Uncountable sets unrelated to real numbers












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The question may be a little general, but are there any other examples of uncountable sets except those related to real numbers?










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    The set of subsets (i.e. power-set) of every countable infinite set.
    $endgroup$
    – Mauro ALLEGRANZA
    Jan 14 at 12:47








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    The set of all functions from $mathbb{N}tomathbb{N}$ is always an interesting example.
    $endgroup$
    – user3482749
    Jan 14 at 12:48






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    I like the first uncountable ordinal
    $endgroup$
    – Questioner
    Jan 14 at 13:52
















0












$begingroup$


The question may be a little general, but are there any other examples of uncountable sets except those related to real numbers?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    The set of subsets (i.e. power-set) of every countable infinite set.
    $endgroup$
    – Mauro ALLEGRANZA
    Jan 14 at 12:47








  • 1




    $begingroup$
    The set of all functions from $mathbb{N}tomathbb{N}$ is always an interesting example.
    $endgroup$
    – user3482749
    Jan 14 at 12:48






  • 1




    $begingroup$
    I like the first uncountable ordinal
    $endgroup$
    – Questioner
    Jan 14 at 13:52














0












0








0





$begingroup$


The question may be a little general, but are there any other examples of uncountable sets except those related to real numbers?










share|cite|improve this question









$endgroup$




The question may be a little general, but are there any other examples of uncountable sets except those related to real numbers?







elementary-set-theory






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asked Jan 14 at 12:44









Cebiş MellimCebiş Mellim

18612




18612








  • 1




    $begingroup$
    The set of subsets (i.e. power-set) of every countable infinite set.
    $endgroup$
    – Mauro ALLEGRANZA
    Jan 14 at 12:47








  • 1




    $begingroup$
    The set of all functions from $mathbb{N}tomathbb{N}$ is always an interesting example.
    $endgroup$
    – user3482749
    Jan 14 at 12:48






  • 1




    $begingroup$
    I like the first uncountable ordinal
    $endgroup$
    – Questioner
    Jan 14 at 13:52














  • 1




    $begingroup$
    The set of subsets (i.e. power-set) of every countable infinite set.
    $endgroup$
    – Mauro ALLEGRANZA
    Jan 14 at 12:47








  • 1




    $begingroup$
    The set of all functions from $mathbb{N}tomathbb{N}$ is always an interesting example.
    $endgroup$
    – user3482749
    Jan 14 at 12:48






  • 1




    $begingroup$
    I like the first uncountable ordinal
    $endgroup$
    – Questioner
    Jan 14 at 13:52








1




1




$begingroup$
The set of subsets (i.e. power-set) of every countable infinite set.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 12:47






$begingroup$
The set of subsets (i.e. power-set) of every countable infinite set.
$endgroup$
– Mauro ALLEGRANZA
Jan 14 at 12:47






1




1




$begingroup$
The set of all functions from $mathbb{N}tomathbb{N}$ is always an interesting example.
$endgroup$
– user3482749
Jan 14 at 12:48




$begingroup$
The set of all functions from $mathbb{N}tomathbb{N}$ is always an interesting example.
$endgroup$
– user3482749
Jan 14 at 12:48




1




1




$begingroup$
I like the first uncountable ordinal
$endgroup$
– Questioner
Jan 14 at 13:52




$begingroup$
I like the first uncountable ordinal
$endgroup$
– Questioner
Jan 14 at 13:52










2 Answers
2






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1












$begingroup$

The set



$$ {(x_n): x_n in {0,1} quad forall n in mathbb N}$$



is uncountable.






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$endgroup$





















    1












    $begingroup$

    There are many.



    The fist one to come to mind is the first uncountable ordinal. It is so unrelated to the real numbers that trying to come up with a useful relationship at all between the two was allegedly what drove Georg Cantor mad. There are, of course, any ordinal above that one as well.



    As mentioned in the comments, the power-set of any infinite set is uncountably infinite.






    share|cite|improve this answer









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      2 Answers
      2






      active

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      2 Answers
      2






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      1












      $begingroup$

      The set



      $$ {(x_n): x_n in {0,1} quad forall n in mathbb N}$$



      is uncountable.






      share|cite|improve this answer









      $endgroup$


















        1












        $begingroup$

        The set



        $$ {(x_n): x_n in {0,1} quad forall n in mathbb N}$$



        is uncountable.






        share|cite|improve this answer









        $endgroup$
















          1












          1








          1





          $begingroup$

          The set



          $$ {(x_n): x_n in {0,1} quad forall n in mathbb N}$$



          is uncountable.






          share|cite|improve this answer









          $endgroup$



          The set



          $$ {(x_n): x_n in {0,1} quad forall n in mathbb N}$$



          is uncountable.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 14 at 12:55









          FredFred

          48.7k11849




          48.7k11849























              1












              $begingroup$

              There are many.



              The fist one to come to mind is the first uncountable ordinal. It is so unrelated to the real numbers that trying to come up with a useful relationship at all between the two was allegedly what drove Georg Cantor mad. There are, of course, any ordinal above that one as well.



              As mentioned in the comments, the power-set of any infinite set is uncountably infinite.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                There are many.



                The fist one to come to mind is the first uncountable ordinal. It is so unrelated to the real numbers that trying to come up with a useful relationship at all between the two was allegedly what drove Georg Cantor mad. There are, of course, any ordinal above that one as well.



                As mentioned in the comments, the power-set of any infinite set is uncountably infinite.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  There are many.



                  The fist one to come to mind is the first uncountable ordinal. It is so unrelated to the real numbers that trying to come up with a useful relationship at all between the two was allegedly what drove Georg Cantor mad. There are, of course, any ordinal above that one as well.



                  As mentioned in the comments, the power-set of any infinite set is uncountably infinite.






                  share|cite|improve this answer









                  $endgroup$



                  There are many.



                  The fist one to come to mind is the first uncountable ordinal. It is so unrelated to the real numbers that trying to come up with a useful relationship at all between the two was allegedly what drove Georg Cantor mad. There are, of course, any ordinal above that one as well.



                  As mentioned in the comments, the power-set of any infinite set is uncountably infinite.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 14 at 12:50









                  ArthurArthur

                  119k7119202




                  119k7119202






























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