Fixed points of injective self-maps

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Is it consistent in $mathsfZF$ that there is a set $X$ with more than $1$ point such that every injective map $f:Xto X$ has a fixed point?







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  • I think that strongly amorphous sets will have this property.
    – Yair Hayut
    20 hours ago










  • @Yair: Yes. That is correct.
    – Asaf Karagila
    19 hours ago










  • Thanks for your comments, Yair and Asaf. Can one of you elaborate a bit and post it as an answer?
    – Dominic van der Zypen
    19 hours ago














up vote
9
down vote

favorite
1












Is it consistent in $mathsfZF$ that there is a set $X$ with more than $1$ point such that every injective map $f:Xto X$ has a fixed point?







share|cite|improve this question



















  • I think that strongly amorphous sets will have this property.
    – Yair Hayut
    20 hours ago










  • @Yair: Yes. That is correct.
    – Asaf Karagila
    19 hours ago










  • Thanks for your comments, Yair and Asaf. Can one of you elaborate a bit and post it as an answer?
    – Dominic van der Zypen
    19 hours ago












up vote
9
down vote

favorite
1









up vote
9
down vote

favorite
1






1





Is it consistent in $mathsfZF$ that there is a set $X$ with more than $1$ point such that every injective map $f:Xto X$ has a fixed point?







share|cite|improve this question











Is it consistent in $mathsfZF$ that there is a set $X$ with more than $1$ point such that every injective map $f:Xto X$ has a fixed point?









share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked 21 hours ago









Dominic van der Zypen

12.4k43066




12.4k43066











  • I think that strongly amorphous sets will have this property.
    – Yair Hayut
    20 hours ago










  • @Yair: Yes. That is correct.
    – Asaf Karagila
    19 hours ago










  • Thanks for your comments, Yair and Asaf. Can one of you elaborate a bit and post it as an answer?
    – Dominic van der Zypen
    19 hours ago
















  • I think that strongly amorphous sets will have this property.
    – Yair Hayut
    20 hours ago










  • @Yair: Yes. That is correct.
    – Asaf Karagila
    19 hours ago










  • Thanks for your comments, Yair and Asaf. Can one of you elaborate a bit and post it as an answer?
    – Dominic van der Zypen
    19 hours ago















I think that strongly amorphous sets will have this property.
– Yair Hayut
20 hours ago




I think that strongly amorphous sets will have this property.
– Yair Hayut
20 hours ago












@Yair: Yes. That is correct.
– Asaf Karagila
19 hours ago




@Yair: Yes. That is correct.
– Asaf Karagila
19 hours ago












Thanks for your comments, Yair and Asaf. Can one of you elaborate a bit and post it as an answer?
– Dominic van der Zypen
19 hours ago




Thanks for your comments, Yair and Asaf. Can one of you elaborate a bit and post it as an answer?
– Dominic van der Zypen
19 hours ago










1 Answer
1






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oldest

votes

















up vote
10
down vote



accepted










As Yair suggests, a strongly amorphous set has this property.



Recall that an amorphous set is a set which cannot be split into two infinite sets. A strongly amorphous set is a set such that in addition to being amorphous, every partition has only finitely many non-singletons. While we didn't require that it is an infinite set, this is the common assumption, so the existence of an amorphous set contradicts the axiom of choice.



One can easily show that an amorphous set cannot be linearly ordered, and that a strongly amorphous set cannot carry a group structure.



Now. If $X$ is strongly amorphous and and $fcolon Xto X$ is a permutation (which it has to be if $f$ is injective, since amorphous sets are Dedekind-finite), then the orbits of $f$ are all finite and form a partition of $X$. Therefore all but finitely many points are moved at all.




One could argue that perhaps this is too extreme of an example. Well, in Cohen's first model with a Dedekind-finite set of reals, the canonical Dedekind-finite set (that of the generic Cohen reals) is linearly ordered, but has the property that any partition into finite parts is almost entirely singletons (i.e. all but finitely many parts are singletons).



The same argument now works on that set as well. If you have a permutation, its orbits must be finite, so almost all of them are singletons.




One can also have analogues of strongly amorphous sets for larger cardinals. For example an $aleph_1$-amorphous set is a set where every subset is countable or co-countable. Being strongly $aleph_1$-amorphous means that all but countably many parts of any partition are singletons.



And it is worth noting that $sf DC_kappa$ is consistent with the existence of a $kappa^+$-amorphous set. So you can't use $sf DC$ principles to avoid this issue.






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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    10
    down vote



    accepted










    As Yair suggests, a strongly amorphous set has this property.



    Recall that an amorphous set is a set which cannot be split into two infinite sets. A strongly amorphous set is a set such that in addition to being amorphous, every partition has only finitely many non-singletons. While we didn't require that it is an infinite set, this is the common assumption, so the existence of an amorphous set contradicts the axiom of choice.



    One can easily show that an amorphous set cannot be linearly ordered, and that a strongly amorphous set cannot carry a group structure.



    Now. If $X$ is strongly amorphous and and $fcolon Xto X$ is a permutation (which it has to be if $f$ is injective, since amorphous sets are Dedekind-finite), then the orbits of $f$ are all finite and form a partition of $X$. Therefore all but finitely many points are moved at all.




    One could argue that perhaps this is too extreme of an example. Well, in Cohen's first model with a Dedekind-finite set of reals, the canonical Dedekind-finite set (that of the generic Cohen reals) is linearly ordered, but has the property that any partition into finite parts is almost entirely singletons (i.e. all but finitely many parts are singletons).



