expression and ZF set theory were theories that were based on the predicate logic. On the other hand, the coming new theory is not based on the predicate logic. It takes a completely different way.
__What a way? It is a way of using only free variables.
__Using only free variables to build a new theory that is almost equivalent to ZF. Is it really possible? Of course, it is not possible so far as we take usual methods. If it is possible easily, someone would have done it long ago.
__We need a special idea to do it --- a new idea that has not appeared in the history of mathematics. This new idea brings a new theory.
__This new idea goes over the frames of usual mathematics. I cannot say it is easy to understand it. If you are not an expert, you need not read the next chapter.
__However, this theory has a very simple and beautiful form. If you believe that truth must live in simplicity and beauty, read chapter 2, please.
__
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__Chapter 2
__
__[ 1 ] Outline
__In this chapter I will tell of the ward theory of regular expression. Readers might expect me to explain the new idea firstly and show the axioms secondly. However, I show firstly the axioms without the explanation of the new idea.
__Even if you don't get this idea beforehand, you would be able to understand almost all of the following axioms. We can get the ward theory of regular expression if we use free variables and convert the ward theory of temporary expression. Axiom-1 ~ Axiom-7 of the ward theory of temporary expression can be easily converted to Axiom-1 ~ Axiom-7 of the ward theory of regular expression. However, as for Axiom-8, we need a new idea for converting.
__Anyhow, you will be able to understand Axiom-1 ~ Axiom-7 easily. Please see the whole system of the ward theory of regular expression at first.
__[ 2 ] Axioms of the ward theory of regular expression
__
__________The Ward Theory of Regular Expression
__In the followings, details are as same as the ward theory of temporary expression. I don't repeat them here.
__However, you should be aware that each italic alphabet such as p, q, r.... is a free variable in the followings.
__
__Axiom-1
________ p__<__p
__Axiom-2
________( p__<__q )__^__( q__<__r )__ =>__ ( p__<__r )
__Axiom-3
________%__<__p
__[ % is a new symbol. It is a sort of function as a constant. you may write it as %(p) instead of % .]
__Axiom-4
________$__<__$
__Axiom-5
________( x__<__p & q )__ <=>__ ( x__<__p )__^__( x__<__q )
__[This p & q is a new function of p and q. you may write it as &(p,q) instead of__p & q .]
__Axiom-6
________( p U q__<__x )__ <=>__ ( p__<__x )__^__( q__<__x )
__[This p U q is a new function of p and q. you may write it as U(p,q) instead of__p U q .]
__Axiom-7
________( p__<__r ) => ( r__=__p U p''' )__^__( p & p'''__=__% )
__[This p''' is a new function of p and r. you may write it as__C(p,r)__instead of__p''' .]
__Definition of @
__( t @ p )__ :::__ [ t__<__p ]__^__[ ~ ( t__=__% ) ]__^__[ ( x__<__t )__ =>
________________( x__=__% )__V__( x = t ) ]
__Definition of function
____( x__=__y )__ =>__ F (x)__=__F (y)
__Definition of semi-function
____( x__ :::__ y )__ =>__ F (x)__=__F (y)
__Axiom-8
________~ ( p__=__%)__ =>__ #(p) @ p
__[ This__#(p)__is a semi-function of p .]
__
__[ 3 ] Definitionism
__I have already shown above the axioms of the ward theory of regular expression.
__You would be able to understand Axiom-1 ~ Axiom-7 easily. These Axioms are given by just converting the ward theory of temporary expression to the ward theory of regular expression.
__However, you should be aware of the following two points.
__(i) This system is not based on predicate logic but on propositional logic.
__(ii) In each of Axiom-3, Axiom-5, Axiom-6 and Axiom-7, a new symbol appears and its meaning is prescribed by the axiom.
__
__You should pay a special attention to the point of (ii). In each of Axiom-3, Axiom-5, Axiom-6 and Axiom-7, the meaning of a new symbol is prescribed by the axiom. Therefore we can regard these axioms as definitions in a broad sense (or extended definitions).
__The ward theory of regular expression takes definitions for very importance. We can say that the ward theory of regular expression is a theory based on extended definitions. This standpoint is compared with formalism since Hilbert and can be named definitionism.
__Axioms of the ward theory of regular expression are not formulas that are assumed to be true nor formulas that are introduced as far as they bring no inconsistency but formulas that prescribe the whole system. Therefore they are as same as other axioms of mathematics (e.g. the axioms of group, the axioms of Non-Euclid space). They are completely different from the axioms of ZF.
__[ 4 ] Semi-function
__We can get Axiom-1 ~ Axiom-7 of the ward theory of regular expression by just converting those of the ward theory of temporary expression. Then, Axiom-8 ?
