A new definition of Cardinality.
in [Logic]
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From: Ostap S. B. M. Bender Jr. on 29 Nov 2009 05:29 On Nov 22, 3:01 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. > > 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. > What is"equinumerous"? How do you define that? "Of the same cardinality"? :-) What is the "transitive closure" of a set? Here is a set: {Apple, PC, Pear, Banana}. What is its "transitive closure"? > > 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 29 Nov 2009 06:29 On Nov 29, 5:29 am, "Ostap S. B. M. Bender Jr." <ostap_bender_1...(a)hotmail.com> wrote: > On Nov 22, 3:01 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. > > > 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. > > What is"equinumerous"? How do you define that? "Of the same > cardinality"? :-) equinumerous bes the existence of a bijection. x equinumerous to y <-> Exist a bijection between x and y. the definition of "bijection" do not mention 'cardinality', so it is not circular. > > What is the "transitive closure" of a set? > > Here is a set: {Apple, PC, Pear, Banana}. What is its "transitive > closure"? That is not a set in ZF, unless you add ur-elements to it, and you should add the constants apple,PC,Pear,Banana. However all of our discussion is about ZF without ur-elements . About the transitive closure of your set, this largely depends on how you will integrate ur-element to a set theory like ZF, but anyhow in most treatments of ur-elements , the transitive closure of your set would be the set itself. Now if you chose the approach of stipulating that every ur-element is an empty object (with modification of Extensionality to exclude them) then all your elements would be subnumerous to any cardinality, and your set will enter as a member into the fifth cardinal (i.e Natural number 4). Regards Zuhair > > > > > 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: Jesse F. Hughes on 29 Nov 2009 11:46 "Ostap S. B. M. Bender Jr." <ostap_bender_1900(a)hotmail.com> writes: > What is the "transitive closure" of a set? He defines that right below! >> >> 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,......} > > Here is a set: {Apple, PC, Pear, Banana}. What is its "transitive > closure"? > If Apple, PC, Pear and Banana are urelements, then Tc({Apple, PC, Pear, Banana}) = {Apple, PC, Pear, Banana}. I'm not sure that Zuhair's theory has urelements, so we can't answer your question unless you tell us which sets Apple, PC, Pear and Banana denote. -- Jesse F. Hughes Me: "Quincy, there's only *one* Truth, isn't there?" Quincy (age 4): "Yeah, and it's *mine*." -- A lesson in postmodernism goes awry.
From: George Greene on 30 Nov 2009 13:05 On Nov 30, 8:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Every set is equipollent to some cardinal kappa. This is a simple > theorem of ZFC. Therefore, NObody is disputing THAT. So why are you even belaboring it? This is called STUPIDITY. ON YOUR part. > Look, you said Rupert was wrong when he claimed that every set is > equipollent to some cardinal kappa. You are A DAMN LIAR. I said NO such thing. I said he was wrong about ZFC determining the cardinality of p(w). And ZFC in fact does NOT determine that p(w) is equipollent to any PARTICULAR kappa! > Now you seem to agree that this > is so, and are instead disputing something else entirely. Nothing has changed. I have always been disputing the SAME thing. Rupert is the one who chose to respond to the dispute by saying "it does so too have a cardinality -- beth-1". THAT WAS STUPID. Your failure to even properly identify the issue under discussion isn't much smarter.
From: Jesse F. Hughes on 30 Nov 2009 13:40
George Greene <greeneg(a)email.unc.edu> writes: > On Nov 30, 8:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Every set is equipollent to some cardinal kappa. This is a simple >> theorem of ZFC. > > Therefore, NObody is disputing THAT. Oh. I guess I can't read. Here's the bit that gave me this odd notion that you doubted this simple theorem. ,---- | > If I assume the axiom of choice, then every set can be well-ordered, | | Right. | | > and is equipollent to some cardinal kappa, | | WRONG. `---- > So why are you even belaboring it? > This is called STUPIDITY. > ON YOUR part. Yes, I must be stupid for misunderstanding that very clear exchange. Rupert said that ZFC proves every set is equipollent to some cardinal kappa and you said he was wrong. I thought that meant you believed he was wrong. I don't know where I got this silly notion. >> Look, you said Rupert was wrong when he claimed that every set is >> equipollent to some cardinal kappa. > > You are A DAMN LIAR. > I said NO such thing. Right. I probably just forged the above exchange. Note: you continued that little exchange as follows: ,---- | 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! `---- But, of course, this is a non-sequitur. ZFC proves that there is a unique cardinal equipollent to P(w). > I said he was wrong about ZFC determining the cardinality of p(w). > And ZFC in fact does NOT determine that p(w) is equipollent to any > PARTICULAR kappa! That may be what you meant, but it sure as hell isn't what you said. More to the point, you mean that ZFC does not prove |P(w)|=aleph_alpha for any ordinal alpha. That is clearly true. But it ain't my fault if what you meant is not what you said. >> Now you seem to agree that this >> is so, and are instead disputing something else entirely. > > Nothing has changed. > I have always been disputing the SAME thing. > Rupert is the one who chose to respond to the dispute by saying > "it does so too have a cardinality -- beth-1". > THAT WAS STUPID. Well, in a perfectly literal sense, Rupert's claim is true and correct (and not terribly informative). P(w) has a cardinality and that cardinality is beth-1 -- by definition of beth-1. Unfortunately, that does not tell us which aleph is the cardinality of P(w), but no one has disputed this obvious fact. > Your failure to even properly identify the issue under discussion > isn't much smarter. I can see that your writing skills are lacking, but let's try your reading comprehension. Pretend, for a moment, that these were not your words, but were written by someone else. How would you interpret them? ,---- | 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! `---- -- "And that's what we do. We put in more troops to get to a position where we can be in some other place. The question is, who ought to make that decision? The Congress or the commanders? And as you know, my position is clear. I'm the commander guy." --- George W. Bush |