http://cantorsattic.info/api.php?action=feedcontributions&user=Pholy&feedformat=atomCantor's Attic - User contributions [en]2019-10-23T05:56:06ZUser contributionsMediaWiki 1.24.4http://cantorsattic.info/index.php?title=Inaccessible&diff=1375Inaccessible2017-05-16T09:54:06Z<p>Pholy: </p>
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<div>{{DISPLAYTITLE: Inaccessible cardinal}}<br />
<br />
Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy, although weaker notions such as the [[worldly]] cardinals can still be viewed as large cardinals. <br />
<br />
A cardinal $\kappa$ is ''inaccessible'', also equivalently called ''strongly inaccessible'', if it is an [[uncountable]] [[regular]] [[strong limit]] cardinal. <br />
<br />
* If $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC and so inaccessible cardinals are [[worldly]], but this is not an equivalence. <br />
* Every inaccessible cardinal $\kappa$ is an [[aleph fixed point]] and a [[beth fixed point]], and consequently $V_\kappa=H_\kappa$. <br />
* (Zermelo) The models of second-order ZFC are precisely the models $\langle V_\kappa,\in\rangle$ for an inaccessible cardinal $\kappa$.<br />
* Solovay proved that if there is an inaccessible cardinal, then there is an inner model of a forcing extension satisfying ZF+DC in which every set of reals is Lebesgue measurable there. (citation)<br />
* Shelah proved that Solovay's use of the inaccessible cardinal is necessary, in the sense that in any model of ZF+DC in which every set of reals is Lebesgue measurable, there is an inner model of ZFC with an inaccessible cardinal.<br />
* Consequently, the consistency of the existence of an inaccessible cardinal with ZFC is equivalent to the impossibility of our constructing a non-measurable set of reals using only ZF+DC. <br />
* The uncountable [[inaccessible#Grothendieck_universe | Grothedieck universes]] are precisely the sets of the form $V_\kappa$ for an inaccessible cardinal $\kappa$. <br />
* The [[inaccessible#universe_axiom | universe axiom]] is equivalent to the assertion that there is a proper class of inaccessible cardinals. <br />
<br />
==Weakly inaccessible cardinal==<br />
<br />
A cardinal $\kappa$ is ''weakly inaccessible'' if it is an [[uncountable]] [[regular]] [[limit]] cardinal. Under the [[GCH]], this is equivalent to inaccessibility, since under GCH every limit cardinal is a strong limit cardinal. So the difference between weak and strong inaccessibility only arises when GCH fails badly. Every inaccessible cardinal is weakly inaccessible, but forcing arguments show that any inaccessible cardinal can become a non-inaccessible weakly inaccessible cardinal in a forcing extension, such as after adding an enormous number of Cohen reals (this forcing is c.c.c. and hence preserves all cardinals and cofinalities and hence also all regular limit cardinals). Meanwhile, every weakly inaccessible cardinal is fully inaccessible in any inner model of GCH, since it will remain a regular limit cardinal in that model and hence also be a strong limit there. In particular, every weakly inaccessible cardinal is inaccessible in the constructible universe $L$. Consequently, although the two large cardinal notions are not provably equivalent, they are equiconsistent. <br />
<br />
==Levy collapse==<br />
<br />
The Levy collapse of an inaccessible cardinal $\kappa$ is the $\lt\kappa$-support product of $\text{Coll}(\omega,\gamma)$ for all $\gamma\lt\kappa$. This forcing collapses all cardinals below $\kappa$ to $\omega$, but since it is $\kappa$-c.c., it preserves $\kappa$ itself, and hence ensures $\kappa=\omega_1$ in the forcing extension.<br />
<br />
==Inaccessible to reals==<br />
<br />
