Difference between revisions of "Measurable"
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A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to "measure" the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$. | A '''measurable cardinal''' $\kappa$ is an [[uncountable]] [[cardinal]] such that it is possible to "measure" the subsets of $\kappa$ using a 2-valued [[measure]] on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial [[elementary embedding]] $j:V\to M$. | ||
| − | Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies ZFC, therefore ZFC cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in ZFC, but in ZF they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable. | + | Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is [[inaccessible]], the $\kappa$th inacessible, the $\kappa$th [[weakly compact]] cardinal, the $\kappa$th [[Ramsey]], and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise ''much'' stronger than existence of cardinals with those properties), in fact under the [[axiom of determinacy]], the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable. |
Measurable cardinals were introduced by Stanislaw Ulam in 1930. | Measurable cardinals were introduced by Stanislaw Ulam in 1930. | ||
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# There exists a $\kappa$-complete (or even just $\sigma$-complete) nonprincipal ultrafilter on $\kappa$. | # There exists a $\kappa$-complete (or even just $\sigma$-complete) nonprincipal ultrafilter on $\kappa$. | ||
# There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point''). | # There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the ''critical point''). | ||
| − | # There exists an ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $( | + | # There exists an ultrafilter $U$ on $\kappa$ such that the [[ultrapower]] $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded and isn't isomorphic to $V$. |
The equivalence between the first two definition is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$. | The equivalence between the first two definition is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$. | ||
| − | To see that the third definition implies the first two, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to | + | To see that the third definition implies the first two, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use [[ultrapower|ultrapower embeddings]]: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe. |
| − | The equivalence of the last definition with the other ones is simply due to the fact that the ultrapower $( | + | The equivalence of the last definition with the other ones is simply due to the fact that the ultrapower $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded if and only if $U$ is $\sigma$-complete, and is isomorphic to $V$ if and only if $U$ is principal. |
== Properties == | == Properties == | ||
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''See also: [[Ultrapower]]'' | ''See also: [[Ultrapower]]'' | ||
| − | If $\kappa$ is measurable, then it has a measure that take every value in [0,1]. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$. | + | If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$. |
Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$. | Every measurable cardinal is [[regular]], and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-[[indescribable]]. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$. | ||
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Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha<\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$. | Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha<\kappa$ such that $V\models\varphi(\alpha,x)$ is [[stationary]] in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$. | ||
| − | Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with ZFC that the first measurable cardinal and the first [[strongly compact]] cardinal are equal. | + | Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of [[strongly compact | strong compactness]] and [[supercompact | supercompactness]] respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first [[strongly compact]] cardinal are equal. |
If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds. | If a measurable $\kappa$ is such that there is $\kappa$ [[strongly compact]] cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not [[supercompact]]. If a measurable $\kappa$ has infinitely many [[Woodin]] cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the [[axiom of projective determinacy]] holds. | ||
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If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed. | If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed. | ||
| − | === Failure of GCH at a measurable === | + | === Failure of $\text{GCH}$ at a measurable === |
Gitik proved that the following statements are equiconsistent: | Gitik proved that the following statements are equiconsistent: | ||
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* There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$ | * There is a measurable cardinal of [[Mitchell rank | Mitchell order]] $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$ | ||
| − | Thus violating GCH at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal. | + | Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal. |
However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it. | However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it. | ||
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If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable. | If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. [[Martin's axiom]] implies that the continuum is not real-valued measurable. | ||
| − | Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is generic extension in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of ZFC. | + | Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is generic extension in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$. |
== See also == | == See also == | ||
Revision as of 14:31, 11 November 2017
A measurable cardinal $\kappa$ is an uncountable cardinal such that it is possible to "measure" the subsets of $\kappa$ using a 2-valued measure on the powerset of $\kappa$, $\mathcal{P}(\kappa)$. There exists several other equivalent definitions: For example, $\kappa$ can also be the critical point of a nontrivial elementary embedding $j:V\to M$.
Every measurable is a large cardinal, i.e. $V_\kappa$ satisfies $\text{ZFC}$, therefore $\text{ZFC}$ cannot prove the existence of a measurable cardinal. In fact $\kappa$ is inaccessible, the $\kappa$th inacessible, the $\kappa$th weakly compact cardinal, the $\kappa$th Ramsey, and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in $\text{ZFC}$, but in $\text{ZF}$ they may not (but their existence remains consistency-wise much stronger than existence of cardinals with those properties), in fact under the axiom of determinacy, the first two uncountable cardinals, $\aleph_1$ and $\aleph_2$, are both measurable.
Measurable cardinals were introduced by Stanislaw Ulam in 1930.
Contents
Definitions
The following definitions are equivalent for every uncountable cardinal $\kappa$:
- There exists a 2-valued measure on $\kappa$.
- There exists a $\kappa$-complete (or even just $\sigma$-complete) nonprincipal ultrafilter on $\kappa$.
- There exists a nontrivial elementary embedding $j:V\to M$ with $M$ a transitive class and such that $\kappa$ is the least ordinal moved (the critical point).
- There exists an ultrafilter $U$ on $\kappa$ such that the ultrapower $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded and isn't isomorphic to $V$.
