Elementary embeddings, elementary substructures,category of sets












2












$begingroup$


I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.






functions category-theory







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$endgroup$








  • 2




    $begingroup$
    The elementary embeddings are all bijections between finite sets and all injections between infinite sets.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:06










  • $begingroup$
    What about elementary substructures in Set?
    $endgroup$
    – user122424
    Apr 4 '14 at 15:08






  • 2




    $begingroup$
    You can easily work that out yourself once you know what the elementary embeddings are.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:19










  • $begingroup$
    Are you sure you are interested in $bf{Set}$? Why not in $bf{Mod}_tau$ for some first order signature $tau$?
    $endgroup$
    – Berci
    Apr 6 '14 at 15:53


















2












$begingroup$


I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.






functions category-theory







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    The elementary embeddings are all bijections between finite sets and all injections between infinite sets.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:06










  • $begingroup$
    What about elementary substructures in Set?
    $endgroup$
    – user122424
    Apr 4 '14 at 15:08






  • 2




    $begingroup$
    You can easily work that out yourself once you know what the elementary embeddings are.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:19










  • $begingroup$
    Are you sure you are interested in $bf{Set}$? Why not in $bf{Mod}_tau$ for some first order signature $tau$?
    $endgroup$
    – Berci
    Apr 6 '14 at 15:53
















2












2








2





$begingroup$


I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.






functions category-theory







share|cite|improve this question











$endgroup$




I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.






functions category-theory




functions category-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited May 31 '18 at 1:34









Andrés E. Caicedo

65.7k8160250




65.7k8160250










asked Apr 4 '14 at 15:03









user122424user122424

1,1582717




1,1582717








  • 2




    $begingroup$
    The elementary embeddings are all bijections between finite sets and all injections between infinite sets.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:06










  • $begingroup$
    What about elementary substructures in Set?
    $endgroup$
    – user122424
    Apr 4 '14 at 15:08






  • 2




    $begingroup$
    You can easily work that out yourself once you know what the elementary embeddings are.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:19










  • $begingroup$
    Are you sure you are interested in $bf{Set}$? Why not in $bf{Mod}_tau$ for some first order signature $tau$?
    $endgroup$
    – Berci
    Apr 6 '14 at 15:53
















  • 2




    $begingroup$
    The elementary embeddings are all bijections between finite sets and all injections between infinite sets.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:06










  • $begingroup$
    What about elementary substructures in Set?
    $endgroup$
    – user122424
    Apr 4 '14 at 15:08






  • 2




    $begingroup$
    You can easily work that out yourself once you know what the elementary embeddings are.
    $endgroup$
    – Zhen Lin
    Apr 4 '14 at 15:19










  • $begingroup$
    Are you sure you are interested in $bf{Set}$? Why not in $bf{Mod}_tau$ for some first order signature $tau$?
    $endgroup$
    – Berci
    Apr 6 '14 at 15:53










2




2




$begingroup$
The elementary embeddings are all bijections between finite sets and all injections between infinite sets.
$endgroup$
– Zhen Lin
Apr 4 '14 at 15:06




$begingroup$
The elementary embeddings are all bijections between finite sets and all injections between infinite sets.
$endgroup$
– Zhen Lin
Apr 4 '14 at 15:06












$begingroup$
What about elementary substructures in Set?
$endgroup$
– user122424
Apr 4 '14 at 15:08




$begingroup$
What about elementary substructures in Set?
$endgroup$
– user122424
Apr 4 '14 at 15:08




2




2




$begingroup$
You can easily work that out yourself once you know what the elementary embeddings are.
$endgroup$
– Zhen Lin
Apr 4 '14 at 15:19




$begingroup$
You can easily work that out yourself once you know what the elementary embeddings are.
$endgroup$
– Zhen Lin
Apr 4 '14 at 15:19












$begingroup$
Are you sure you are interested in $bf{Set}$? Why not in $bf{Mod}_tau$ for some first order signature $tau$?
$endgroup$
– Berci
Apr 6 '14 at 15:53






$begingroup$
Are you sure you are interested in $bf{Set}$? Why not in $bf{Mod}_tau$ for some first order signature $tau$?
$endgroup$
– Berci
Apr 6 '14 at 15:53












1 Answer
1






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oldest

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$begingroup$

Since a set is a structure in the empty language - there's no additional structure besides the underlying carrier set itself - things are quite simple to describe.



Specifically, we have full quantifier elimination, and the quantifier-free formulas can basically only say finitely many facts about size. This analysis - which I've left a bit vague, since it's a good exercise - shows the following:




  • If $A,B$ are infinite sets, then the elementary embeddings from $A$ to $B$ are precisely the injections.


  • If $A,B$ are finite sets, then the elementary embeddings from $A$ to $B$ are precisely the bijections.



Since an elementary substructure is just a substructure for which the inclusion map is an elementary embedding, this also characterizes the elementary substructures: $A$ is an elementary substructure of $B$ iff $Asubseteq B$ and either $A=B$ or $A$ (and hence $B$ as well) is infinite.



Finally, note that we also get a characterization of elementary equivalence: $Aequiv B$ iff $A,B$ either have the same cardinality or are each infinite. In particular, two sets are elementarily equivalent iff there is an elementary embedding of one of them into the other (since cardinalities are comparable - note that this uses a weak form of the axiom of choice!).



So what we see in the "pure set situation" is a near-complete collapse of logical notions: elementary equivalence/embedding/substructure mean almost the bare minimum of what they have to mean. The situation is far more interesting with even a little bit of structure involved, and this situation should really be thought of as pathological.