A cardinal $\kappa$ is ''inaccessible to reals'' if it is inaccessible in $L[x]$ for every real $x$. For example, after the Levy collapse of an inaccessible cardinal $\kappa$, which forces $\kappa=\omega_1$ in the extension, the cardinal $\kappa$ is of course no longer inaccessible, but it remains inaccessible to reals. <br />
<br />
==Universes==<br />
<br />
When $\kappa$ is inaccessible, then $V_\kappa$ provides a highly natural transitive model of set theory, a universe in which one can view a large part of classical mathematics as taking place. In what appears to be an instance of convergent evolution, the same universe concept arose in category theory out of the desire to provide a hierarchy of notions of smallness, so that one may form such categories as the category of all small groups, or small rings or small categories, without running into the difficulties of [[Russell's paradox]]. Namely, a ''Grothendieck universe'' is a transitive set $W$ that is closed under pairing, power set and unions. That is, <br />
* (transitivity) If $b\in a\in W$, then $b\in W$. <br />
* (pairing) If $a,b\in W$, then $\{a,b\}\in W$. <br />
* (power set) If $a\in W$, then $P(a)\in W$.<br />
* (union) If $a\in W$, then $\cup a\in W$. <br />
It follows by a simple inductive argument that the Grothendieck universes are precisely the sets of the form $V_\kappa$, where $\kappa$ is either [[zero | $0$]], [[omega | $\omega$]] or an inaccessible cardinal. <br />
<br />
The ''Grothendieck universe axiom'' is the assertion that every set is an element of a Grothendieck universe. This is equivalent to the assertion that the inaccessible cardinals form a proper class. <br />
<br />
== Degrees of inaccessibility ==<br />
<br />
A cardinal $\kappa$ is ''$1$-inaccessible'' if it is inaccessible and a limit of inaccessible cardinals. In other words, $\kappa$ is $1$-inaccessible if $\kappa$ is the $\kappa^{\rm th}$ inaccessible cardinal, that is, if $\kappa$ is a fixed point in the enumeration of all inaccessible cardinals. Equivalently, $\kappa$ is $1$-inaccessible if $V_\kappa$ is a universe and satisfies the universe axiom. <br />
<br />
More generally, $\kappa$ is $\alpha$-inaccessible if it is inaccessible and for every $\beta\lt\alpha$ it is a limit of $\beta$-inaccessible cardinals.<br />
<br />
==Hyper-inaccessible==<br />
<br />
A cardinal $\kappa$ is ''hyperinaccessible'' if it is $\kappa$-inaccessible. One may similarly define that $\kappa$ is $\alpha$-hyperinaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$, it is a limit of $\beta$-hyperinaccessible cardinals. Continuing, $\kappa$ is ''hyperhyperinaccessible'' if $\kappa$ is $\kappa$-hyperinaccessible. <br />
<br />
More generally, $\kappa$ is ''hyper${}^\alpha$-inaccessible'' if it is hyperinaccessible and for every $\beta\lt\alpha$ it is $\kappa$-hyper${}^\beta$-inaccessible, where $\kappa$ is ''$\alpha$-hyper${}^\beta$-inaccessible'' if it is hyper${}^\beta$-inaccessible and for every $\gamma<\alpha$, it is a limit of $\gamma$-hyper${}^\beta$-inaccessible cardinals.<br />
<br />
Every [[Mahlo]] cardinal $\kappa$ is hyper${}^\kappa$-inaccessible.</div>Pholyhttp://cantorsattic.info/index.php?title=Ineffable&diff=1374Ineffable2017-05-12T12:15:25Z<p>Pholy: </p>
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<div>{{DISPLAYTITLE: Ineffable cardinal}}<br />
<br />
Ineffable cardinals were introduced by Jensen and Kunen in <cite>JensenKunen1969:Ineffable</cite> and arose out of their study of $\diamondsuit$ principles. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ is stationary. Equivalently an uncountable regular $\kappa$ is ineffable if and only if for every function $F:[\kappa]^2\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^2$ is constant <cite>JensenKunen1969:Ineffable</cite>. This second characterization strengthens a characterization of weakly compact cardinals which requires that there exist such an $H$ of size $\kappa$.<br />
