A new definition of Cardinality.
in [Logic]
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From: zuhair on 24 Nov 2009 02:15 On Nov 24, 12:55 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Nov 23, 10:37 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > > > On Nov 22, 9:20 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > On Nov 23, 11:37 am, 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... > > > > > 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. > > > > That is certainly not a problem in ZFC but I certainly don't see right > > > now how to do it without regularity or choice. You should look into > > > that one carefully, I should think. > > > The sketch of the proof goes like that: > > > First we prove that for every set x there exist a set of exactly all > > sets that are > > hereditarily strictly subnumerous to it,lets denote that later set by > > H_(<x) > > > so we have: for every x there exist H_(<x) > > > H_(<x) = {y | y strictly subnumerous to x and > > for all z (z e Tc(y) -> z strictly subnumerous to > > x)} > > > as a theorem of ZF. > > Well, 'tis surely a theorem of ZFC, but yeah, why would it be a > theorem of ZF? Let's get clear about that first. > > And you were saying you could do it in ZF-regularity. > > But, yeah, I would like to know how you do it in ZF. That is my personal guess, I don't have a proof of that yet. By the way can you help me regarding this issue, and post the complete proof of this theorem in ZFC, I think this would be of great help. Zuhair
From: Rupert on 24 Nov 2009 03:23 On Nov 24, 6:15 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Nov 24, 12:55 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > > On Nov 23, 10:37 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > On Nov 22, 9:20 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > On Nov 23, 11:37 am, 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... > > > > > > 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. > > > > > That is certainly not a problem in ZFC but I certainly don't see right > > > > now how to do it without regularity or choice. You should look into > > > > that one carefully, I should think. > > > > The sketch of the proof goes like that: > > > > First we prove that for every set x there exist a set of exactly all > > > sets that are > > > hereditarily strictly subnumerous to it,lets denote that later set by > > > H_(<x) > > > > so we have: for every x there exist H_(<x) > > > > H_(<x) = {y | y strictly subnumerous to x and > > > for all z (z e Tc(y) -> z strictly subnumerous to > > > x)} > > > > as a theorem of ZF. > > > Well, 'tis surely a theorem of ZFC, but yeah, why would it be a > > theorem of ZF? Let's get clear about that first. > > > And you were saying you could do it in ZF-regularity. > > > But, yeah, I would like to know how you do it in ZF. > > That is my personal guess, I don't have a proof of that yet. > By the way can you help me regarding this issue, and post the complete > proof of this theorem in ZFC, I think this would be of great help. > > Zuhair- Hide quoted text - > > - Show quoted text - If I assume the axiom of choice, then every set can be well-ordered, and is equipollent to some cardinal kappa, a cardinal being a von Neumann ordinal not equipollent to any smaller ordinal. Now let R(0) be the empty set, and for any R(alpha), let R(alpha+1) be the set of all subsets of R(alpha) of cardinality less than kappa, and for any limit ordinal beta, let R(beta) be the union of all R(alpha) for alpha<beta. Iterate this up to R(kappa). In ZF I can prove this is a set, using transfinite recursion and suchlike, and this is the set you're after. Fair enough? But without the axiom of choice I know longer know that the set can be well-ordered... and so it is not really clear what sort of transfinite recursion to use. Do you get that? Do you want me to explain in more detail?
From: George Greene on 24 Nov 2009 10:15 On Nov 24, 12:55 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > But, yeah, I would like to know how you do it in ZF. You can't do it in ZFC either. ZFC DOES NOT assign a cardinality (in the usual sense) to p(w).
From: George Greene on 24 Nov 2009 10:20 On Nov 24, 3:23 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > If I assume the axiom of choice, then every set can be well-ordered, Right. > and is equipollent to some cardinal kappa, WRONG. Ex[whatever] does NOT mean that there exists a UNIQUE x such that whatever! There are MANY DIFFERENT well-orderings of p(w) under ZFC and they correspond TO DIFFERENT ordinals! > a cardinal being a von > Neumann ordinal not equipollent to any smaller ordinal. ZFC does NOT assign a cardinality to p(w) or any other uncountable set except maybe the alephs. Under ZFC, beth-1 (=p(w)=p(aleph-0) =2^beth-0) "can" take ANY aleph-n (where n is natural) as a cardinality! Remember, there was that whole conunundrum about THE CONTINUUM HYPOTHESIS ?!?!?!!?
From: zuhair on 24 Nov 2009 15:10
On Nov 24, 10:15 am, George Greene <gree...(a)email.unc.edu> wrote: > On Nov 24, 12:55 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > But, yeah, I would like to know how you do it in ZF. > > You can't do it in ZFC either. > ZFC DOES NOT assign a cardinality (in the usual sense) > to p(w). Dear George Greene, please answer the following questions: Is the following a theorem of ZFC or not? For all s Exist x for all y ( y e x <-> (y strictly subnumerous to s and for all z (z e Tc(y) -> z strictly subnumerous to s)). were Tc(y) stands for the transitive closure of y. in case it is a theorem of ZFC, then is it a theorem of ZF? i.e. without choice, and if it is a theorem of ZF, then is it a theorem of ZF minus regularity? Simple questions that demand simple answer and proofs of these answers! Zuhair |