What is the cardinality of a lone element?











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I know that elements which are not sets do not have cardinality, but hear me out.



Suppose that $A={u,{{v,w},x,{y,{z}}}}$. The nesting of the sets allows you to uniquely identify particular elements by there "depth" in $A$. For example $z$ is the only element $ein^4A$ ($in^4$ is an abuse of notation meant to suggest that $e$ is an element of an element of an element of an element of $A$).



However, this type of statement is not sufficient for uniquely identifying any element other than $z$, as, for example, the set of all $ein^2 A$ contains ${v,w}$, $x$, and${y,{z}}$.



It is still possible to distinguish between each of these elements, though; $ein^2A and |e|neq2$ defines $x$, $ein^2A and nexists bin e:|b|=1$ defines ${u,w}$, and $ein^2A and exists bin e:|b|=1$ defines ${y,{z}}$.



The statement used to identify $x$ is true because $xin^2A$ and $|x|neq2$, the latter because $x$ is not a set, and therefore cannot have a cardinality of 2. Similarly, ${y,{z}}in^2A$, and $|{z}|=1$.



Now, since I've used cardinality to differentiate the elements $ein^2A$ from one another, I should like to define the cardinality of each such element. While this is no problem for the sets ${u,v}$, ${y,{z}}$, and the singleton set ${z}in^3A$, it is an issue for the elements $x$ and $y$. The statement $|x|neq1$, while true, cannot be evaluated if $x$ does not have cardinality. Since ${x}neq x$, the cardinality of $x$ cannot be 1.



One possibility is that the cardinality of $x$ is 0, but this runs the risk of equating the element $x$ with the empty set, which would violate the axioms of ZFC if $x$ is considered to be a set. On the other hand, you could say that the empty set is the only set whose cardinality is zero, and that $x$ is not a set: thus, $|x|=0notimplies x=emptyset$. But this runs into the problem of defining what a set is.



So, what is the best way to define the cardinality of an element? If there isn't one, then is there a good way to describe how a lone element is unlike a set, in set theoretic terms?










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  • Every set except the empty set has elements. And if I understand your question correctly, you imply that $u,v,w,x,y,z$ have no elements. So they must all be the empty set, with cardinality zero. Or am I missing something?
    – TonyK
    Nov 18 at 2:35










  • $u$,$v$,$w$,$x$,$y$, and $z$ are not sets, they are constants, variables, vectors etc. For simplicity's sake, you can treat each of $u$,$v$,$w$,$x$,$y$, and $z$ as a separate real number.
    – R. Burton
    Nov 18 at 3:09















up vote
2
down vote

favorite












I know that elements which are not sets do not have cardinality, but hear me out.



Suppose that $A={u,{{v,w},x,{y,{z}}}}$. The nesting of the sets allows you to uniquely identify particular elements by there "depth" in $A$. For example $z$ is the only element $ein^4A$ ($in^4$ is an abuse of notation meant to suggest that $e$ is an element of an element of an element of an element of $A$).



However, this type of statement is not sufficient for uniquely identifying any element other than $z$, as, for example, the set of all $ein^2 A$ contains ${v,w}$, $x$, and${y,{z}}$.



It is still possible to distinguish between each of these elements, though; $ein^2A and |e|neq2$ defines $x$, $ein^2A and nexists bin e:|b|=1$ defines ${u,w}$, and $ein^2A and exists bin e:|b|=1$ defines ${y,{z}}$.



The statement used to identify $x$ is true because $xin^2A$ and $|x|neq2$, the latter because $x$ is not a set, and therefore cannot have a cardinality of 2. Similarly, ${y,{z}}in^2A$, and $|{z}|=1$.



Now, since I've used cardinality to differentiate the elements $ein^2A$ from one another, I should like to define the cardinality of each such element. While this is no problem for the sets ${u,v}$, ${y,{z}}$, and the singleton set ${z}in^3A$, it is an issue for the elements $x$ and $y$. The statement $|x|neq1$, while true, cannot be evaluated if $x$ does not have cardinality. Since ${x}neq x$, the cardinality of $x$ cannot be 1.