<br />
If $\kappa$ is ineffable, then $\diamondsuit_\kappa$ holds and there cannot be a slim $\kappa$-Kurepa tree <cite>JensenKunen1969:Ineffable</cite> . A $\kappa$-Kurepa tree is a tree of height $\kappa$ having levels of size less than $\kappa$ and at least $\kappa^+$-many branches. A $\kappa$-Kurepa tree is slim if every infinite level $\alpha$ has size at most $|\alpha|$. <br />
<br />
<br />
==Ineffable cardinals and the constructible universe==<br />
<br />
Ineffable cardinals are downward absolute to $L$. In $L$, an inaccessible cardinal $\kappa$ is ineffable if and only if there are no slim $\kappa$-Kurepa trees. Thus, for inaccessible cardinals, in $L$, ineffability is completely characterized using slim Kurepa trees. <cite>JensenKunen1969:Ineffable</cite><br />
<br />
If [[zero sharp | $0^\sharp$]] exists, then every Silver indiscernible is ineffable in $L$. <cite>Jech2003:SetTheory</cite><br />
<br />
==Relations with other large cardinals==<br />
<br />
* [[Measurable]] cardinals are ineffable and stationary limits of ineffable cardinals. <br />
* [[Erdos|$\omega$-Erd&#337;s]] cardinals are stationary limits of ineffable cardinals, but not ineffable since they are $\Pi_1^1$-describable. <cite>Jech2003:SetTheory</cite> <br />
* Ineffable cardinals are $\Pi^1_2$-[[indescribable]] <cite>JensenKunen1969:Ineffable</cite>. <br />
* Ineffable cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> (<cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> for proof)<br />
<br />
==Weakly ineffable cardinal==<br />
<br />
Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in <cite>JensenKunen1969:Ineffable</cite> as a weakening of ineffable cardinals. An uncountable regular cardinal $\kappa$ is weakly ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ has size $\kappa$. If $\kappa$ is weakly ineffable, then $\diamondsuit_\kappa$ holds.<br />
<br />
* Weakly ineffable cardinals are downward absolute to $L$. <cite>JensenKunen1969:Ineffable</cite><br />
* Weakly ineffable cardinals are $\Pi_1^1$-[[indescribable]]. <cite>JensenKunen1969:Ineffable</cite><br />
* Ineffable cardinals are limits of weakly ineffable cardinals. <br />
* Weakly ineffable cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> (<cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> for proof)<br />
<br />
==Subtle cardinal==<br />
<br />
Subtle cardinals were introduced by Jensen and Kunen in <cite>JensenKunen1969:Ineffable</cite> as a weakening of weakly ineffable cardinals. A uncountable regular cardinal $\kappa$ is subtle if for every for every $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ and every closed unbounded $C\subseteq\kappa$ there are $\alpha<\beta$ in $C$ such that $A_\beta\cap\alpha=A_\alpha$. If $\kappa$ is subtle, then $\diamondsuit_\kappa$ holds.<br />
<br />
* Subtle cardinals are downward absolute to $L$. <cite>JensenKunen1969:Ineffable</cite> <br />
* Weakly ineffable cardinals are limits of subtle cardinals. <cite>JensenKunen1969:Ineffable</cite> <br />
* Subtle cardinals are limits of [[totally indescribable]] cardinals. <cite>JensenKunen1969:Ineffable</cite> <br />
* The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable.<br />
* [[Erdos|$\alpha$-Erd&#337;s]] cardinals are subtle. <cite>JensenKunen1969:Ineffable</cite><br />
<br />
==Completely ineffable cardinal==<br />
<br />
Completely ineffable cardinals were introduced in <cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite> as a strenthening of ineffable cardinals. Define that a collection $R\subseteq P(\kappa)$ is a stationary<br />
class if<br />
* $R\neq\emptyset$,<br />
* for all $A\in R$, $A$ is stationary in $\kappa$,<br />
* if $A\in R$ and $B\supseteq A$, then $B\in R$.<br />
A cardinal $\kappa$ is completely ineffable if there is a stationary class $R$ such that for every $A\in R$ and $F:[A]^2\to<br />
