Erdős cardinal

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Template:Harvs.

A cardinal $\kappa$ is called $\alpha$-Erdős if for every function $f:[\kappa {]}^{<\omega }\to \{0,1\}$, there is a set of order type $\alpha$ that is homogeneous for $f$. In the notation of the partition calculus, $\kappa$ is $\alpha$-Erdős if

$\kappa \to (\alpha {)}^{<\omega }$.

Under this definition, any cardinal larger than the least $\alpha$-Erdős cardinal is $\alpha$-Erdős.

The existence of zero sharp implies that the constructible universe $L$ satisfies "for every countable ordinal $\alpha$, there is an $\alpha$-Erdős cardinal". In fact, for every indiscernible $\kappa$, $L_{\kappa }$ satisfies "for every ordinal $\alpha$, there is an $\alpha$-Erdős cardinal in $\mathrm{C}\mathrm{o}\mathrm{l}\mathrm{l}(\omega ,\alpha )$" (the Lévy collapse to make $\alpha$ countable).

However, the existence of an ${\omega }_1$-Erdős cardinal implies existence of zero sharp. If $f$ is the satisfaction relation for $L$ (using ordinal parameters), then the existence of zero sharp is equivalent to there being an ${\omega }_1$-Erdős ordinal with respect to $f$. Thus, the existence of an ${\omega }_1$-Erdős cardinal implies that the axiom of constructibility is false.

The least $\omega$-Erdős cardinal is not weakly compact,^{p. 39.} nor is the least ${\omega }_1$-Erdős cardinal.^{p. 39}

If $\kappa$ is $\alpha$-Erdős, then it is $\alpha$-Erdős in every transitive model satisfying "$\alpha$ is countable."

Dodd's Notion of Erdős Cardinals

For a limit ordinal $\alpha$, a cardinal $\kappa$ is less often called $\alpha$-Erdős if for every closed unbounded $C\subseteq \kappa$ and every function $f:[C{]}^{<\omega }\to \kappa$ such that $f(x)<min(x)$ for all $x\in [C{]}^{<\omega }$, there is a set $H\subseteq C$ of order-type $\alpha$ that is homogeneous for $f$.^{p. 138.}

An equivalent definition is that $\kappa$ is $\alpha$-Erdős if for any $A\subseteq \kappa$, there is a set $I$ of order-type $\alpha$ of order-indiscernibles for the structure $(L_{\kappa }[A];\in ,A)$ such that:

• for every $\beta \in I$, $(L_{\beta }[A];\in ,A)\prec (L_{\kappa }[A];\in ,A)$, and
• for every $\gamma <\kappa$, the set $I\setminus \gamma$ forms a set of order-indiscernibles for the structure $(L_{\kappa }[A];\in ,A,\xi {)}_{\xi <\gamma }$.

The least cardinal $\kappa$ to satisfy the partition relation $\kappa \to (\alpha {)}^{<\omega }$ is still $\alpha$-Erdős under this definition. Every $\omega$-Erdős cardinal is an inaccessible limit of ineffable cardinals.

See also

• List of large cardinal properties

References

• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite book

Citations

Template:Settheory-stub