Natural models of Ackermann Set Theory
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Consider Ackermann's Set Theory (as described here, §6, including the axiom of regularity for sets), henceforth denoted AST, as a theory in the language $leftlangle in,mathbf{V} rightrangle$ where $mathbf{V}$ is a constant symbol intended to represent the von Neumann universe in models of AST.
In Natural Models of Ackermann's Set Theory, Rudolf Grewe studies models of AST of the form $(V_{alpha},in,V_{beta})$ for ordinals $beta<alpha$, so-called natural models of AST.
Grewe proves that for ordinals $alpha>beta$ where $alpha$ is limit, the structure $(V_{alpha},in,V_{beta})$ is a model of AST if and only if $V_{beta}$ is not $leftlangle in rightrangle$-definable in $V_{alpha}$ with parameters in $V_{beta}$. Equivalently, the ordinal $beta$ must not be definable in $V_{alpha}$ with parameters in $V_{beta}$.
This doesn't seem to be a very good thing to know if one wants to produce natural models of AST since the undefinability of an ordinal is a meta-statement. However this makes it easy to see what cannot be a model of AST.
Later in the article, Grewe proves that in fact $V_{beta}$ must be a model of ZF. If $kappa$ is a cardinal which is smallest to satisfy some large cardinal property $P$ which is a $leftlangle in rightrangle$-sentence, the ordinal $kappa$ will always be definable in $V_{alpha}$ without parameters by this very minimality. Therefore no such $V_{kappa}$ is a candidate to interpret V in our natural model.
Of course the undefinability in $leftlangle in rightrangle$ of the interpretation of $mathbf{V}$ in a model of AST is an important feature of the theory, but I find this prevalence in the case of natural models to be troubling.
I have two questions pertaining to this, one of which is vague:
- Is there a classical large cardinal axiom P such that ZFC+P proves that AST has a natural model?
- Is the importance of the undefinability in $mathbf{V}$ of some rank $V_{beta}$ in the von Neumann hierarchy a specific feature in AST, or is it a recurring theme in set-theory that has lied beyond my amateurish gaze for now?
set-theory model-theory
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
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add a comment |
$begingroup$
Consider Ackermann's Set Theory (as described here, §6, including the axiom of regularity for sets), henceforth denoted AST, as a theory in the language $leftlangle in,mathbf{V} rightrangle$ where $mathbf{V}$ is a constant symbol intended to represent the von Neumann universe in models of AST.
In Natural Models of Ackermann's Set Theory, Rudolf Grewe studies models of AST of the form $(V_{alpha},in,V_{beta})$ for ordinals $beta<alpha$, so-called natural models of AST.
Grewe proves that for ordinals $alpha>beta$ where $alpha$ is limit, the structure $(V_{alpha},in,V_{beta})$ is a model of AST if and only if $V_{beta}$ is not $leftlangle in rightrangle$-definable in $V_{alpha}$ with parameters in $V_{beta}$. Equivalently, the ordinal $beta$ must not be definable in $V_{alpha}$ with parameters in $V_{beta}$.
This doesn't seem to be a very good thing to know if one wants to produce natural models of AST since the undefinability of an ordinal is a meta-statement. However this makes it easy to see what cannot be a model of AST.
Later in the article, Grewe proves that in fact $V_{beta}$ must be a model of ZF. If $kappa$ is a cardinal which is smallest to satisfy some large cardinal property $P$ which is a $leftlangle in rightrangle$-sentence, the ordinal $kappa$ will always be definable in $V_{alpha}$ without parameters by this very minimality. Therefore no such $V_{kappa}$ is a candidate to interpret V in our natural model.
Of course the undefinability in $leftlangle in rightrangle$ of the interpretation of $mathbf{V}$ in a model of AST is an important feature of the theory, but I find this prevalence in the case of natural models to be troubling.
I have two questions pertaining to this, one of which is vague:
- Is there a classical large cardinal axiom P such that ZFC+P proves that AST has a natural model?
- Is the importance of the undefinability in $mathbf{V}$ of some rank $V_{beta}$ in the von Neumann hierarchy a specific feature in AST, or is it a recurring theme in set-theory that has lied beyond my amateurish gaze for now?
set-theory model-theory
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
$endgroup$
add a comment |
$begingroup$
Consider Ackermann's Set Theory (as described here, §6, including the axiom of regularity for sets), henceforth denoted AST, as a theory in the language $leftlangle in,mathbf{V} rightrangle$ where $mathbf{V}$ is a constant symbol intended to represent the von Neumann universe in models of AST.
