Reining in the Axiom of Power Set in ZF












3












$begingroup$


Given the powerset operator $mathit P$, we have the following mapping



$tag 1 mathcal Phi: mathbb N to Phi(mathbb N) $
$quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad , n mapsto mathit P^n(mathbb N)$




What happens if we take away the Axiom of Power Set in $ZFpm C$ and replace
it with $text{(1)}$? Would this contradict the other axioms?











share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Incidentally, there are some very interesting results on the "amount of powerset" (or really, "powerset-along-replacement") we need in various situations. Most famously, Harvey Friedman showed that more and more iterations of powerset are needed to prove determinacy further and further up the Borel hierarchy.
    $endgroup$
    – Noah Schweber
    Jan 11 at 22:08






  • 1




    $begingroup$
    On formatting: You can use pairs of L& R dollar-signs to "display" a line instead of typing quad quad quad ..., and to put un-formatted text (like "because") in a displayed line, enclose it within text {...}, like text { because }.
    $endgroup$
    – DanielWainfleet
    Jan 14 at 0:47










  • $begingroup$
    @DanielWainfleet I've felt silly at times trying to (perfectly) align stuff since the final result might depend on the display device. But anyway, not exactly sure what you mean. Please share some links with useful mathstackexchange formatting technique so I can copy n paste it in the future.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 1:27










  • $begingroup$
    @NoahSchweber Another interesting paper, 'Large irredundant sets in operator algebras' arxiv.org/abs/1808.01511?context=math I don't know the formulation for the $text{♢-Axiom}$ (stronger than the continuum hypothesis), but my guess is all modern day physics can be expressed/modeled in $V_{omega+omega} + text{♢}$. I wonder how this holds up: "...every open set of Minkowski spacetime is associated with a C*-algebra" /wikipedia.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 14:55












  • $begingroup$
    Did something not work with the Ask Question page that you felt the need to ask a completely disjoint question on a page where you already accepted one answer?
    $endgroup$
    – Asaf Karagila
    Jan 14 at 15:44
















3












$begingroup$


Given the powerset operator $mathit P$, we have the following mapping



$tag 1 mathcal Phi: mathbb N to Phi(mathbb N) $
$quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad , n mapsto mathit P^n(mathbb N)$




What happens if we take away the Axiom of Power Set in $ZFpm C$ and replace
it with $text{(1)}$? Would this contradict the other axioms?











share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Incidentally, there are some very interesting results on the "amount of powerset" (or really, "powerset-along-replacement") we need in various situations. Most famously, Harvey Friedman showed that more and more iterations of powerset are needed to prove determinacy further and further up the Borel hierarchy.
    $endgroup$
    – Noah Schweber
    Jan 11 at 22:08






  • 1




    $begingroup$
    On formatting: You can use pairs of L& R dollar-signs to "display" a line instead of typing quad quad quad ..., and to put un-formatted text (like "because") in a displayed line, enclose it within text {...}, like text { because }.
    $endgroup$
    – DanielWainfleet
    Jan 14 at 0:47










  • $begingroup$
    @DanielWainfleet I've felt silly at times trying to (perfectly) align stuff since the final result might depend on the display device. But anyway, not exactly sure what you mean. Please share some links with useful mathstackexchange formatting technique so I can copy n paste it in the future.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 1:27










  • $begingroup$
    @NoahSchweber Another interesting paper, 'Large irredundant sets in operator algebras' arxiv.org/abs/1808.01511?context=math I don't know the formulation for the $text{♢-Axiom}$ (stronger than the continuum hypothesis), but my guess is all modern day physics can be expressed/modeled in $V_{omega+omega} + text{♢}$. I wonder how this holds up: "...every open set of Minkowski spacetime is associated with a C*-algebra" /wikipedia.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 14:55












  • $begingroup$
    Did something not work with the Ask Question page that you felt the need to ask a completely disjoint question on a page where you already accepted one answer?
    $endgroup$
    – Asaf Karagila
    Jan 14 at 15:44














3












3








3





$begingroup$


Given the powerset operator $mathit P$, we have the following mapping



$tag 1 mathcal Phi: mathbb N to Phi(mathbb N) $
$quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad , n mapsto mathit P^n(mathbb N)$




What happens if we take away the Axiom of Power Set in $ZFpm C$ and replace
it with $text{(1)}$? Would this contradict the other axioms?











share|cite|improve this question











$endgroup$




Given the powerset operator $mathit P$, we have the following mapping



$tag 1 mathcal Phi: mathbb N to Phi(mathbb N) $
$quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad quad , n mapsto mathit P^n(mathbb N)$




What happens if we take away the Axiom of Power Set in $ZFpm C$ and replace
it with $text{(1)}$? Would this contradict the other axioms?