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    $begingroup$

    Since a set is a structure in the empty language - there's no additional structure besides the underlying carrier set itself - things are quite simple to describe.



    Specifically, we have full quantifier elimination, and the quantifier-free formulas can basically only say finitely many facts about size. This analysis - which I've left a bit vague, since it's a good exercise - shows the following:




    • If $A,B$ are infinite sets, then the elementary embeddings from $A$ to $B$ are precisely the injections.


    • If $A,B$ are finite sets, then the elementary embeddings from $A$ to $B$ are precisely the bijections.



    Since an elementary substructure is just a substructure for which the inclusion map is an elementary embedding, this also characterizes the elementary substructures: $A$ is an elementary substructure of $B$ iff $Asubseteq B$ and either $A=B$ or $A$ (and hence $B$ as well) is infinite.



    Finally, note that we also get a characterization of elementary equivalence: $Aequiv B$ iff $A,B$ either have the same cardinality or are each infinite. In particular, two sets are elementarily equivalent iff there is an elementary embedding of one of them into the other (since cardinalities are comparable - note that this uses a weak form of the axiom of choice!).



    So what we see in the "pure set situation" is a near-complete collapse of logical notions: elementary equivalence/embedding/substructure mean almost the bare minimum of what they have to mean. The situation is far more interesting with even a little bit of structure involved, and this situation should really be thought of as pathological.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Since a set is a structure in the empty language - there's no additional structure besides the underlying carrier set itself - things are quite simple to describe.



      Specifically, we have full quantifier elimination, and the quantifier-free formulas can basically only say finitely many facts about size. This analysis - which I've left a bit vague, since it's a good exercise - shows the following:




      • If $A,B$ are infinite sets, then the elementary embeddings from $A$ to $B$ are precisely the injections.


      • If $A,B$ are finite sets, then the elementary embeddings from $A$ to $B$ are precisely the bijections.



      Since an elementary substructure is just a substructure for which the inclusion map is an elementary embedding, this also characterizes the elementary substructures: $A$ is an elementary substructure of $B$ iff $Asubseteq B$ and either $A=B$ or $A$ (and hence $B$ as well) is infinite.



      Finally, note that we also get a characterization of elementary equivalence: $Aequiv B$ iff $A,B$ either have the same cardinality or are each infinite. In particular, two sets are elementarily equivalent iff there is an elementary embedding of one of them into the other (since cardinalities are comparable - note that this uses a weak form of the axiom of choice!).



      So what we see in the "pure set situation" is a near-complete collapse of logical notions: elementary equivalence/embedding/substructure mean almost the bare minimum of what they have to mean. The situation is far more interesting with even a little bit of structure involved, and this situation should really be thought of as pathological.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Since a set is a structure in the empty language - there's no additional structure besides the underlying carrier set itself - things are quite simple to describe.



        Specifically, we have full quantifier elimination, and the quantifier-free formulas can basically only say finitely many facts about size. This analysis - which I've left a bit vague, since it's a good exercise - shows the following:




        • If $A,B$ are infinite sets, then the elementary embeddings from $A$ to $B$ are precisely the injections.


        • If $A,B$ are finite sets, then the elementary embeddings from $A$ to $B$ are precisely the bijections.



        Since an elementary substructure is just a substructure for which the inclusion map is an elementary embedding, this also characterizes the elementary substructures: $A$ is an elementary substructure of $B$ iff $Asubseteq B$ and either $A=B$ or $A$ (and hence $B$ as well) is infinite.



        Finally, note that we also get a characterization of elementary equivalence: $Aequiv B$ iff $A,B$ either have the same cardinality or are each infinite. In particular, two sets are elementarily equivalent iff there is an elementary embedding of one of them into the other (since cardinalities are comparable - note that this uses a weak form of the axiom of choice!).



        So what we see in the "pure set situation" is a near-complete collapse of logical notions: elementary equivalence/embedding/substructure mean almost the bare minimum of what they have to mean. The situation is far more interesting with even a little bit of structure involved, and this situation should really be thought of as pathological.






        share|cite|improve this answer









        $endgroup$



        Since a set is a structure in the empty language - there's no additional structure besides the underlying carrier set itself - things are quite simple to describe.



        Specifically, we have full quantifier elimination, and the quantifier-free formulas can basically only say finitely many facts about size. This analysis - which I've left a bit vague, since it's a good exercise - shows the following:




        • If $A,B$ are infinite sets, then the elementary embeddings from $A$ to $B$ are precisely the injections.


        • If $A,B$ are finite sets, then the elementary embeddings from $A$ to $B$ are precisely the bijections.



        Since an elementary substructure is just a substructure for which the inclusion map is an elementary embedding, this also characterizes the elementary substructures: $A$ is an elementary substructure of $B$ iff $Asubseteq B$ and either $A=B$ or $A$ (and hence $B$ as well) is infinite.



        Finally, note that we also get a characterization of elementary equivalence: $Aequiv B$ iff $A,B$ either have the same cardinality or are each infinite. In particular, two sets are elementarily equivalent iff there is an elementary embedding of one of them into the other (since cardinalities are comparable - note that this uses a weak form of the axiom of choice!).



        So what we see in the "pure set situation" is a near-complete collapse of logical notions: elementary equivalence/embedding/substructure mean almost the bare minimum of what they have to mean. The situation is far more interesting with even a little bit of structure involved, and this situation should really be thought of as pathological.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 16 '18 at 19:34









        Noah SchweberNoah Schweber

        127k10151290




        127k10151290






























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