__In order to get Axiom-8 of the ward theory of regular expression, you must understand the semi-function beforehand. This is a new idea that has never appeared in the history of mathematics. Then, I shall give a precise explanation for it in the following.
__What is a semi-function?
__A semi-function resembles to a function but both are somewhat different. The idea of a semi-function is weaker (broader) than that of a function.
__Today we know the idea of function. It is expressed as such a formula.
____________ ( x__=__y )____ =>__ F (x)__=__F (y)____________------ ( * )
__If this formula is true for any x, y as free variables, then F (x)__is regarded as a function of x.
__Today we also know that the idea of definition. It is expressed as such a formula.
______________Z__ :::__ G(x)
__________(The left side of__:::__is defined by the right side.)
__By the way, let's suppose the following formula.
____________ ( x__:::__y )__ =>__ F (x)__=__F (y)________ ------ ( ** )
__If F satisfies this formula for any x, y as free variables, then F is named semi-function of x. A semi-function is prescribed in this way.
__[5] Functions and semi-functions
__Functions and semi-functions are different. Well, what gives this difference?
__Look at ( * ) and ( ** ) above. Difference appears at the place of " = " and " ::: " in the left side of " => " . This difference brings the difference of functions and semi-functions.
__How do " = " and " ::: " differ?
__" ::: " is, of course, the symbol of definition. For example,
__________ x__:::__y
__This formula expresses that sign x and sign y are same as a sign.
__Meanwhile, " = " is the symbol of equivalence as a ward. For example,
__________ x__=__y
__This formula expresses that x and y are the same ward. In other words, if x and y are the same ward, then the sign x and the sign y behave as the same sign in the universe of ward theory.
__If x and y are the same sign, then the sign x and the sign y behave as the same sign not only in the universe of ward theory but also out of the universe of ward theory.
__For example, suppose the following two formula.
__________ x__:::__0.999999....
__________ y__:::__1.000000....
__In this case, " x = y " is true because x and y get the same value as a real number. However, " x__:::__y " is not true because you can put x in front of y on a special line, which is not the line of real numbers. (This special line exists out of real numbers.)
__Figuratively saying, " ::: " is compared to the world of multi-dimensional space and " = " is compared to the world of a projection. In a case, x and y are different three-dimensional shapes and have the same projection on a plane. If you regard this plane as the universe of ward theory, x and y is the same in this universe. Therefore the following formula is true.
__________ x__=__y
__However, x and y is not the same out of this universe. Therefore the following formula is not true.
__________ x__:::__y
__To sum up, the symbol " = " expresses the equivalence in the universe of ward theory while the symbol " ::: " expresses the equivalence out of the universe of ward theory.
__ [Comments] To say precisely, "out of universe of ward theory " means "out of the universe of ward theory and in the universe of propositional logic ".
__Now we should consider the relation of functions and semi-functions.
__Of course, the following formula is true.____
____________ ( x__:::__y )__=>__( x__=__y)
__However, we can not say that the following formula is true.
____________ ( x__=__y )__=>__( x__:::__y)
__Remember previous ( * ) and ( ** ), please. You will find that a semi-function resembles a function but is weaker (broader) than a function.
__A function and a semi-function have such a relation.
__
__[6] The meanings of the semi-function
__We have already got semi-functions and now we should consider their meanings.
__What does the semi-function mean?
__When__F (x)__is a semi-function,__F (x)__has the following properties.
__ (i) If x and y are the same sign (i.e.__x__:::__y ), then of course__F (x)__and__F (y)__are the same sign. The same sign has the same meaning. Therefore we can get a unique value__F (x)__for any sign x.
__ In this sense,__F (x)__does exist. (If__F (x)__does not exist,__ F (x)__has no value.)
__ (ii) If x and y are the same ward (i.e.__x__=__y ) but are not the same sign, then__F (x)__and__F (y)__need not to be the same. Therefore we can not get a unique value__F (x)__for any ward x.
__ In this sense,__F (x)__doesn't fix.
__To sum up, a semi-function__F (x)__does exist but doesn't fix. It is floating for one variable ward x and therefore is not invisible. On the other hand it brings a unique value__F (x)__for one variable sign x in propositional logic and therefore does exist.
__In other words, a semi-function is what does exist but is not invisible. And therefore the meanings of the semi-function are almost as same as the meanings of the function's existence.
__ZF does, on the predicate logic, use the words "a function exists." However, the ward theory of regular expression is a theory without the predicate logic. The ward theory of regular expression can not use a word "exist". It can not use existential quantifiers nor universal quantifiers. Nevertheless, it can get the idea of existence if it takes up the idea of semi-functions.
__Moreover, if it takes up the idea of semi-functions, it can give definitions to existential quantifiers and universal quantifiers.
__This will be shown in the followings.
__[ 7 ] Semi-functions and quantifiers
__In ZF, quantifiers are given only as non-defined term. However, quantifiers are prescribed strictly in the ward theory of regular expression.