set-theory axioms






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 14 at 17:15







CopyPasteIt

















asked Jan 9 at 11:50









CopyPasteItCopyPasteIt

4,3771828




4,3771828








  • 1




    $begingroup$
    Incidentally, there are some very interesting results on the "amount of powerset" (or really, "powerset-along-replacement") we need in various situations. Most famously, Harvey Friedman showed that more and more iterations of powerset are needed to prove determinacy further and further up the Borel hierarchy.
    $endgroup$
    – Noah Schweber
    Jan 11 at 22:08






  • 1




    $begingroup$
    On formatting: You can use pairs of L& R dollar-signs to "display" a line instead of typing quad quad quad ..., and to put un-formatted text (like "because") in a displayed line, enclose it within text {...}, like text { because }.
    $endgroup$
    – DanielWainfleet
    Jan 14 at 0:47










  • $begingroup$
    @DanielWainfleet I've felt silly at times trying to (perfectly) align stuff since the final result might depend on the display device. But anyway, not exactly sure what you mean. Please share some links with useful mathstackexchange formatting technique so I can copy n paste it in the future.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 1:27










  • $begingroup$
    @NoahSchweber Another interesting paper, 'Large irredundant sets in operator algebras' arxiv.org/abs/1808.01511?context=math I don't know the formulation for the $text{♢-Axiom}$ (stronger than the continuum hypothesis), but my guess is all modern day physics can be expressed/modeled in $V_{omega+omega} + text{♢}$. I wonder how this holds up: "...every open set of Minkowski spacetime is associated with a C*-algebra" /wikipedia.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 14:55












  • $begingroup$
    Did something not work with the Ask Question page that you felt the need to ask a completely disjoint question on a page where you already accepted one answer?
    $endgroup$
    – Asaf Karagila
    Jan 14 at 15:44














  • 1




    $begingroup$
    Incidentally, there are some very interesting results on the "amount of powerset" (or really, "powerset-along-replacement") we need in various situations. Most famously, Harvey Friedman showed that more and more iterations of powerset are needed to prove determinacy further and further up the Borel hierarchy.
    $endgroup$
    – Noah Schweber
    Jan 11 at 22:08






  • 1




    $begingroup$
    On formatting: You can use pairs of L& R dollar-signs to "display" a line instead of typing quad quad quad ..., and to put un-formatted text (like "because") in a displayed line, enclose it within text {...}, like text { because }.
    $endgroup$
    – DanielWainfleet
    Jan 14 at 0:47










  • $begingroup$
    @DanielWainfleet I've felt silly at times trying to (perfectly) align stuff since the final result might depend on the display device. But anyway, not exactly sure what you mean. Please share some links with useful mathstackexchange formatting technique so I can copy n paste it in the future.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 1:27










  • $begingroup$
    @NoahSchweber Another interesting paper, 'Large irredundant sets in operator algebras' arxiv.org/abs/1808.01511?context=math I don't know the formulation for the $text{♢-Axiom}$ (stronger than the continuum hypothesis), but my guess is all modern day physics can be expressed/modeled in $V_{omega+omega} + text{♢}$. I wonder how this holds up: "...every open set of Minkowski spacetime is associated with a C*-algebra" /wikipedia.
    $endgroup$
    – CopyPasteIt
    Jan 14 at 14:55












  • $begingroup$
    Did something not work with the Ask Question page that you felt the need to ask a completely disjoint question on a page where you already accepted one answer?
    $endgroup$
    – Asaf Karagila
    Jan 14 at 15:44








1




1




$begingroup$
Incidentally, there are some very interesting results on the "amount of powerset" (or really, "powerset-along-replacement") we need in various situations. Most famously, Harvey Friedman showed that more and more iterations of powerset are needed to prove determinacy further and further up the Borel hierarchy.
$endgroup$
– Noah Schweber
Jan 11 at 22:08




$begingroup$
Incidentally, there are some very interesting results on the "amount of powerset" (or really, "powerset-along-replacement") we need in various situations. Most famously, Harvey Friedman showed that more and more iterations of powerset are needed to prove determinacy further and further up the Borel hierarchy.
$endgroup$
– Noah Schweber
Jan 11 at 22:08




1




1




$begingroup$
On formatting: You can use pairs of L& R dollar-signs to "display" a line instead of typing quad quad quad ..., and to put un-formatted text (like "because") in a displayed line, enclose it within text {...}, like text { because }.
$endgroup$
– DanielWainfleet
Jan 14 at 0:47




$begingroup$
On formatting: You can use pairs of L& R dollar-signs to "display" a line instead of typing quad quad quad ..., and to put un-formatted text (like "because") in a displayed line, enclose it within text {...}, like text { because }.
$endgroup$
– DanielWainfleet
Jan 14 at 0:47












