Difference between revisions of "Erdos"

From Cantor's Attic
Jump to: navigation, search
m (At most to at least)
(Restructured)
Line 2: Line 2:
 
[[Category:Large cardinal axioms]]
 
[[Category:Large cardinal axioms]]
 
[[Category:Partition property]]
 
[[Category:Partition property]]
The $\alpha$-Erd&#337;s cardinals were introduced by Erd&#337;s and Hajnal in <cite>ErdosHajnal1958:ErdosCardinals</cite> and arose out of their study of partition relations. A cardinal $\kappa(\alpha)$ is $\alpha$-Erd&#337;s if it is the least cardinal $\kappa$ such that $\kappa\rightarrow (\alpha)^{\lt\omega}_2$ (if any such cardinal exists).  
+
The $\alpha$-Erd&#337;s cardinals were introduced by Erd&#337;s and Hajnal in <cite>ErdosHajnal1958:ErdosCardinals</cite> and arose out of their study of partition relations. A cardinal $\kappa$ is $\alpha$-Erd&#337;s for an infinite limit ordinal $\alpha$ if it is the least cardinal $\kappa$ such that $\kappa\rightarrow (\alpha)^{\lt\omega}_2$ (if any such cardinal exists).  
  
For infinite cardinals $\kappa$ and $\lambda$, the [[partition property]] $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. Here $[X]^n$ is the set of all $n$-elements subsets of $X$. The more general partition property $\kappa\to(\lambda)^{\lt\omega}_\gamma$ asserts that for every function $F:[\kappa]^{\lt\omega}\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant for every $n$, although the value of $F$ on $[H]^n$ may be different for different $n$. Indeed, if $\kappa$ is $\alpha$-Erd&#337;s for some infinite ordinal $\alpha$, then $\kappa\rightarrow (\alpha)^{\lt\omega}_\lambda$ for all $\lambda<\kappa$ (Silver's PhD thesis). A cardinal $\kappa$ is [[Ramsey]] precisely when it is $\kappa$-Erd&#337;s. If $\alpha<\beta$, then $\kappa(\alpha)<\kappa(\beta)$ <cite>ErdosHajnal1958:ErdosCardinals</cite>.  
+
For infinite cardinals $\kappa$ and $\lambda$, the [[partition property]] $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. Here $[X]^n$ is the set of all $n$-elements subsets of $X$. The more general partition property $\kappa\to(\lambda)^{\lt\omega}_\gamma$ asserts that for every function $F:[\kappa]^{\lt\omega}\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant for every $n$, although the value of $F$ on $[H]^n$ may be different for different $n$. Indeed, if $\kappa$ is $\alpha$-Erd&#337;s for some infinite ordinal $\alpha$, then $\kappa\rightarrow (\alpha)^{\lt\omega}_\lambda$ for all $\lambda<\kappa$ (Silver's PhD thesis). A cardinal $\kappa$ is [[Ramsey]] precisely when it is $\kappa$-Erd&#337;s.
 +
 
 +
The $\alpha$-Erd&#337;s cardinal is precisely the least cardinal $\kappa$ such that for any language $\mathcal{L}$ of size less than $\kappa$ and any structure $\mathcal{M}$ with language $\mathcal{L}$ and domain $\kappa$, there is a set of indescernibles for $\mathcal{M}$ of order-type $\alpha$.
 +
 
 +
A cardinal $\kappa$ is called Erd&#337;s if and only if it is $\alpha$-Erd&#337;s for some infinite limit ordinal $\alpha$. Because there exists at most one Erd&#337;s cardinal, the notations $\eta_\alpha$ and $\kappa(\alpha)$ are sometimes used to denote the $\alpha$-Erd&#337;s cardinal.
  
 
==Erd&#337;s cardinals and the constructible universe==
 
==Erd&#337;s cardinals and the constructible universe==
Line 11: Line 15:
  
 
==Facts==
 
==Facts==
*Every $\alpha$-Erd&#337;s cardinal is inaccessible. (Silver's PhD thesis)
+
*Every Erd&#337;s cardinal is [[inaccessible]]. (Silver's PhD thesis)
*An $\alpha$-Erd&#337;s cardinal $\kappa$ is [[weakly compact]] iff $\alpha=\kappa$ (that is, $\kappa$ is [[Ramsey]]). <cite>Kanamori2009:HigherInfinite</cite>
+
*Every Erd&#337;s cardinal is [[subtle]]. <cite>JensenKunen1969:Ineffable</cite>
*An $\alpha$-Erd&#337;s cardinal is [[subtle]]. <cite>JensenKunen1969:Ineffable</cite>
+
*An Erd&#337;s cardinal $\kappa$ is [[weakly compact]] iff $\kappa$ is [[Ramsey#almost Ramsey|almost Ramsey]], iff $\kappa$ is [[Ramsey]]. <cite>Kanamori2009:HigherInfinite</cite>
*The $\omega$-Erd&#337;s cardinal is a stationary limit of ineffable cardinals. <cite>Jech2003:SetTheory</cite>
+
*$\eta_\omega$ is a stationary limit of [[ineffable]] cardinals. <cite>Jech2003:SetTheory</cite>
*The existence of the $\omega$-Erd&#337;s cardinal implies the consistency of a proper class of [[Ramsey#iterable|$n$-iterable]] cardinals for every $1\leq n<\omega$. <cite>Gitman2011:RamseyLikeCardinals</cite>
+
*The existence of $\eta_\omega$ implies the consistency of a proper class of [[Ramsey#iterable|$n$-iterable]] cardinals for every $1\leq n<\omega$. <cite>Gitman2011:RamseyLikeCardinals</cite>
*The $\alpha$-Erd&#337;s cardinal is less than the $\beta$-Erd&#337;s cardinal whenever $\alpha<\beta$, and the $\alpha$-Erd&#337;s cardinal is at least $\alpha$. <cite>Kanamori2009:HigherInfinite</cite>
+
*$\eta_\alpha<\eta_\beta$ whenever $\alpha<\beta$ and $\eta_\alpha\geq\alpha$. <cite>Kanamori2009:HigherInfinite</cite>
 