2$, there is $H\in R$ such that $F\upharpoonright [H]^2$ is constant.<br />
<br />
* Completely ineffable cardinals are downward absolute to $L$. <cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite><br />
* Completely ineffable cardinals are limits of ineffable cardinals. <cite>AbramsonHarringtonKleinbergZwicker1977:FlippingProperties</cite><br />
<br />
==$n$-ineffable cardinal==<br />
The $n$-ineffable cardinals for $2\leq n<\omega$ were introduced by Baumgartner in <cite>Baumgartner1975:Ineffability</cite> as a strengthening of ineffable cardinals. A cardinal is $n$-ineffable if for every function $F:[\kappa]^n\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^n$ is constant. A cardinal $\kappa$ is totally ineffable if it is $n$-ineffable for every $n$. <br />
* $2$-ineffable cardinals are exactly the ineffable cardinals.<br />
* an $n+1$-ineffable cardinal is a stationary limit of $n$-ineffable cardinals. <cite>Baumgartner1975:Ineffability</cite><br />
* a $1$-iterable cardinal is a stationary limit of totally ineffable cardinals. (this follows from material in <cite>Gitman2011:RamseyLikeCardinals</cite>)<br />
<br />
{{References}}</div>Pholyhttp://cantorsattic.info/index.php?title=Inaccessible&diff=1373Inaccessible2017-05-02T12:28:51Z<p>Pholy: </p>
<hr />
<div>{{DISPLAYTITLE: Inaccessible cardinal}}<br />
<br />
Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy, although weaker notions such as the [[worldly]] cardinals can still be viewed as large cardinals. <br />
<br />
A cardinal $\kappa$ is ''inaccessible'', also equivalently called ''strongly inaccessible'', if it is an [[uncountable]] [[regular]] [[strong limit]] cardinal. <br />
<br />
* If $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC and so inaccessible cardinals are [[worldly]], but this is not an equivalence. <br />
* Every inaccessible cardinal $\kappa$ is an [[aleph fixed point]] and a [[beth fixed point]], and consequently $V_\kappa=H_\kappa$. <br />
* (Zermelo) The models of second-order ZFC are precisely the models $\langle V_\kappa,\in\rangle$ for an inaccessible cardinal $\kappa$.<br />
* Solovay proved that if there is an inaccessible cardinal, then there is an inner model of a forcing extension satisfying ZF+DC in which every set of reals is Lebesgue measurable there. (citation)<br />
* Shelah proved that Solovay's use of the inaccessible cardinal is necessary, in the sense that in any model of ZF+DC in which every set of reals is Lebesgue measurable, there is an inner model of ZFC with an inaccessible cardinal.<br />
* Consequently, the consistency of the existence of an inaccessible cardinal with ZFC is equivalent to the impossibility of our constructing a non-measurable set of reals using only ZF+DC. <br />
* The uncountable [[inaccessible#Grothendieck_universe | Grothedieck universes]] are precisely the sets of the form $V_\kappa$ for an inaccessible cardinal $\kappa$. <br />
* The [[inaccessible#universe_axiom | universe axiom]] is equivalent to the assertion that there is a proper class of inaccessible cardinals. <br />
<br />
==Weakly inaccessible cardinal==<br />
<br />
A cardinal $\kappa$ is ''weakly inaccessible'' if it is an [[uncountable]] [[regular]] [[limit]] cardinal. Under the [[GCH]], this is equivalent to inaccessibility, since under GCH every limit cardinal is a strong limit cardinal. So the difference between weak and strong inaccessibility only arises when GCH fails badly. Every inaccessible cardinal is weakly inaccessible, but forcing arguments show that any inaccessible cardinal can become a non-inaccessible weakly inaccessible cardinal in a forcing extension, such as after adding an enormous number of Cohen reals (this forcing is c.c.c. and hence preserves all cardinals and cofinalities and hence also all regular limit cardinals). Meanwhile, every weakly inaccessible cardinal is fully inaccessible in any inner model of GCH, since it will remain a regular limit cardinal in that model and hence also be a strong limit there. In particular, every weakly inaccessible cardinal is inaccessible in the constructible universe $L$. Consequently, although the two large cardinal notions are not provably equivalent, they are equiconsistent. <br />
