The relation between the limit cardinal $alpha$ and a sequence of cardinal numbers strictly less than $alpha$












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For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$






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    What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
    – Asaf Karagila♦
    2 hours ago


















0














For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$






set-theory







share|cite|improve this question


















  • 1




    What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
    – Asaf Karagila♦
    2 hours ago
















0












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0







For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$






set-theory







share|cite|improve this question













For a limit cardinal $alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$






set-theory




set-theory






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asked 3 hours ago









ali

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  • 1




    What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
    – Asaf Karagila♦
    2 hours ago
















  • 1




    What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
    – Asaf Karagila♦
    2 hours ago










1




1




What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
– Asaf Karagila♦
2 hours ago






What do the $dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?)
– Asaf Karagila♦
2 hours ago












1 Answer
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Your question suggests you are looking for a countable sequence; then the answer is: NO.
$aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
You may want to study the notion `cofinality of a cardinal number'.






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    Your question suggests you are looking for a countable sequence; then the answer is: NO.
    $aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
    You may want to study the notion `cofinality of a cardinal number'.






    share|cite|improve this answer


























      0














      Your question suggests you are looking for a countable sequence; then the answer is: NO.
      $aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
      You may want to study the notion `cofinality of a cardinal number'.






      share|cite|improve this answer
























        0












        0








        0






        Your question suggests you are looking for a countable sequence; then the answer is: NO.
        $aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
        You may want to study the notion `cofinality of a cardinal number'.






        share|cite|improve this answer












        Your question suggests you are looking for a countable sequence; then the answer is: NO.
        $aleph_{omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals.
        You may want to study the notion `cofinality of a cardinal number'.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 1 hour ago









        hartkp

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