A new definition of Cardinality.
in [Logic]
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From: zuhair on 22 Nov 2009 11:12 Hi all, As far as I know, all the definitions of cardinality have limited in a way or another, lets take them one after the other: 1) Von Neumann's Cardinals: A cardinal is the least of all equinumerous ordinals. 2) Frege-Russell Cardinals: A cardinal is an equivalence class of sets under equivalence relation "bijection". 3) Scott-Potter Cardinals: A cardinal is a class of all equinumerous sets from a common level. Now lets come to discuss each one of them: 1) Von Neumann's cardinals has the limitation of being dependent on choice, without choice one cannot know what is the cardinality of Power (omega) for example.Accordingly in any theory which do not have the axiom of choice among its axioms most of its set would be of indeterminable cardinality, which is a big draw back. 2) Frege-Russell cardinals contradict Z set theory, since their existence would imply the existence of the set of all sets, which is in contradiction with Z. However in NBG and MK class theories, we can define Frege-Russell cardinals, but by then they would be proper classes and not set, which is a great draw back, since proper classes cannot be members of other classes. In NF and related theories Frege-Russell cardinals are sets, but these theories generally depend on the concept of stratification of formulas, which is a complex concept, and even finite axiomatization of NF and NFU and related theories is a complicated approach, and at the end it also resort to stratification for most of its inferences. All that make these cardinals undesirable. 3) Scott-Potter Cardinals: depend on the concept of "level" which depends on the concept of type (Scott) and the iterative concept (Potter), both concepts of which are complex and difficult to work with. I would like to suggest the following definition: 4) The cardinality of any set x is: The class of all sets that are equinumerous to x were every member of their transitive closure is strictly subnumerous to x. So for any set x, any y is a member of the cardinality of x, if and only if, y is equinumerous to x and every member of the transitive closure of y is strictly subnumerous to x. In symbols: Define(cardinality(x)):- z=cardinality(x) <-> for all y (y e z <-> (y equi-numerous to x & for all m (m e Tc(y)->m strictly subnumerous to x))) Were Tc(y) stands for the 'transitive closure of y' defined in the standard manner. Tc(y)=U{y,Uy,UUy,UUUy,......} We can actually better define these cardinals through defining the concept of "hereditary sets" Define(hereditary): x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) So a cardinal can be defined in the following manner: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". Or simply A Cardinal is a class of all equinumerous hereditary sets. So cardinality of x would be written shortly as: Cardinality(x) = {y| y is hereditary & y equinumerous to x} Now it can be proven in Z that those cardinals would be 'sets', so they are not proper classes! which makes them easy to work with. These cardinals don't require choice. They don't require complex concepts like "stratification,type, iteration" They simply depend on the basic concept used to compare set sizes, which is the presence of injections. To me this definition seems to be simpler, more general, and it works with or without choice , with or without regularity. So at the end I shall write the definition of cardinal again: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) Cardinality(x) = {y| y is hereditary & y equinumerous to x} Zuhair
From: zuhair on 22 Nov 2009 12:25 Hi all, As far as I know, all the definitions of cardinality are limited in a way or another, lets take them one after the other: 1) Von Neumann's Cardinals: A cardinal is the least of all equinumerous ordinals. 2) Frege-Russell Cardinals: A cardinal is an equivalence class of sets under equivalence relation "bijection". 3) Scott-Potter Cardinals: A cardinal is a class of all equinumerous sets from a common level. Now lets come to discuss each one of them: 1) Von Neumann's cardinals has the limitation of being dependent on choice, without choice one cannot know what is the cardinality of Power(omega) for example.Accordingly in any theory which do not have the axiom of choice among its axioms most of its sets would be of indeterminable cardinality, which is a big draw back. 2) Frege-Russell cardinals contradict Z set theory, since their existence would imply the existence of the set of all sets, which is in contradiction with Z. However in NBG and MK class theories, we can define Frege-Russell cardinals, but by then they would be proper classes and not sets, which is a great draw back, since proper classes cannot be members of other classes, and they are hard to manipulate. In NF and related theories, Frege-Russell cardinals are sets, but these theories generally depend on the concept of stratification of formulas, which is a complex concept, and even finite axiomatization of NF and NFU and related theories is a complicated approach, and at the end it also resort to stratification for most of its inferences. All that make these cardinals undesirable. 