Is the least inaccessible cardinal equivalent to the first aleph fixed point? [duplicate]












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This question already has an answer here:




  • Fixed point, union of $aleph_0, aleph_{aleph_0},aleph_{aleph_{aleph_0}}dots$ [see Noah's comment on notation]

    1 answer



  • Ordinals that satisfy $alpha = aleph_alpha$ with cofinality $kappa$

    2 answers



  • $aleph$ function fixed points below a weakly inaccessible cardinal are a club set

    1 answer




Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous one. My question is:




Is $I$ the first fixed point of the function $alpha mapsto aleph_alpha$?




My background for this question is that while I was reading about ordinal collapsing functions (specifically about collapsing large cardinals) I found that by collapsing the least inaccessible cardinal ($I$) you got the first omega fixed point, which gave me the question, as if $I$ is the first fixed point then it is also a fixed point of the collapsing function, which wouldn’t be as useful.






functions set-theory cardinals large-cardinals fixedpoints







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 Jan 27 at 9:16


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 2




    $begingroup$
    No. The first fixed point is the limit of $aleph_0,aleph_{aleph_0},dots$, which means it has cofinality $aleph_0$ and is not regular.
    $endgroup$
    – Wojowu
    Jan 27 at 9:15










  • $begingroup$
    There are probably quite a few more of these around.
    $endgroup$
    – Asaf Karagila♦
    Jan 27 at 9:17










  • $begingroup$
    I tried looking for some, as I knew there would be them, but I couldn’t find any...
    $endgroup$
    – L. McDonald
    Jan 27 at 9:18
















1












$begingroup$



This question already has an answer here:




  • Fixed point, union of $aleph_0, aleph_{aleph_0},aleph_{aleph_{aleph_0}}dots$ [see Noah's comment on notation]

    1 answer



  • Ordinals that satisfy $alpha = aleph_alpha$ with cofinality $kappa$

    2 answers



  • $aleph$ function fixed points below a weakly inaccessible cardinal are a club set

    1 answer




Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous one. My question is:




Is $I$ the first fixed point of the function $alpha mapsto aleph_alpha$?




My background for this question is that while I was reading about ordinal collapsing functions (specifically about collapsing large cardinals) I found that by collapsing the least inaccessible cardinal ($I$) you got the first omega fixed point, which gave me the question, as if $I$ is the first fixed point then it is also a fixed point of the collapsing function, which wouldn’t be as useful.






functions set-theory cardinals large-cardinals fixedpoints







share|cite|improve this question









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marked as duplicate by Asaf Karagila♦ cardinals
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 Jan 27 at 9:16


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 2




    $begingroup$
    No. The first fixed point is the limit of $aleph_0,aleph_{aleph_0},dots$, which means it has cofinality $aleph_0$ and is not regular.
    $endgroup$
    – Wojowu
    Jan 27 at 9:15










  • $begingroup$
    There are probably quite a few more of these around.
    $endgroup$
    – Asaf Karagila♦
    Jan 27 at 9:17










  • $begingroup$
    I tried looking for some, as I knew there would be them, but I couldn’t find any...
    $endgroup$
    – L. McDonald
    Jan 27 at 9:18














1












1








1





$begingroup$



This question already has an answer here:




  • Fixed point, union of $aleph_0, aleph_{aleph_0},aleph_{aleph_{aleph_0}}dots$ [see Noah's comment on notation]

    1 answer



  • Ordinals that satisfy $alpha = aleph_alpha$ with cofinality $kappa$

    2 answers



  • $aleph$ function fixed points below a weakly inaccessible cardinal are a club set

    1 answer




Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous one. My question is:




Is $I$ the first fixed point of the function $alpha mapsto aleph_alpha$?




