not well founded $omega$-models of ZFC

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I would like to know how to obtain not well-founded $omega$-models of ZFC.



Are there any books about it? Other references to the literature are also welcome.






reference-request set-theory







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  • I don't understand the votes to close - this is a reasonable thing to wonder about, and it is hard to find resources on it.
    – Noah Schweber
    Jan 31 '17 at 22:29






  • 1




    Additionally it's bizarre that this has been closed as "unclear" - it's perfectly clear. I wonder how many of those who voted to close are actually familiar with the topic . . .
    – Noah Schweber
    Mar 7 '17 at 18:16















up vote
3
down vote

favorite












I would like to know how to obtain not well-founded $omega$-models of ZFC.



Are there any books about it? Other references to the literature are also welcome.






reference-request set-theory







share|cite|improve this question























  • I don't understand the votes to close - this is a reasonable thing to wonder about, and it is hard to find resources on it.
    – Noah Schweber
    Jan 31 '17 at 22:29






  • 1




    Additionally it's bizarre that this has been closed as "unclear" - it's perfectly clear. I wonder how many of those who voted to close are actually familiar with the topic . . .
    – Noah Schweber
    Mar 7 '17 at 18:16













up vote
3
down vote

favorite









up vote
3
down vote

favorite











I would like to know how to obtain not well-founded $omega$-models of ZFC.



Are there any books about it? Other references to the literature are also welcome.






reference-request set-theory







share|cite|improve this question















I would like to know how to obtain not well-founded $omega$-models of ZFC.



Are there any books about it? Other references to the literature are also welcome.






reference-request set-theory




reference-request set-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 30 at 0:47









hardmath

28.3k94693




28.3k94693










asked Jan 31 '17 at 8:07









luis

646




646











  • I don't understand the votes to close - this is a reasonable thing to wonder about, and it is hard to find resources on it.
    – Noah Schweber
    Jan 31 '17 at 22:29






  • 1




    Additionally it's bizarre that this has been closed as "unclear" - it's perfectly clear. I wonder how many of those who voted to close are actually familiar with the topic . . .
    – Noah Schweber
    Mar 7 '17 at 18:16

















  • I don't understand the votes to close - this is a reasonable thing to wonder about, and it is hard to find resources on it.
    – Noah Schweber
    Jan 31 '17 at 22:29






  • 1




    Additionally it's bizarre that this has been closed as "unclear" - it's perfectly clear. I wonder how many of those who voted to close are actually familiar with the topic . . .
    – Noah Schweber
    Mar 7 '17 at 18:16
















I don't understand the votes to close - this is a reasonable thing to wonder about, and it is hard to find resources on it.
– Noah Schweber
Jan 31 '17 at 22:29




I don't understand the votes to close - this is a reasonable thing to wonder about, and it is hard to find resources on it.
– Noah Schweber
Jan 31 '17 at 22:29




1




1




Additionally it's bizarre that this has been closed as "unclear" - it's perfectly clear. I wonder how many of those who voted to close are actually familiar with the topic . . .
– Noah Schweber
Mar 7 '17 at 18:16





Additionally it's bizarre that this has been closed as "unclear" - it's perfectly clear. I wonder how many of those who voted to close are actually familiar with the topic . . .
– Noah Schweber
Mar 7 '17 at 18:16











1 Answer
1






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5
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Once you know that the consistency of "There is a transitive model of ZFC" implies the consistency of "There is an $omega$-model of ZFC", assuming the existence of the former will provide you with the existence of the latter.



Of course, it is consistent that there are no $omega$-models, even if ZFC is consistent. So just obtaining them out of a model of ZFC is impossible.



But here is a nice way to get what you want. Start with a transitive model of ZFC, $M$. Fix some regular $kappa>omega$, and now add a generic ultrafilter and consider the generic ultrapower of $M$. It will have critical point $kappa$, so it remains an $omega$-model, but it will be ill-founded, as long as you didn't use a precipitous ideal for your forcing.






share|cite|improve this answer
















  • 1




    +1. Another (unrelated of course) way to get an illfounded $omega$-model is via a Barwise Compactness argument: this lets us get an $omega$-model with wellfounded part of height $omega_1^CK$, or indeed of height $alpha$ for any admissible ordinal $alpha$.
    – Noah Schweber
    Jan 31 '17 at 22:28











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
5
down vote



accepted










Once you know that the consistency of "There is a transitive model of ZFC" implies the consistency of "There is an $omega$-model of ZFC", assuming the existence of the former will provide you with the existence of the latter.



