A new definition of Cardinality.
in [Logic]
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From: Rupert on 22 Nov 2009 19:16 On Nov 23, 10:01 am, 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. > > 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 But the question arises: can you prove in ZF that every set has a cardinality, on this definition? Not quite obvious to me just at the moment... I have an idea that it is known that you need choice or regularity in order to define cardinals...
From: zuhair on 22 Nov 2009 19:37 On Nov 22, 7:16 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Nov 23, 10:01 am, 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. > > > 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 > > But the question arises: can you prove in ZF that every set has a > cardinality, on this definition? Not quite obvious to me just at the > moment... Well, to say the truth I am not sure either. But from my prior discussions I got the impression that working in ZF minus Regularity minus Choice, it is provable that: for any set there exist a set of exactly all sets that are hereditarily strictly sub-numerous to it. I remember the prove needed some form of induction or so. Then from Separation on that set it is easy to define the cardinality of x. Zuhair > > I have an idea that it is known that you need choice or regularity in > order to define cardinals...
From: zuhair on 22 Nov 2009 19:45 On Nov 22, 7:16 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Nov 23, 10:01 am, 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. > > > 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 > > But the question arises: can you prove in ZF that every set has a > cardinality, on this definition? Not quite obvious to me just at the > moment... > > I have an idea that it is known that you need choice or regularity in > order to define cardinals... Possibly that we would help: http://groups.google.com/group/sci.logic/browse_thread/thread/a4131a62df6fb8ed/431334f073da57c8?hl=en&q=hereditarily#431334f073da57c8 Zuhair
From: zuhair on 22 Nov 2009 19:48 On Nov 22, 7:45 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Nov 22, 7:16 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > On Nov 23, 10:01 am, 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. > > > > 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 > > > But the question arises: can you prove in ZF that every set has a > > cardinality, on this definition? Not quite obvious to me just at the > > moment... > > > I have an idea that it is known that you need choice or regularity in > > order to define cardinals... > > Possibly that we would help: > > http://groups.google.com/group/sci.logic/browse_thread/thread/a4131a6... > > Zuhair The above link actually speaks of ZFC, but I don't know really if this actually depends on choice or not? or weather regularity is needed or not? It seems that I got the wrong impression then. Zuhair
From: zuhair on 22 Nov 2009 19:51
On Nov 22, 7:16 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Nov 23, 10:01 am, 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. > > > 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 > > But the question arises: can you prove in ZF that every set has a > cardinality, on this definition? Not quite obvious to me just at the > moment... > > I have an idea that it is known that you need choice or regularity in > order to define cardinals... I think you mean in theories like ZF or Scott-Potters', however in NF and related theories, Neither choice nor regularity are need to define cardinals, and I do have the vague sense that this definition of mine here do not depend on them either?! Zuhair |