3) Scott-Potter Cardinals: depend on the concept of "level" which depends on the concept of type (Scott) and the iterative concept (Potter), both concepts of which are complex and difficult to work with, besides they are not concepts related to the basic concepts we compare set sizes with. I would like to suggest the following definition: 4) The cardinality of any set x is: The class of all sets that are equinumerous to x were every member of their transitive closure is strictly subnumerous to x. So for any set x, any y is a member of the cardinality of x, if and only if, y is equinumerous to x and every member of the transitive closure of y is strictly subnumerous to x. In symbols: Define(cardinality(x)):- z=cardinality(x) <-> for all y (y e z <-> (y equi-numerous to x & for all m (m e Tc(y)->m strictly subnumerous to x))) Were Tc(y) stands for the 'transitive closure of y' defined in the standard manner. Tc(y)=U{y,Uy,UUy,UUUy,......} We can actually better define these cardinals through defining the concept of "hereditary sets" Define(hereditary): x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) So a cardinal can be defined in the following manner: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". Or simply A Cardinal is a class of all equinumerous hereditary sets. So cardinality of x would be written shortly as: Cardinality(x) = {y| y is hereditary & y equinumerous to x} Now it can be proven in Z that those cardinals would be 'sets', so they are not proper classes! which makes them easy to manipulate. These cardinals don't require choice. They don't require complex concepts like "stratification,type, iteration" They simply depend on the basic concept used to compare set sizes, which is the presence or absence of injections between the compared sets. To me this definition seems to be simpler, more general, and it works with or without choice, with or without regularity. So at the end I shall write the definition of cardinal again: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) Cardinality(x) = {y| y is hereditary & y equinumerous to x} Zuhair
From: zuhair on 22 Nov 2009 12:27 Hi all, As far as I know, all the definitions of cardinality are limited in a way or another, lets take them one after the other: 1) Von Neumann's Cardinals: A cardinal is the least of all equinumerous ordinals. 2) Frege-Russell Cardinals: A cardinal is an equivalence class of sets under equivalence relation "bijection". 3) Scott-Potter Cardinals: A cardinal is a class of all equinumerous sets from a common level. Now lets come to discuss each one of them: 1) Von Neumann's cardinals has the limitation of being dependent on choice, without choice one cannot know what is the cardinality of Power(omega) for example.Accordingly in any theory which do not have the axiom of choice among its axioms most of its sets would be of indeterminable cardinality, which is a big draw back. 2) Frege-Russell cardinals contradict Z set theory, since their existence would imply the existence of the set of all sets, which is in contradiction with Z. However in NBG and MK class theories, we can define Frege-Russell cardinals, but by then they would be proper classes and not sets, which is a great draw back, since proper classes cannot be members of other classes, and they are hard to manipulate. In NF and related theories, Frege-Russell cardinals are sets, but these theories generally depend on the concept of stratification of formulas, which is a complex concept, and even finite axiomatization of NF and NFU and related theories is a complicated approach, and at the end it also resort to stratification for most of its inferences. All that make these cardinals undesirable. 3) Scott-Potter Cardinals: depend on the concept of "level" which depends on the concept of type (Scott) and the iterative concept (Potter), both concepts of which are complex and difficult to work with, besides they are not concepts related to the basic concepts we use to compare set sizes. I would like to suggest the following definition: 4) The cardinality of any set x is: The class of all sets that are equinumerous to x were every member of their transitive closure is strictly subnumerous to x. So for any set x, any y is a member of the cardinality of x, if and only if, y is equinumerous to x and every member of the transitive closure of y is strictly subnumerous to x. In symbols: Define(cardinality(x)):- z=cardinality(x) <-> for all y (y e z <-> (y equi-numerous to x & for all m (m e Tc(y)->m strictly subnumerous to x))) Were Tc(y) stands for the 'transitive closure of y' defined in the standard manner. Tc(y)=U{y,Uy,UUy,UUUy,......