<br />
==Levy collapse==<br />
<br />
The Levy collapse of an inaccessible cardinal $\kappa$ is the $\lt\kappa$-support product of $\text{Coll}(\omega,\gamma)$ for all $\gamma\lt\kappa$. This forcing collapses all cardinals below $\kappa$ to $\omega$, but since it is $\kappa$-c.c., it preserves $\kappa$ itself, and hence ensures $\kappa=\omega_1$ in the forcing extension.<br />
<br />
==Inaccessible to reals==<br />
<br />
A cardinal $\kappa$ is ''inaccessible to reals'' if it is inaccessible in $L[x]$ for every real $x$. For example, after the Levy collapse of an inaccessible cardinal $\kappa$, which forces $\kappa=\omega_1$ in the extension, the cardinal $\kappa$ is of course no longer inaccessible, but it remains inaccessible to reals. <br />
<br />
==Universes==<br />
<br />
When $\kappa$ is inaccessible, then $V_\kappa$ provides a highly natural transitive model of set theory, a universe in which one can view a large part of classical mathematics as taking place. In what appears to be an instance of convergent evolution, the same universe concept arose in category theory out of the desire to provide a hierarchy of notions of smallness, so that one may form such categories as the category of all small groups, or small rings or small categories, without running into the difficulties of [[Russell's paradox]]. Namely, a ''Grothendieck universe'' is a transitive set $W$ that is closed under pairing, power set and unions. That is, <br />
* (transitivity) If $b\in a\in W$, then $b\in W$. <br />
* (pairing) If $a,b\in W$, then $\{a,b\}\in W$. <br />
* (power set) If $a\in W$, then $P(a)\in W$.<br />
* (union) If $a\in W$, then $\cup a\in W$. <br />
It follows by a simple inductive argument that the Grothendieck universes are precisely the sets of the form $V_\kappa$, where $\kappa$ is either [[zero | $0$]], [[omega | $\omega$]] or an inaccessible cardinal. <br />
<br />
The ''Grothendieck universe axiom'' is the assertion that every set is an element of a Grothendieck universe. This is equivalent to the assertion that the inaccessible cardinals form a proper class. <br />
<br />
== Degrees of inaccessibility ==<br />
<br />
A cardinal $\kappa$ is ''$1$-inaccessible'' if it is inaccessible and a limit of inaccessible cardinals. In other words, $\kappa$ is $1$-inaccessible if $\kappa$ is the $\kappa^{\rm th}$ inaccessible cardinal, that is, if $\kappa$ is a fixed point in the enumeration of all inaccessible cardinals. Equivalently, $\kappa$ is $1$-inaccessible if $V_\kappa$ is a universe and satisfies the universe axiom. <br />
<br />
More generally, $\kappa$ is $\alpha$-inaccessible if it is inaccessible and for every $\beta\lt\alpha$ it is a limit of $\beta$-inaccessible cardinals.<br />
<br />
==Hyper-inaccessible==<br />
<br />
A cardinal $\kappa$ is ''hyperinaccessible'' if it is $\kappa$-inaccessible. One may similarly define that $\kappa$ is $\alpha$-hyperinaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$, it is a limit of $\beta$-hyperinaccessible cardinals. Continuing, $\kappa$ is ''hyperhyperinaccessible'' if $\kappa$ is $\kappa$-hyperinaccessible. <br />
<br />
More generally, $\kappa$ is ''hyper${}^\alpha$-inaccessible'' if for every $\beta\lt\kappa$ it is $\kappa$-hyper${}^\beta$-inaccessible.<br />
<br />
Every [[Mahlo]] cardinal $\kappa$ is hyper${}^\kappa$-inaccessible.</div>Pholy