    The same argument now works on that set as well. If you have a permutation, its orbits must be finite, so almost all of them are singletons.




    One can also have analogues of strongly amorphous sets for larger cardinals. For example an $aleph_1$-amorphous set is a set where every subset is countable or co-countable. Being strongly $aleph_1$-amorphous means that all but countably many parts of any partition are singletons.



    And it is worth noting that $sf DC_kappa$ is consistent with the existence of a $kappa^+$-amorphous set. So you can't use $sf DC$ principles to avoid this issue.






    share|cite|improve this answer

























      up vote
      10
      down vote



      accepted










      As Yair suggests, a strongly amorphous set has this property.



      Recall that an amorphous set is a set which cannot be split into two infinite sets. A strongly amorphous set is a set such that in addition to being amorphous, every partition has only finitely many non-singletons. While we didn't require that it is an infinite set, this is the common assumption, so the existence of an amorphous set contradicts the axiom of choice.



      One can easily show that an amorphous set cannot be linearly ordered, and that a strongly amorphous set cannot carry a group structure.



      Now. If $X$ is strongly amorphous and and $fcolon Xto X$ is a permutation (which it has to be if $f$ is injective, since amorphous sets are Dedekind-finite), then the orbits of $f$ are all finite and form a partition of $X$. Therefore all but finitely many points are moved at all.




      One could argue that perhaps this is too extreme of an example. Well, in Cohen's first model with a Dedekind-finite set of reals, the canonical Dedekind-finite set (that of the generic Cohen reals) is linearly ordered, but has the property that any partition into finite parts is almost entirely singletons (i.e. all but finitely many parts are singletons).



      The same argument now works on that set as well. If you have a permutation, its orbits must be finite, so almost all of them are singletons.




      One can also have analogues of strongly amorphous sets for larger cardinals. For example an $aleph_1$-amorphous set is a set where every subset is countable or co-countable. Being strongly $aleph_1$-amorphous means that all but countably many parts of any partition are singletons.



      And it is worth noting that $sf DC_kappa$ is consistent with the existence of a $kappa^+$-amorphous set. So you can't use $sf DC$ principles to avoid this issue.






      share|cite|improve this answer























        up vote
        10
        down vote



        accepted







        up vote
        10
        down vote



        accepted






        As Yair suggests, a strongly amorphous set has this property.



        Recall that an amorphous set is a set which cannot be split into two infinite sets. A strongly amorphous set is a set such that in addition to being amorphous, every partition has only finitely many non-singletons. While we didn't require that it is an infinite set, this is the common assumption, so the existence of an amorphous set contradicts the axiom of choice.



        One can easily show that an amorphous set cannot be linearly ordered, and that a strongly amorphous set cannot carry a group structure.



        Now. If $X$ is strongly amorphous and and $fcolon Xto X$ is a permutation (which it has to be if $f$ is injective, since amorphous sets are Dedekind-finite), then the orbits of $f$ are all finite and form a partition of $X$. Therefore all but finitely many points are moved at all.




        One could argue that perhaps this is too extreme of an example. Well, in Cohen's first model with a Dedekind-finite set of reals, the canonical Dedekind-finite set (that of the generic Cohen reals) is linearly ordered, but has the property that any partition into finite parts is almost entirely singletons (i.e. all but finitely many parts are singletons).



        The same argument now works on that set as well. If you have a permutation, its orbits must be finite, so almost all of them are singletons.




        One can also have analogues of strongly amorphous sets for larger cardinals. For example an $aleph_1$-amorphous set is a set where every subset is countable or co-countable. Being strongly $aleph_1$-amorphous means that all but countably many parts of any partition are singletons.



        And it is worth noting that $sf DC_kappa$ is consistent with the existence of a $kappa^+$-amorphous set. So you can't use $sf DC$ principles to avoid this issue.






        share|cite|improve this answer













        As Yair suggests, a strongly amorphous set has this property.



        Recall that an amorphous set is a set which cannot be split into two infinite sets. A strongly amorphous set is a set such that in addition to being amorphous, every partition has only finitely many non-singletons. While we didn't require that it is an infinite set, this is the common assumption, so the existence of an amorphous set contradicts the axiom of choice.



        One can easily show that an amorphous set cannot be linearly ordered, and that a strongly amorphous set cannot carry a group structure.



        Now. If $X$ is strongly amorphous and and $fcolon Xto X$ is a permutation (which it has to be if $f$ is injective, since amorphous sets are Dedekind-finite), then the orbits of $f$ are all finite and form a partition of $X$. Therefore all but finitely many points are moved at all.




        One could argue that perhaps this is too extreme of an example. Well, in Cohen's first model with a Dedekind-finite set of reals, the canonical Dedekind-finite set (that of the generic Cohen reals) is linearly ordered, but has the property that any partition into finite parts is almost entirely singletons (i.e. all but finitely many parts are singletons).



        The same argument now works on that set as well. If you have a permutation, its orbits must be finite, so almost all of them are singletons.




        One can also have analogues of strongly amorphous sets for larger cardinals. For example an $aleph_1$-amorphous set is a set where every subset is countable or co-countable. Being strongly $aleph_1$-amorphous means that all but countably many parts of any partition are singletons.



        And it is worth noting that $sf DC_kappa$ is consistent with the existence of a $kappa^+$-amorphous set. So you can't use $sf DC$ principles to avoid this issue.







        share|cite|improve this answer













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        share|cite|improve this answer











        answered 19 hours ago









        Asaf Karagila

        19.9k672168




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