One possibility is that the cardinality of $x$ is 0, but this runs the risk of equating the element $x$ with the empty set, which would violate the axioms of ZFC if $x$ is considered to be a set. On the other hand, you could say that the empty set is the only set whose cardinality is zero, and that $x$ is not a set: thus, $|x|=0notimplies x=emptyset$. But this runs into the problem of defining what a set is.



So, what is the best way to define the cardinality of an element? If there isn't one, then is there a good way to describe how a lone element is unlike a set, in set theoretic terms?










share|cite|improve this question
























  • Every set except the empty set has elements. And if I understand your question correctly, you imply that $u,v,w,x,y,z$ have no elements. So they must all be the empty set, with cardinality zero. Or am I missing something?
    – TonyK
    Nov 18 at 2:35










  • $u$,$v$,$w$,$x$,$y$, and $z$ are not sets, they are constants, variables, vectors etc. For simplicity's sake, you can treat each of $u$,$v$,$w$,$x$,$y$, and $z$ as a separate real number.
    – R. Burton
    Nov 18 at 3:09













up vote
2
down vote

favorite









up vote
2
down vote

favorite











I know that elements which are not sets do not have cardinality, but hear me out.



Suppose that $A={u,{{v,w},x,{y,{z}}}}$. The nesting of the sets allows you to uniquely identify particular elements by there "depth" in $A$. For example $z$ is the only element $ein^4A$ ($in^4$ is an abuse of notation meant to suggest that $e$ is an element of an element of an element of an element of $A$).



However, this type of statement is not sufficient for uniquely identifying any element other than $z$, as, for example, the set of all $ein^2 A$ contains ${v,w}$, $x$, and${y,{z}}$.



It is still possible to distinguish between each of these elements, though; $ein^2A and |e|neq2$ defines $x$, $ein^2A and nexists bin e:|b|=1$ defines ${u,w}$, and $ein^2A and exists bin e:|b|=1$ defines ${y,{z}}$.



The statement used to identify $x$ is true because $xin^2A$ and $|x|neq2$, the latter because $x$ is not a set, and therefore cannot have a cardinality of 2. Similarly, ${y,{z}}in^2A$, and $|{z}|=1$.



Now, since I've used cardinality to differentiate the elements $ein^2A$ from one another, I should like to define the cardinality of each such element. While this is no problem for the sets ${u,v}$, ${y,{z}}$, and the singleton set ${z}in^3A$, it is an issue for the elements $x$ and $y$. The statement $|x|neq1$, while true, cannot be evaluated if $x$ does not have cardinality. Since ${x}neq x$, the cardinality of $x$ cannot be 1.



One possibility is that the cardinality of $x$ is 0, but this runs the risk of equating the element $x$ with the empty set, which would violate the axioms of ZFC if $x$ is considered to be a set. On the other hand, you could say that the empty set is the only set whose cardinality is zero, and that $x$ is not a set: thus, $|x|=0notimplies x=emptyset$. But this runs into the problem of defining what a set is.



So, what is the best way to define the cardinality of an element? If there isn't one, then is there a good way to describe how a lone element is unlike a set, in set theoretic terms?










share|cite|improve this question















I know that elements which are not sets do not have cardinality, but hear me out.



Suppose that $A={u,{{v,w},x,{y,{z}}}}$. The nesting of the sets allows you to uniquely identify particular elements by there "depth" in $A$. For example $z$ is the only element $ein^4A$ ($in^4$ is an abuse of notation meant to suggest that $e$ is an element of an element of an element of an element of $A$).



However, this type of statement is not sufficient for uniquely identifying any element other than $z$, as, for example, the set of all $ein^2 A$ contains ${v,w}$, $x$, and${y,{z}}$.



It is still possible to distinguish between each of these elements, though; $ein^2A and |e|neq2$ defines $x$, $ein^2A and nexists bin e:|b|=1$ defines ${u,w}$, and $ein^2A and exists bin e:|b|=1$ defines ${y,{z}}$.



The statement used to identify $x$ is true because $xin^2A$ and $|x|neq2$, the latter because $x$ is not a set, and therefore cannot have a cardinality of 2. Similarly, ${y,{z}}in^2A$, and $|{z}|=1$.