In Natural Models of Ackermann's Set Theory, Rudolf Grewe studies models of AST of the form $(V_{alpha},in,V_{beta})$ for ordinals $beta<alpha$, so-called natural models of AST.
Grewe proves that for ordinals $alpha>beta$ where $alpha$ is limit, the structure $(V_{alpha},in,V_{beta})$ is a model of AST if and only if $V_{beta}$ is not $leftlangle in rightrangle$-definable in $V_{alpha}$ with parameters in $V_{beta}$. Equivalently, the ordinal $beta$ must not be definable in $V_{alpha}$ with parameters in $V_{beta}$.
This doesn't seem to be a very good thing to know if one wants to produce natural models of AST since the undefinability of an ordinal is a meta-statement. However this makes it easy to see what cannot be a model of AST.
Later in the article, Grewe proves that in fact $V_{beta}$ must be a model of ZF. If $kappa$ is a cardinal which is smallest to satisfy some large cardinal property $P$ which is a $leftlangle in rightrangle$-sentence, the ordinal $kappa$ will always be definable in $V_{alpha}$ without parameters by this very minimality. Therefore no such $V_{kappa}$ is a candidate to interpret V in our natural model.
Of course the undefinability in $leftlangle in rightrangle$ of the interpretation of $mathbf{V}$ in a model of AST is an important feature of the theory, but I find this prevalence in the case of natural models to be troubling.
I have two questions pertaining to this, one of which is vague:
- Is there a classical large cardinal axiom P such that ZFC+P proves that AST has a natural model?
- Is the importance of the undefinability in $mathbf{V}$ of some rank $V_{beta}$ in the von Neumann hierarchy a specific feature in AST, or is it a recurring theme in set-theory that has lied beyond my amateurish gaze for now?
set-theory model-theory
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
$endgroup$
Consider Ackermann's Set Theory (as described here, §6, including the axiom of regularity for sets), henceforth denoted AST, as a theory in the language $leftlangle in,mathbf{V} rightrangle$ where $mathbf{V}$ is a constant symbol intended to represent the von Neumann universe in models of AST.
In Natural Models of Ackermann's Set Theory, Rudolf Grewe studies models of AST of the form $(V_{alpha},in,V_{beta})$ for ordinals $beta<alpha$, so-called natural models of AST.
Grewe proves that for ordinals $alpha>beta$ where $alpha$ is limit, the structure $(V_{alpha},in,V_{beta})$ is a model of AST if and only if $V_{beta}$ is not $leftlangle in rightrangle$-definable in $V_{alpha}$ with parameters in $V_{beta}$. Equivalently, the ordinal $beta$ must not be definable in $V_{alpha}$ with parameters in $V_{beta}$.
This doesn't seem to be a very good thing to know if one wants to produce natural models of AST since the undefinability of an ordinal is a meta-statement. However this makes it easy to see what cannot be a model of AST.
Later in the article, Grewe proves that in fact $V_{beta}$ must be a model of ZF. If $kappa$ is a cardinal which is smallest to satisfy some large cardinal property $P$ which is a $leftlangle in rightrangle$-sentence, the ordinal $kappa$ will always be definable in $V_{alpha}$ without parameters by this very minimality. Therefore no such $V_{kappa}$ is a candidate to interpret V in our natural model.
Of course the undefinability in $leftlangle in rightrangle$ of the interpretation of $mathbf{V}$ in a model of AST is an important feature of the theory, but I find this prevalence in the case of natural models to be troubling.
I have two questions pertaining to this, one of which is vague:
- Is there a classical large cardinal axiom P such that ZFC+P proves that AST has a natural model?
- Is the importance of the undefinability in $mathbf{V}$ of some rank $V_{beta}$ in the von Neumann hierarchy a specific feature in AST, or is it a recurring theme in set-theory that has lied beyond my amateurish gaze for now?
set-theory model-theory
set-theory model-theory
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
edited Dec 17 '18 at 18:45
nombre
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
asked Dec 17 '18 at 16:17
nombrenombre
2,669913
2,669913
add a comment |
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