} We can actually better define these cardinals through defining the concept of "hereditary sets" Define(hereditary): x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) So a cardinal can be defined in the following manner: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". Or simply A Cardinal is a class of all equinumerous hereditary sets. So cardinality of x would be written shortly as: Cardinality(x) = {y| y is hereditary & y equinumerous to x} Now it can be proven in Z that those cardinals would be 'sets', so they are not proper classes! which makes them easy to manipulate. These cardinals don't require choice. They don't require complex concepts like "stratification,type, iteration" They simply depend on the basic concept used to compare set sizes, which is the presence or absence of injections between the compared sets. To me this definition seems to be simpler, more general, and it works with or without choice, with or without regularity. So at the end I shall write the definition of cardinal again: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) Cardinality(x) = {y| y is hereditary & y equinumerous to x} Zuhair
From: zuhair on 22 Nov 2009 18:01 Hi all, As far as I know, all the definitions of cardinality are limited in a way or another, lets take them one after the other: 1) Von Neumann's Cardinals: A cardinal is the least of all equinumerous ordinals. 2) Frege-Russell Cardinals: A cardinal is an equivalence class of sets under equivalence relation "bijection". 3) Scott-Potter Cardinals: A cardinal is a class of all equinumerous sets from a common level. Now lets come to discuss each one of them: 1) Von Neumann's cardinals has the limitation of being dependent on choice, without choice one cannot know what is the cardinality of Power(omega) for example.Accordingly in any theory which do not have the axiom of choice among its axioms most of its sets would be of indeterminable cardinality, which is a big draw back. 2) Frege-Russell cardinals contradict Z set theory, since their existence would imply the existence of the set of all sets, which is in contradiction with Z. However in NBG and MK class theories, we can define Frege-Russell cardinals, but by then they would be proper classes and not sets, which is a great draw back, since proper classes cannot be members of other classes, and they are hard to work with. In NF and related theories, Frege-Russell cardinals are sets, but these theories generally depend on the concept of stratification of formulas, which is a complex concept, and even finite axiomatization of NF and NFU and related theories is a complicated approach, and at the end it also resort to stratification for most of its inferences. All that make these cardinals undesirable. 3) Scott-Potter Cardinals: depend on the concept of "level" which depends on the concept of type (Scott) and the iterative concept (Potter), both concepts of which are complex and difficult to work with, besides they are not the basic concepts we use to compare set sizes. I would like to suggest the following definition: 4) The cardinality of any set x is: The class of all sets that are equinumerous to x were every member of their transitive closure is strictly subnumerous to x. So for any set x, any y is a member of the cardinality of x, if and only if, y is equinumerous to x and every member of the transitive closure of y is strictly subnumerous to x. In symbols: Define(cardinality(x)):- z=cardinality(x) <-> for all y (y e z <-> (y equi-numerous to x & for all m (m e Tc(y)->m strictly subnumerous to x))) Were Tc(y) stands for the 'transitive closure of y' defined in the standard manner. Tc(y)=U{y,Uy,UUy,UUUy,......} We can actually better define these cardinals through defining the concept of "hereditary sets" Define(hereditary): x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) So a cardinal can be defined in the following manner: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". Or simply A Cardinal is a class of all equinumerous hereditary sets. So cardinality of x would be written shortly as: Cardinality(x) = {y| y is hereditary & y equinumerous to x} Now it can be proven in ZF that those cardinals would be 'sets', so they are not proper classes! which makes them easy to handle. These cardinals don't require choice. They don't require complex concepts like "stratification,type, iteration" They simply depend on the basic concept used to compare set sizes, which is the presence or absence of injections between the compared sets. To me this definition seems to be simpler, more general, and it works with or without choice, with or without regularity. So at the end I shall write the definition of cardinal again: A Cardinal is an equivalence class of hereditary sets under equivalence relation "bijection". x is hereditary <-> for all y (y e Tc(x) -> y strictly subnumerous to x) Cardinality(x) = {y| y is hereditary & y equinumerous to x} Zuhair