Now, since I've used cardinality to differentiate the elements $ein^2A$ from one another, I should like to define the cardinality of each such element. While this is no problem for the sets ${u,v}$, ${y,{z}}$, and the singleton set ${z}in^3A$, it is an issue for the elements $x$ and $y$. The statement $|x|neq1$, while true, cannot be evaluated if $x$ does not have cardinality. Since ${x}neq x$, the cardinality of $x$ cannot be 1.



One possibility is that the cardinality of $x$ is 0, but this runs the risk of equating the element $x$ with the empty set, which would violate the axioms of ZFC if $x$ is considered to be a set. On the other hand, you could say that the empty set is the only set whose cardinality is zero, and that $x$ is not a set: thus, $|x|=0notimplies x=emptyset$. But this runs into the problem of defining what a set is.



So, what is the best way to define the cardinality of an element? If there isn't one, then is there a good way to describe how a lone element is unlike a set, in set theoretic terms?







elementary-set-theory definition






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edited Nov 18 at 2:28









Andrés E. Caicedo

64.3k8157244




64.3k8157244










asked Nov 18 at 1:16









R. Burton

1367




1367












  • Every set except the empty set has elements. And if I understand your question correctly, you imply that $u,v,w,x,y,z$ have no elements. So they must all be the empty set, with cardinality zero. Or am I missing something?
    – TonyK
    Nov 18 at 2:35










  • $u$,$v$,$w$,$x$,$y$, and $z$ are not sets, they are constants, variables, vectors etc. For simplicity's sake, you can treat each of $u$,$v$,$w$,$x$,$y$, and $z$ as a separate real number.
    – R. Burton
    Nov 18 at 3:09


















  • Every set except the empty set has elements. And if I understand your question correctly, you imply that $u,v,w,x,y,z$ have no elements. So they must all be the empty set, with cardinality zero. Or am I missing something?
    – TonyK
    Nov 18 at 2:35










  • $u$,$v$,$w$,$x$,$y$, and $z$ are not sets, they are constants, variables, vectors etc. For simplicity's sake, you can treat each of $u$,$v$,$w$,$x$,$y$, and $z$ as a separate real number.
    – R. Burton
    Nov 18 at 3:09
















Every set except the empty set has elements. And if I understand your question correctly, you imply that $u,v,w,x,y,z$ have no elements. So they must all be the empty set, with cardinality zero. Or am I missing something?
– TonyK
Nov 18 at 2:35




Every set except the empty set has elements. And if I understand your question correctly, you imply that $u,v,w,x,y,z$ have no elements. So they must all be the empty set, with cardinality zero. Or am I missing something?
– TonyK
Nov 18 at 2:35












$u$,$v$,$w$,$x$,$y$, and $z$ are not sets, they are constants, variables, vectors etc. For simplicity's sake, you can treat each of $u$,$v$,$w$,$x$,$y$, and $z$ as a separate real number.
– R. Burton
Nov 18 at 3:09




$u$,$v$,$w$,$x$,$y$, and $z$ are not sets, they are constants, variables, vectors etc. For simplicity's sake, you can treat each of $u$,$v$,$w$,$x$,$y$, and $z$ as a separate real number.
– R. Burton
Nov 18 at 3:09










1 Answer
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In ZF(C) and most other set theories, every object is a set, so in particular each of the elements $u,v,w,x,y,z$ in your question are sets, which have cardinalities by virtue of the fact that they are sets.



There are some set theories such as ZFA and NFU that admit atoms (a.k.a. urelements), which are objects that are not sets. In these settings, you would typically not define $|a|$ when $a$ is an atom.



You could define $|a|=0$ if $a$ is an atom, since atoms don't have any elements. In this case, you're right that the sentence $forall x,, |x|=0 Rightarrow x = varnothing$ is no longer true. When dealing with atoms, though, it is common to have some way of discerning between sets and atoms, e.g. via a unary predicate $S$ or a definition of a class $V$ of (non-atomic) sets, in which case the sentence $forall x in V,, |x| = 0 Rightarrow x=varnothing$ is true. Thus anything you'd want to prove about sets in a regular set theory without atoms, you just relativise to the class $V$ of all sets.