The equivalence between the first two definition is due to the fact that if $\mu$ is a 2-valued measure on $\kappa$, then $U=\{X\subset\kappa|\mu(X)=1\}$ is a nonprincipal ultrafilter (since $\mu$ is 2-valued) and is also $\sigma$-complete because of $\mu$'s $\sigma$-additivity. Similarly, if $U$ is a $\sigma$-complete nonprincipal ultrafilter on $\kappa$, then $\mu:\mathcal{P}(\kappa)\to[0,1]$ defined by $\mu(X)=1$ whenever $X\in U$, $\mu(X)=0$ otherwise is a 2-valued measure on $\kappa$.
To see that the third definition implies the first two, one can show that if $j:V\to M$ is a nontrivial elementary embedding, then the set $\mathcal{U}=\{x\subset\kappa|\kappa\in j(x)\})$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and in fact a normal fine measure. To show the converse, one needs to use ultrapower embeddings: if $U$ is a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then the canonical ultrapower embedding $j:V\to\text{Ult}_U(V)$ is a nontrivial elementary embedding of the universe.
The equivalence of the last definition with the other ones is simply due to the fact that the ultrapower $(\text{Ult}_U(V),\in_U)$ of the universe is well-founded if and only if $U$ is $\sigma$-complete, and is isomorphic to $V$ if and only if $U$ is principal.
Properties
See also: Ultrapower
If $\kappa$ is measurable, then it has a measure that take every value in $[0,1]$. Also there must be a normal fine measure on $\mathcal{P}_\kappa(\kappa)$.
Every measurable cardinal is regular, and (under AC) bears most large cardinal properties weaker than it. It is in particular $\Pi^2_1$-indescribable. However the least measurable cardinal is not $\Sigma^2_1$-indescribable. Independently of the truth of AC, the existence of a measurable cardinal implies the consistency of the existence of large cardinals with the said properties, even if that measurable is merely $\omega_1$.
If $\kappa$ is measurable and $\lambda<\kappa$ then it cannot be true that $\kappa<2^\lambda$. Under AC this means that $\kappa$ is a strong limit (and since it is regular, it must be strongly inaccessible, hence it cannot be $\omega_1$).
If there exists a measurable cardinal then $0^\#$ exists, and therefore $V\neq L$. In fact, the sharp of every real number exists, and therefore $\mathbf{\Pi}^1_1$-determinacy holds. Furthermore, assuming the axiom of determinacy, the cardinals $\omega_1$, $\omega_2$, $\omega_{\omega+1}$ and $\omega_{\omega+2}$ are measurable, also in $L(\mathbb{R})$ every regular cardinal smaller than $\Theta$ is measurable.
Every measurable has the following reflection property: let $j:V\to M$ be a nontrivial elementary embedding with critical point $\kappa$. If $x\in V_\kappa$ and $M\models\varphi(\kappa,x)$ for some first-order formula $\varphi$, then the set of all ordinals $\alpha<\kappa$ such that $V\models\varphi(\alpha,x)$ is stationary in $\kappa$ and has the same measure as $\kappa$ itself by any 2-valued measure on $\kappa$.
Measurability of $\kappa$ is equivalent with $\kappa$-strong compactness of $\kappa$, and also with $\kappa$-supercompactness of $\kappa$ (fragments of strong compactness and supercompactness respectively.) It is also consistent with $\text{ZFC}$ that the first measurable cardinal and the first strongly compact cardinal are equal.
If a measurable $\kappa$ is such that there is $\kappa$ strongly compact cardinals below it, then it is strongly compact. If it is a limit of strongly compact cardinals, then it is strongly compact yet not supercompact. If a measurable $\kappa$ has infinitely many Woodin cardinals below it, then the axiom of determinacy holds in $L(\mathbb{R})$, also the axiom of projective determinacy holds.
If $\kappa$ is measurable in a ground model, then it is measurable in any forcing extension of that ground model whose notion of forcing has cardinality strictly smaller than $\kappa$. Prikry showed however that every measurable can be collapsed to a cardinal of cofinality $\omega$ and no other cardinal is collapsed.
Failure of $\text{GCH}$ at a measurable
Gitik proved that the following statements are equiconsistent:
- The generalized continuum hypothesis fails at a measurable cardinal $\kappa$, i.e. $2^\kappa > \kappa^+$
- The singular cardinal hypothesis fails, i.e. there is a strong limit singular $\kappa$ such that $2^\kappa > \kappa^+$
- There is a measurable cardinal of Mitchell order $\kappa^{++}$, i.e. $o(\kappa)=\kappa^{++}$
Thus violating $\text{GCH}$ at a measurable (or violating the SCH at any strong limit cardinal) is strictly stronger consistency-wise than the existence of a measurable cardinal.
However, if the generalized continuum hypothesis fails at a measurable, then it fails at $\kappa$ many cardinals below it.
Real-valued measurable cardinal
A cardinal $\kappa$ is real-valued measurable if there exists a $\kappa$-additive measure on $\kappa$. The smallest cardinal $\kappa$ carrying a $\sigma$-additive 2-valued measure must also carry a $\kappa$-additive measure, and is therefore real-valued measurable, also it is strongly inaccessible under AC.
If a real-valued measurable cardinal is not measurable, then it must be smaller than (or equal to) $2^{\aleph_0}$. Martin's axiom implies that the continuum is not real-valued measurable.
Solovay showed that the existence of a measurable cardinal is equiconsistent with the existence of a real-valued measurable cardinal. More precisely, he showed that if there is a measurable then there is generic extension in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable, and conversely if there exists a real-valued measurable then it is measurable in some model of $\text{ZFC}$.
See also
Read more
- Jech, Thomas - Set theory
- Bering A., Edgar - A brief introduction to measurable cardinals