From: zuhair on 22 Nov 2009 18:29
On Nov 22, 6:01 pm, zuhair <zaljo...(a)gmail.com> wrote: > Hi all, > > As far as I know, all the definitions of cardinality are limited in a > way or another, lets take them one after the other: > > 1) Von Neumann's Cardinals: > > A cardinal is the least of all equinumerous ordinals. > > 2) Frege-Russell Cardinals: > > A cardinal is an equivalence class of sets under equivalence relation > "bijection". > > 3) Scott-Potter Cardinals: > > A cardinal is a class of all equinumerous sets from a common level. > > Now lets come to discuss each one of them: > > 1) Von Neumann's cardinals has the limitation of being dependent on > choice, without choice one cannot know what is the cardinality of > Power(omega) for example.Accordingly in any theory which do not have > the axiom of choice among its axioms most of its sets would be of > indeterminable cardinality, which is a big draw back. In addition to that, the concept of Cardinality has nothing to do whatsoever with the concept of "order", so defining cardinals as subclass of ordinals seems to be strange, though practical if choice is assumed. > > 2) Frege-Russell cardinals contradict Z set theory, since their > existence would imply the existence of the set of all sets, which is > in contradiction with Z. > > However in NBG and MK class theories, we can define > Frege-Russell cardinals, but by then they would be proper classes and > not sets, which is a great draw back, since proper classes cannot be > members of other classes, and they are hard to work with. > > In NF and related theories, Frege-Russell cardinals are sets, but > these theories generally depend on the concept of stratification > of formulas, which is a complex concept, and even finite > axiomatization of NF and NFU and related theories is a complicated > approach, and at the end it also resort to stratification for most of > its inferences. All that make these cardinals undesirable. > > 3) Scott-Potter Cardinals: depend on the concept of "level" which > depends on the concept of type (Scott) and the iterative concept > (Potter), both concepts of which are complex and difficult to work > with, besides they are not the basic > concepts we use to compare set sizes. > > I would like to suggest the following definition: > > 4) The cardinality of any set x is: The class of all sets > that are equinumerous to x were every member of their transitive > closure is strictly subnumerous to x. > > So for any set x, any y is a member of the cardinality of x, > if and only if, y is equinumerous to x and every member of the > transitive closure of y is strictly subnumerous to x. > > In symbols: > > Define(cardinality(x)):- > > z=cardinality(x) <-> > for all y (y e z <-> > (y equi-numerous to x & > for all m (m e Tc(y)->m strictly subnumerous to x))) > > Were Tc(y) stands for the 'transitive closure of y' defined > in the standard manner. > > Tc(y)=U{y,Uy,UUy,UUUy,......} > > We can actually better define these cardinals through defining the > concept of "hereditary sets" > > Define(hereditary): > x is hereditary <-> > for all y (y e Tc(x) -> y strictly subnumerous to x) > > So a cardinal can be defined in the following manner: > > A Cardinal is an equivalence class of hereditary sets under > equivalence relation "bijection". > > Or simply > > A Cardinal is a class of all equinumerous hereditary sets. > > So cardinality of x would be written shortly as: > > Cardinality(x) = {y| y is hereditary & y equinumerous to x} > > Now it can be proven in ZF that those cardinals would be 'sets', so > they are not proper classes! which makes them easy to handle. > > These cardinals don't require choice. > > They don't require complex concepts like "stratification,type, > iteration" > > They simply depend on the basic concept used to compare set sizes, > which is the presence or absence of injections between the compared > sets. > > To me this definition seems to be simpler, more general, and it works > with or without choice, with or without regularity. > > So at the end I shall write the definition of cardinal again: > > A Cardinal is an equivalence class of hereditary sets under > equivalence relation "bijection". > > x is hereditary <-> > for all y (y e Tc(x) -> y strictly subnumerous to x) > > Cardinality(x) = {y| y is hereditary & y equinumerous to x} > > Zuhair |