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    In ZF(C) and most other set theories, every object is a set, so in particular each of the elements $u,v,w,x,y,z$ in your question are sets, which have cardinalities by virtue of the fact that they are sets.



    There are some set theories such as ZFA and NFU that admit atoms (a.k.a. urelements), which are objects that are not sets. In these settings, you would typically not define $|a|$ when $a$ is an atom.



    You could define $|a|=0$ if $a$ is an atom, since atoms don't have any elements. In this case, you're right that the sentence $forall x,, |x|=0 Rightarrow x = varnothing$ is no longer true. When dealing with atoms, though, it is common to have some way of discerning between sets and atoms, e.g. via a unary predicate $S$ or a definition of a class $V$ of (non-atomic) sets, in which case the sentence $forall x in V,, |x| = 0 Rightarrow x=varnothing$ is true. Thus anything you'd want to prove about sets in a regular set theory without atoms, you just relativise to the class $V$ of all sets.






    share|cite|improve this answer

























      up vote
      3
      down vote



      accepted










      In ZF(C) and most other set theories, every object is a set, so in particular each of the elements $u,v,w,x,y,z$ in your question are sets, which have cardinalities by virtue of the fact that they are sets.



      There are some set theories such as ZFA and NFU that admit atoms (a.k.a. urelements), which are objects that are not sets. In these settings, you would typically not define $|a|$ when $a$ is an atom.



      You could define $|a|=0$ if $a$ is an atom, since atoms don't have any elements. In this case, you're right that the sentence $forall x,, |x|=0 Rightarrow x = varnothing$ is no longer true. When dealing with atoms, though, it is common to have some way of discerning between sets and atoms, e.g. via a unary predicate $S$ or a definition of a class $V$ of (non-atomic) sets, in which case the sentence $forall x in V,, |x| = 0 Rightarrow x=varnothing$ is true. Thus anything you'd want to prove about sets in a regular set theory without atoms, you just relativise to the class $V$ of all sets.






      share|cite|improve this answer























        up vote
        3
        down vote



        accepted







        up vote
        3
        down vote



        accepted






        In ZF(C) and most other set theories, every object is a set, so in particular each of the elements $u,v,w,x,y,z$ in your question are sets, which have cardinalities by virtue of the fact that they are sets.



        There are some set theories such as ZFA and NFU that admit atoms (a.k.a. urelements), which are objects that are not sets. In these settings, you would typically not define $|a|$ when $a$ is an atom.



        You could define $|a|=0$ if $a$ is an atom, since atoms don't have any elements. In this case, you're right that the sentence $forall x,, |x|=0 Rightarrow x = varnothing$ is no longer true. When dealing with atoms, though, it is common to have some way of discerning between sets and atoms, e.g. via a unary predicate $S$ or a definition of a class $V$ of (non-atomic) sets, in which case the sentence $forall x in V,, |x| = 0 Rightarrow x=varnothing$ is true. Thus anything you'd want to prove about sets in a regular set theory without atoms, you just relativise to the class $V$ of all sets.






        share|cite|improve this answer












        In ZF(C) and most other set theories, every object is a set, so in particular each of the elements $u,v,w,x,y,z$ in your question are sets, which have cardinalities by virtue of the fact that they are sets.



        There are some set theories such as ZFA and NFU that admit atoms (a.k.a. urelements), which are objects that are not sets. In these settings, you would typically not define $|a|$ when $a$ is an atom.



        You could define $|a|=0$ if $a$ is an atom, since atoms don't have any elements. In this case, you're right that the sentence $forall x,, |x|=0 Rightarrow x = varnothing$ is no longer true. When dealing with atoms, though, it is common to have some way of discerning between sets and atoms, e.g. via a unary predicate $S$ or a definition of a class $V$ of (non-atomic) sets, in which case the sentence $forall x in V,, |x| = 0 Rightarrow x=varnothing$ is true. Thus anything you'd want to prove about sets in a regular set theory without atoms, you just relativise to the class $V$ of all sets.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 18 at 1:27









        Clive Newstead

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        49.5k472132






























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