$begingroup$
@DanielWainfleet I've felt silly at times trying to (perfectly) align stuff since the final result might depend on the display device. But anyway, not exactly sure what you mean. Please share some links with useful mathstackexchange formatting technique so I can copy n paste it in the future.
$endgroup$
– CopyPasteIt
Jan 14 at 1:27




$begingroup$
@DanielWainfleet I've felt silly at times trying to (perfectly) align stuff since the final result might depend on the display device. But anyway, not exactly sure what you mean. Please share some links with useful mathstackexchange formatting technique so I can copy n paste it in the future.
$endgroup$
– CopyPasteIt
Jan 14 at 1:27












$begingroup$
@NoahSchweber Another interesting paper, 'Large irredundant sets in operator algebras' arxiv.org/abs/1808.01511?context=math I don't know the formulation for the $text{♢-Axiom}$ (stronger than the continuum hypothesis), but my guess is all modern day physics can be expressed/modeled in $V_{omega+omega} + text{♢}$. I wonder how this holds up: "...every open set of Minkowski spacetime is associated with a C*-algebra" /wikipedia.
$endgroup$
– CopyPasteIt
Jan 14 at 14:55






$begingroup$
@NoahSchweber Another interesting paper, 'Large irredundant sets in operator algebras' arxiv.org/abs/1808.01511?context=math I don't know the formulation for the $text{♢-Axiom}$ (stronger than the continuum hypothesis), but my guess is all modern day physics can be expressed/modeled in $V_{omega+omega} + text{♢}$. I wonder how this holds up: "...every open set of Minkowski spacetime is associated with a C*-algebra" /wikipedia.
$endgroup$
– CopyPasteIt
Jan 14 at 14:55














$begingroup$
Did something not work with the Ask Question page that you felt the need to ask a completely disjoint question on a page where you already accepted one answer?
$endgroup$
– Asaf Karagila
Jan 14 at 15:44




$begingroup$
Did something not work with the Ask Question page that you felt the need to ask a completely disjoint question on a page where you already accepted one answer?
$endgroup$
– Asaf Karagila
Jan 14 at 15:44










1 Answer
1






active

oldest

votes


















2












$begingroup$

I assume that by (1) you mean the existence of the operator $Phi$ that you defined above. As you say, the existence of such a $Phi$ is a consequence of the Powerset Axiom and of the other axioms of $ZF$, so if (1) contradicts the other axioms, the Powerset Axiom does as well.



Moreover, it seems to me that $V_{omega_1}$ is a model of $ZFC-P+(1)$, so the consistency strength of this theory is strictly less that that of $ZFC$.






share|cite|improve this answer









$endgroup$









  • 5




    $begingroup$
    Just $V_{omega+omega}$ is already enough.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 12:18






  • 1




    $begingroup$
    @AsafKaragila Thanks for your comment. Since I am not extremely familiar with this topic and I wanted to be sure I opted to go for a regular cardinal, in order to be "safer". If I may add something to this, my answer would be wrong if we considered collection instead of replacement, right? What should we use as a model in that case?
    $endgroup$
    – Leo163
    Jan 9 at 12:51






  • 4




    $begingroup$
    Correct, for Replacement we use $H(kappa)$, the set of sets which are hereditarily smaller than $kappa$.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 13:18












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1 Answer
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active

oldest

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









2












$begingroup$

I assume that by (1) you mean the existence of the operator $Phi$ that you defined above. As you say, the existence of such a $Phi$ is a consequence of the Powerset Axiom and of the other axioms of $ZF$, so if (1) contradicts the other axioms, the Powerset Axiom does as well.



Moreover, it seems to me that $V_{omega_1}$ is a model of $ZFC-P+(1)$, so the consistency strength of this theory is strictly less that that of $ZFC$.






share|cite|improve this answer









$endgroup$









  • 5




    $begingroup$
    Just $V_{omega+omega}$ is already enough.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 12:18






  • 1




    $begingroup$
    @AsafKaragila Thanks for your comment. Since I am not extremely familiar with this topic and I wanted to be sure I opted to go for a regular cardinal, in order to be "safer". If I may add something to this, my answer would be wrong if we considered collection instead of replacement, right? What should we use as a model in that case?
    $endgroup$
    – Leo163
    Jan 9 at 12:51






  • 4




    $begingroup$
    Correct, for Replacement we use $H(kappa)$, the set of sets which are hereditarily smaller than $kappa$.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 13:18
















2












$begingroup$

I assume that by (1) you mean the existence of the operator $Phi$ that you defined above. As you say, the existence of such a $Phi$ is a consequence of the Powerset Axiom and of the other axioms of $ZF$, so if (1) contradicts the other axioms, the Powerset Axiom does as well.