__Now I show Axiom-8 here again.
________~ ( p = % )__ =>__ # (p) @ p
__The converse of this formula is the following.
________~ ( p = % )__ <=__ # (p) @ p
__This formula is true, of course. (You can prove it easily if you just understand the definition of @.)
__These two formulas above bring the following formula.
________~ ( p = % )__ <=>__ # (p) @ p
__In other words, following two propositions are equivalent.
____"p is not null ward "
____" p includes__# (p)__as an atom"
__Pay attention to__# (p). This is an atom and also a semi-function and therefore it does exist but is not invisible.
__Now, we can give a definition for existential quantifier "+". (The left side of " ::: " is defined by the right side.)
__Definition ( for the existential quantifier)
________ [ (+x)__ x @ p ]__ :::__ ~ ( p = % )
__On the other hand, I have already shown the following formula.
________ ~ ( p = % )__ <=>__ # (p) @ p
__These two formulas bring the following formula. This formula is a theorem.
__Theorem ( for the existential quantifier)
________ [ (+x)__ x @ p ]__ <=>__ #(p) @ p
__This theorem means that the following two propositions are equivalent.
____"An atom x exists in p."
____"A semi-function__#(p)__as an atom is included in p."
__Thus the existential quantifier is prescribed by a definition and a theorem.
__In addition, when you get the definition and the theorem for the existential quantifier, you can also get the definition for the universal quantifier easily. Its way is completely as same as that of predicate logic. (Therefore I don't write it here. Sorry.)
__Moreover, you can also get the following theorems. (It is not difficult to prove them.)
__Theorem ( by universal quantifier)
________(*x)[ x @ p__ =>__ x @ q ]__ <=>__ p__<__q
________(*x)[ x @ p__ V__ x @ q ]__ <=>__ p U q__=__$
________(*x)[ x @ p__ ^__ x @ q ]__ <=>__ p & q__=__$
__[Attention]
__A sign that is placed next to a quantifier must be an atom. For example, x in the formula above must be an atom.
__If a sign that is placed next to a quantifier is not an atom, this formula is not valid. For example, if x in the formula above contains two atoms, the left side of equivalence is false.
__[ 8 ] Summary
__I have already shown the ward theory of regular expression.__This is a definite system without a little doubt.
__ If we take it for a basis and add to it the axiom of infinity etc., then we can get the foundation of mathematics. ( I have told this in chapter 1.)
__Thus we can build the mathematics of today like ZF does. Consequences brought by ZF and consequences brought by the ward theory of regular expression are almost same, and in particular are completely same as far as we take up only natural numbers (under the cardinal number of countable).
__However, both are different in some parts.
__Let's consider real numbers. The axiom of real numbers in the ward theory of regular expression has a figure which is completely different from that of ZF or that of mathematics today. The ward theory of regular expression brings its axiom of real numbers, under which real numbers must be infinitesimals instead of points. This axiom consequently brings its analysis, which is not the usual analysis but the infinitesimal analysis.
__In the next chapter I will refer to the axiom of real numbers in the ward theory of regular expression.
__
__CHAPTER 3
__
__I'd like to show the axiom of real numbers in the ward theory of regular expression precisely but show it only briefly on this homepage. Excuse me.
__(1) Axiom of real numbers
__In short, the axiom of real numbers means the following.
____"Any ward that has a infinite cardinal number can be split into two wards, each of which has the same cardinal number."
__Suppose natural numbers, which can be split into odd numbers and even numbers.
__Now, we can repeat this operation n times and split the first ward into many wards ( "many" means 2 to the n-th power).
__If we don' t take this operation step by step (n-times) but take this operation at once with a certain semi-function, we can split the first ward into many infinitesimals. ( "many" means a continuous cardinal number.)
__Thus we can get infinitesimals as real numbers.
__
__(2) Continuum hypothesis
__If the first ward has the cardinal number that is smaller than continuum, the same operation will be limited halfway. If we consider this problem, we are led to continuum hypothesis.
__Such a problem is a higher mathematical problem and therefore I shall tell of it no more. I only tell that the problem of continuum hypothesis is very simple if we take the ward theory of regular expression. You can easily find that why we don't have a mid cardinal number naturally and also why we can have it.
__(3) Without atom
__In the universe of the ward theory with the axiom of real numbers, Axiom-8 is not necessary. (This is a great fact.)
__In other words, we can build a universe of Axiom-1 ~ Axiom-7 and the axiom of real numbers without Axiom-8. There are only many infinitesimals and no atom in this universe.
__This brings a worldview which is different from that of mathematics today. However, this worldview is not inconsistent with our intuition. Our three dimensional space or time need not have minimum elements while matters such as water has minimum elements (molecules, atoms, quarks, etc.).