+
*The consistency strength of the existence of an Erd&#337;s cardinal is stronger than that of the existence of an $n$-iterable cardinal for every $n<\omega$ and weaker than that of the existence of [[Zero sharp|$0^{\#}$]].
 
+
  
 
{{References}}
 
{{References}}

Revision as of 11:55, 5 January 2018

The $\alpha$-Erdős cardinals were introduced by Erdős and Hajnal in [1] and arose out of their study of partition relations. A cardinal $\kappa$ is $\alpha$-Erdős for an infinite limit ordinal $\alpha$ if it is the least cardinal $\kappa$ such that $\kappa\rightarrow (\alpha)^{\lt\omega}_2$ (if any such cardinal exists).

For infinite cardinals $\kappa$ and $\lambda$, the partition property $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. Here $[X]^n$ is the set of all $n$-elements subsets of $X$. The more general partition property $\kappa\to(\lambda)^{\lt\omega}_\gamma$ asserts that for every function $F:[\kappa]^{\lt\omega}\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant for every $n$, although the value of $F$ on $[H]^n$ may be different for different $n$. Indeed, if $\kappa$ is $\alpha$-Erdős for some infinite ordinal $\alpha$, then $\kappa\rightarrow (\alpha)^{\lt\omega}_\lambda$ for all $\lambda<\kappa$ (Silver's PhD thesis). A cardinal $\kappa$ is Ramsey precisely when it is $\kappa$-Erdős.

The $\alpha$-Erdős cardinal is precisely the least cardinal $\kappa$ such that for any language $\mathcal{L}$ of size less than $\kappa$ and any structure $\mathcal{M}$ with language $\mathcal{L}$ and domain $\kappa$, there is a set of indescernibles for $\mathcal{M}$ of order-type $\alpha$.

A cardinal $\kappa$ is called Erdős if and only if it is $\alpha$-Erdős for some infinite limit ordinal $\alpha$. Because there exists at most one Erdős cardinal, the notations $\eta_\alpha$ and $\kappa(\alpha)$ are sometimes used to denote the $\alpha$-Erdős cardinal.

Erdős cardinals and the constructible universe

  • $\omega_1$-Erdős cardinals imply that $0^\sharp$ exists and hence there cannot be $\omega_1$-Erdős cardinals in $L$. [2]
  • $\alpha$-Erdős cardinals are downward absolute to $L$ for $L$-countable $\alpha$. More generally, $\alpha$-Erdős cardinals are downward absolute to any transitive model of ZFC for $M$-countable $\alpha$. [3]

Facts

  • Every Erdős cardinal is inaccessible. (Silver's PhD thesis)
  • Every Erdős cardinal is subtle. [4]
  • An Erdős cardinal $\kappa$ is weakly compact iff $\kappa$ is almost Ramsey, iff $\kappa$ is Ramsey. [5]
  • $\eta_\omega$ is a stationary limit of ineffable cardinals. [6]
  • The existence of $\eta_\omega$ implies the consistency of a proper class of $n$-iterable cardinals for every $1\leq n<\omega$. [7]
  • $\eta_\alpha<\eta_\beta$ whenever $\alpha<\beta$ and $\eta_\alpha\geq\alpha$. [5]
  • The consistency strength of the existence of an Erdős cardinal is stronger than that of the existence of an $n$-iterable cardinal for every $n<\omega$ and weaker than that of the existence of $0^{\#}$.

References

  1. Erdős, Paul and Hajnal, Andras. On the structure of set-mappings. Acta Math Acad Sci Hungar 9:111--131, 1958. MR   bibtex
  2. Silver, Jack. Some applications of model theory in set theory. Ann Math Logic 3(1):45--110, 1971. MR   bibtex
  3. Silver, Jack. A large cardinal in the constructible universe. Fund Math 69:93--100, 1970. MR   bibtex
  4. Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www   bibtex
  5. Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www   bibtex
  6. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
  7. Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www   arχiv   MR   bibtex
Main library