Moreover, it seems to me that $V_{omega_1}$ is a model of $ZFC-P+(1)$, so the consistency strength of this theory is strictly less that that of $ZFC$.






share|cite|improve this answer









$endgroup$









  • 5




    $begingroup$
    Just $V_{omega+omega}$ is already enough.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 12:18






  • 1




    $begingroup$
    @AsafKaragila Thanks for your comment. Since I am not extremely familiar with this topic and I wanted to be sure I opted to go for a regular cardinal, in order to be "safer". If I may add something to this, my answer would be wrong if we considered collection instead of replacement, right? What should we use as a model in that case?
    $endgroup$
    – Leo163
    Jan 9 at 12:51






  • 4




    $begingroup$
    Correct, for Replacement we use $H(kappa)$, the set of sets which are hereditarily smaller than $kappa$.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 13:18














2












2








2





$begingroup$

I assume that by (1) you mean the existence of the operator $Phi$ that you defined above. As you say, the existence of such a $Phi$ is a consequence of the Powerset Axiom and of the other axioms of $ZF$, so if (1) contradicts the other axioms, the Powerset Axiom does as well.



Moreover, it seems to me that $V_{omega_1}$ is a model of $ZFC-P+(1)$, so the consistency strength of this theory is strictly less that that of $ZFC$.






share|cite|improve this answer









$endgroup$



I assume that by (1) you mean the existence of the operator $Phi$ that you defined above. As you say, the existence of such a $Phi$ is a consequence of the Powerset Axiom and of the other axioms of $ZF$, so if (1) contradicts the other axioms, the Powerset Axiom does as well.



Moreover, it seems to me that $V_{omega_1}$ is a model of $ZFC-P+(1)$, so the consistency strength of this theory is strictly less that that of $ZFC$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 9 at 12:11









Leo163Leo163

1,795512




1,795512








  • 5




    $begingroup$
    Just $V_{omega+omega}$ is already enough.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 12:18






  • 1




    $begingroup$
    @AsafKaragila Thanks for your comment. Since I am not extremely familiar with this topic and I wanted to be sure I opted to go for a regular cardinal, in order to be "safer". If I may add something to this, my answer would be wrong if we considered collection instead of replacement, right? What should we use as a model in that case?
    $endgroup$
    – Leo163
    Jan 9 at 12:51






  • 4




    $begingroup$
    Correct, for Replacement we use $H(kappa)$, the set of sets which are hereditarily smaller than $kappa$.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 13:18














  • 5




    $begingroup$
    Just $V_{omega+omega}$ is already enough.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 12:18






  • 1




    $begingroup$
    @AsafKaragila Thanks for your comment. Since I am not extremely familiar with this topic and I wanted to be sure I opted to go for a regular cardinal, in order to be "safer". If I may add something to this, my answer would be wrong if we considered collection instead of replacement, right? What should we use as a model in that case?
    $endgroup$
    – Leo163
    Jan 9 at 12:51






  • 4




    $begingroup$
    Correct, for Replacement we use $H(kappa)$, the set of sets which are hereditarily smaller than $kappa$.
    $endgroup$
    – Asaf Karagila
    Jan 9 at 13:18








5




5




$begingroup$
Just $V_{omega+omega}$ is already enough.
$endgroup$
– Asaf Karagila
Jan 9 at 12:18




$begingroup$
Just $V_{omega+omega}$ is already enough.
$endgroup$
– Asaf Karagila
Jan 9 at 12:18




1




1




$begingroup$
@AsafKaragila Thanks for your comment. Since I am not extremely familiar with this topic and I wanted to be sure I opted to go for a regular cardinal, in order to be "safer". If I may add something to this, my answer would be wrong if we considered collection instead of replacement, right? What should we use as a model in that case?
$endgroup$
– Leo163
Jan 9 at 12:51




$begingroup$
@AsafKaragila Thanks for your comment. Since I am not extremely familiar with this topic and I wanted to be sure I opted to go for a regular cardinal, in order to be "safer". If I may add something to this, my answer would be wrong if we considered collection instead of replacement, right? What should we use as a model in that case?
$endgroup$
– Leo163
Jan 9 at 12:51




4




4




$begingroup$
Correct, for Replacement we use $H(kappa)$, the set of sets which are hereditarily smaller than $kappa$.
$endgroup$
– Asaf Karagila
Jan 9 at 13:18




$begingroup$
Correct, for Replacement we use $H(kappa)$, the set of sets which are hereditarily smaller than $kappa$.
$endgroup$
– Asaf Karagila
Jan 9 at 13:18


















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