Of course, it is consistent that there are no $omega$-models, even if ZFC is consistent. So just obtaining them out of a model of ZFC is impossible.



But here is a nice way to get what you want. Start with a transitive model of ZFC, $M$. Fix some regular $kappa>omega$, and now add a generic ultrafilter and consider the generic ultrapower of $M$. It will have critical point $kappa$, so it remains an $omega$-model, but it will be ill-founded, as long as you didn't use a precipitous ideal for your forcing.






share|cite|improve this answer
















  • 1




    +1. Another (unrelated of course) way to get an illfounded $omega$-model is via a Barwise Compactness argument: this lets us get an $omega$-model with wellfounded part of height $omega_1^CK$, or indeed of height $alpha$ for any admissible ordinal $alpha$.
    – Noah Schweber
    Jan 31 '17 at 22:28















up vote
5
down vote



accepted










Once you know that the consistency of "There is a transitive model of ZFC" implies the consistency of "There is an $omega$-model of ZFC", assuming the existence of the former will provide you with the existence of the latter.



Of course, it is consistent that there are no $omega$-models, even if ZFC is consistent. So just obtaining them out of a model of ZFC is impossible.



But here is a nice way to get what you want. Start with a transitive model of ZFC, $M$. Fix some regular $kappa>omega$, and now add a generic ultrafilter and consider the generic ultrapower of $M$. It will have critical point $kappa$, so it remains an $omega$-model, but it will be ill-founded, as long as you didn't use a precipitous ideal for your forcing.






share|cite|improve this answer
















  • 1




    +1. Another (unrelated of course) way to get an illfounded $omega$-model is via a Barwise Compactness argument: this lets us get an $omega$-model with wellfounded part of height $omega_1^CK$, or indeed of height $alpha$ for any admissible ordinal $alpha$.
    – Noah Schweber
    Jan 31 '17 at 22:28













up vote
5
down vote



accepted







up vote
5
down vote



accepted






Once you know that the consistency of "There is a transitive model of ZFC" implies the consistency of "There is an $omega$-model of ZFC", assuming the existence of the former will provide you with the existence of the latter.



Of course, it is consistent that there are no $omega$-models, even if ZFC is consistent. So just obtaining them out of a model of ZFC is impossible.



But here is a nice way to get what you want. Start with a transitive model of ZFC, $M$. Fix some regular $kappa>omega$, and now add a generic ultrafilter and consider the generic ultrapower of $M$. It will have critical point $kappa$, so it remains an $omega$-model, but it will be ill-founded, as long as you didn't use a precipitous ideal for your forcing.






share|cite|improve this answer












Once you know that the consistency of "There is a transitive model of ZFC" implies the consistency of "There is an $omega$-model of ZFC", assuming the existence of the former will provide you with the existence of the latter.



Of course, it is consistent that there are no $omega$-models, even if ZFC is consistent. So just obtaining them out of a model of ZFC is impossible.



But here is a nice way to get what you want. Start with a transitive model of ZFC, $M$. Fix some regular $kappa>omega$, and now add a generic ultrafilter and consider the generic ultrapower of $M$. It will have critical point $kappa$, so it remains an $omega$-model, but it will be ill-founded, as long as you didn't use a precipitous ideal for your forcing.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 31 '17 at 8:21









Asaf Karagila♦

294k31410737




294k31410737







  • 1




    +1. Another (unrelated of course) way to get an illfounded $omega$-model is via a Barwise Compactness argument: this lets us get an $omega$-model with wellfounded part of height $omega_1^CK$, or indeed of height $alpha$ for any admissible ordinal $alpha$.
    – Noah Schweber
    Jan 31 '17 at 22:28













  • 1




    +1. Another (unrelated of course) way to get an illfounded $omega$-model is via a Barwise Compactness argument: this lets us get an $omega$-model with wellfounded part of height $omega_1^CK$, or indeed of height $alpha$ for any admissible ordinal $alpha$.
    – Noah Schweber
    Jan 31 '17 at 22:28








1




1




+1. Another (unrelated of course) way to get an illfounded $omega$-model is via a Barwise Compactness argument: this lets us get an $omega$-model with wellfounded part of height $omega_1^CK$, or indeed of height $alpha$ for any admissible ordinal $alpha$.
– Noah Schweber
Jan 31 '17 at 22:28





+1. Another (unrelated of course) way to get an illfounded $omega$-model is via a Barwise Compactness argument: this lets us get an $omega$-model with wellfounded part of height $omega_1^CK$, or indeed of height $alpha$ for any admissible ordinal $alpha$.
– Noah Schweber
Jan 31 '17 at 22:28


















 

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