My background for this question is that while I was reading about ordinal collapsing functions (specifically about collapsing large cardinals) I found that by collapsing the least inaccessible cardinal ($I$) you got the first omega fixed point, which gave me the question, as if $I$ is the first fixed point then it is also a fixed point of the collapsing function, which wouldn’t be as useful.






functions set-theory cardinals large-cardinals fixedpoints







share|cite|improve this question









$endgroup$





This question already has an answer here:




  • Fixed point, union of $aleph_0, aleph_{aleph_0},aleph_{aleph_{aleph_0}}dots$ [see Noah's comment on notation]

    1 answer



  • Ordinals that satisfy $alpha = aleph_alpha$ with cofinality $kappa$

    2 answers



  • $aleph$ function fixed points below a weakly inaccessible cardinal are a club set

    1 answer




Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous one. My question is:




Is $I$ the first fixed point of the function $alpha mapsto aleph_alpha$?




My background for this question is that while I was reading about ordinal collapsing functions (specifically about collapsing large cardinals) I found that by collapsing the least inaccessible cardinal ($I$) you got the first omega fixed point, which gave me the question, as if $I$ is the first fixed point then it is also a fixed point of the collapsing function, which wouldn’t be as useful.





This question already has an answer here:




  • Fixed point, union of $aleph_0, aleph_{aleph_0},aleph_{aleph_{aleph_0}}dots$ [see Noah's comment on notation]

    1 answer



  • Ordinals that satisfy $alpha = aleph_alpha$ with cofinality $kappa$

    2 answers



  • $aleph$ function fixed points below a weakly inaccessible cardinal are a club set

    1 answer







functions set-theory cardinals large-cardinals fixedpoints




functions set-theory cardinals large-cardinals fixedpoints






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 27 at 9:04









L. McDonaldL. McDonald

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marked as duplicate by Asaf Karagila♦ cardinals
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 Jan 27 at 9:16


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Asaf Karagila♦ cardinals
Users with the  cardinals badge can single-handedly close cardinals questions as duplicates and reopen them as needed.

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 Jan 27 at 9:16


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 2




    $begingroup$
    No. The first fixed point is the limit of $aleph_0,aleph_{aleph_0},dots$, which means it has cofinality $aleph_0$ and is not regular.
    $endgroup$
    – Wojowu
    Jan 27 at 9:15










  • $begingroup$
    There are probably quite a few more of these around.
    $endgroup$
    – Asaf Karagila♦
    Jan 27 at 9:17










  • $begingroup$
    I tried looking for some, as I knew there would be them, but I couldn’t find any...
    $endgroup$
    – L. McDonald
    Jan 27 at 9:18














  • 2




    $begingroup$
    No. The first fixed point is the limit of $aleph_0,aleph_{aleph_0},dots$, which means it has cofinality $aleph_0$ and is not regular.
    $endgroup$
    – Wojowu
    Jan 27 at 9:15










  • $begingroup$
    There are probably quite a few more of these around.
    $endgroup$
    – Asaf Karagila♦
    Jan 27 at 9:17










  • $begingroup$
    I tried looking for some, as I knew there would be them, but I couldn’t find any...
    $endgroup$
    – L. McDonald
    Jan 27 at 9:18








2




2




$begingroup$
No. The first fixed point is the limit of $aleph_0,aleph_{aleph_0},dots$, which means it has cofinality $aleph_0$ and is not regular.
$endgroup$
– Wojowu
Jan 27 at 9:15




$begingroup$
No. The first fixed point is the limit of $aleph_0,aleph_{aleph_0},dots$, which means it has cofinality $aleph_0$ and is not regular.
$endgroup$
– Wojowu
Jan 27 at 9:15












$begingroup$
There are probably quite a few more of these around.
$endgroup$
– Asaf Karagila♦
Jan 27 at 9:17




$begingroup$
There are probably quite a few more of these around.
$endgroup$
– Asaf Karagila♦
Jan 27 at 9:17












$begingroup$
I tried looking for some, as I knew there would be them, but I couldn’t find any...
$endgroup$
– L. McDonald
Jan 27 at 9:18




$begingroup$
I tried looking for some, as I knew there would be them, but I couldn’t find any...
$endgroup$
– L. McDonald
Jan 27 at 9:18










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