__This worldview concludes that there are two types of numbers. [The type (i) belongs to the universe with Axiom-8 and the type (ii) belongs to the universe without Axiom-8.]
____(i)__natural numbesr__/__digital__ /__at intervals__/__points
____(ii)__real numbers____/__analogue /__continuous__/__infinitesimals____
For example, when we get a number "1", this "1" can be one of two types.
____ (i)__1 as a natural number. This can be written only as "1"
____ (ii)__1 as a real number. This can be written only as "1.00000000...."
____[Attention] "1" and "1.00000000...." are different. Each belongs to each universe.
__This worldview will be very suitable for quantum theory. Why quanta can get numbers only at intervals? The ward theory of regular expression can answer that it is because there are two types of numbers and sometimes a quantum get only a number of (i) above.__
__Comments
__Ward theory and set theory are different and bring different worldview. You might not be able to accept ward theory easily. Consider again with a blank brain, please.
__Note
__On this homepage I used substitutive symbols in stead of regular symbols. For example, I used " * " and " + " instead of reversal "A" and "E". However, in case you have Math B font in your PC, you had better use this font. If you want to do so, express this font for all alphabets (including capitals, symbols, numbers) and pick up some suitable letters to replace the substitutive symbols. The way to replace letters is depends on your software (word processor etc.).
__
[ Appendix ]
Theorems of Ward Theory
__I will show theorems of ward theory in the followings. They are elementary theorems. All of them have already proved.
__I use the ward theory of temporary expression instead of the ward theory of regular expression. We can, however, of course use the ward theory of regular expression.
__
[ Attention ]
__You should pay attention to the following points.
__(i) "Th" means "Theorem"
__(ii) A comma "," means and of propositions. For example,
______ X = Y ,__Y = Z____=>__ X = Z
<>__means
______( X = Y )__^__( Y = Z )__ =>__ X = Z
__(iii) If all the quantifiers in a theorem are universal quantifiers, quantifier are omitted. For exmple,
______________X = X
__means
_________(*X )_____X = X
__
------------------------------------------------
__[Th.31A]__
__________X = X
__[Th.31B]
__________X = Y__ =>____Y = X
__[Th.31C]
__________X = Y,__ Y = Z__ =>__ X = Z
__[Th.32A]
__________X = Y,__ Y__<__A__ =>__ X__<__A
__[Th.32B]
__________X = Y,__ B__<__Y__ =>__ B__<__Y
__[Th.33]
__________ [(+A )(*X ) A__<__X ]__ =>__ A = %
__[Th.34]
__________(*X ) [ X__<__%__ =>__ X = % ]
__[Th.35]
__________(*X ) [ $__<__X__ =>__ X = $ ]
__[Th.36]
__________A__&__B__ =__ B__&__A
__[Th.37]
__________A__U__B__ =__ B__U__A
__[Th.38]
__________( A & B ) & C__= A & ( B & C )
__[Th.39]
__________( A U B ) U C__= A U ( B U C )
__[Th.40]
__________( A U B ) & C__= ( A & C ) U ( B & C )
__[Th.41]
__________( A & B ) U C__= ( A U C ) & ( B U C )
__[Th.42]
__________A & B__= B__ <=>__ B__<__A__
__[Th.43]
__________A U B__= A__ <=>__ B__<__A
__[Th.44]
__________X__<__A__ =>__ X & B__<__A & B
__[Th.45 ~ 47]__ lacks of theorem numbers
__[Th.48]
__________(*A ) [ ~( A = % )__ =>__
______________(+a )(+B ) [A = a U B ,__ a & B__= %,__a @ B ]
__[Th.49]
__________a @ A,__ A & B__= %__ =>__ a & B__= %
__[Th.50]
__________a @ A,__b @ B,__A & B__= %__ =>__ ~( a = b )
__[Th.51]
__________A @ $,__B @ $,__~( A = B )__ =>__ A & B = %
__[Th.52]
__________a @ A__ =>__ a @ $
__[Th.53]
__________A__<__C ,__X @ C____=>__ X @ A__V__X @ A'''
__[Th.54]
__________X @ A,__A__<__B__ =>__ X @ B
__[Th.55]
__________X @ A & B__ <=>[ X @ A__^__X @ B ]
__[Th.56]
__________X @ A U B__ <=>[ X @ A__V__X @ B ]
__[Th.57]
__________A__<__B__ <=>__ (*X ) [ X @ A__ =>__ X @ B ]
__[Th.58]
__________( A & B )'''__=__A'''__U__B'''
__[Th.59]
__________( A U B )'''__=__A'''__&__B'''
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Written by______________________H. N. Greentree (Japan)
E-mail address :__________________ nando@js2.so-net.ne.jp
______________________________________________________________[The End]