Erdos cardinal

☆ Save On Wikipedia ↗

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958).

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

κ → ( α ) < ω {\displaystyle \kappa \rightarrow (\alpha )^{<\omega }} {\displaystyle \kappa \rightarrow (\alpha )^{<\omega }}.

Under this definition, any cardinal larger than the least α {\displaystyle \alpha } {\displaystyle \alpha }-Erdős cardinal is α {\displaystyle \alpha } {\displaystyle \alpha }-Erdős.

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

However, the existence of an ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}}-Erdős cardinal implies existence of zero sharp. If f {\displaystyle f} {\displaystyle f} is the satisfaction relation for L {\displaystyle L} {\displaystyle L} (using ordinal parameters), then the existence of zero sharp is equivalent to there being an ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}}-Erdős ordinal with respect to f {\displaystyle f} {\displaystyle f}. Thus, the existence of an ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}}-Erdős cardinal implies that the axiom of constructibility is false.

The least ω {\displaystyle \omega } {\displaystyle \omega }-Erdős cardinal is not weakly compact,[1]p. 39. nor is the least ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}}-Erdős cardinal.[1]p. 39

If κ {\displaystyle \kappa } {\displaystyle \kappa } is α {\displaystyle \alpha } {\displaystyle \alpha }-Erdős, then it is α {\displaystyle \alpha } {\displaystyle \alpha }-Erdős in every transitive model satisfying " α {\displaystyle \alpha } {\displaystyle \alpha } is countable."

Dodd's notion of Erdős cardinals

For a limit ordinal α {\displaystyle \alpha } {\displaystyle \alpha }, a cardinal κ {\displaystyle \kappa } {\displaystyle \kappa } is less often called α {\displaystyle \alpha } {\displaystyle \alpha }-Erdős if for every closed unbounded C ⊆ κ {\displaystyle C\subseteq \kappa } {\displaystyle C\subseteq \kappa } and every function f : [ C ] < ω → κ {\displaystyle f:[C]^{<\omega }\rightarrow \kappa } {\displaystyle f:[C]^{<\omega }\rightarrow \kappa } such that f ( x ) < min ( x ) {\displaystyle f(x)<\min(x)} {\displaystyle f(x)<\min(x)} for all x ∈ [ C ] < ω {\displaystyle x\in [C]^{<\omega }} {\displaystyle x\in [C]^{<\omega }}, there is a set H ⊆ C {\displaystyle H\subseteq C} {\displaystyle H\subseteq C} of order-type α {\displaystyle \alpha } {\displaystyle \alpha } that is homogeneous for f {\displaystyle f} {\displaystyle f}.[2]p. 138.

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

  • for every β ∈ I {\displaystyle \beta \in I} {\displaystyle \beta \in I}, ( L β [ A ] ; ∈ , A ) ≺ ( L κ [ A ] ; ∈ , A ) {\displaystyle (L_{\beta }[A];\in ,A)\prec (L_{\kappa }[A];\in ,A)} {\displaystyle (L_{\beta }[A];\in ,A)\prec (L_{\kappa }[A];\in ,A)}, and
  • for every γ < κ {\displaystyle \gamma <\kappa } {\displaystyle \gamma <\kappa }, the set I ∖ γ {\displaystyle I\setminus \gamma } {\displaystyle I\setminus \gamma } forms a set of order-indiscernibles for the structure ( L κ [ A ] ; ∈ , A , ξ ) ξ < γ {\displaystyle (L_{\kappa }[A];\in ,A,\xi )_{\xi <\gamma }} {\displaystyle (L_{\kappa }[A];\in ,A,\xi )_{\xi <\gamma }}.

The least cardinal κ {\displaystyle \kappa } {\displaystyle \kappa } to satisfy the partition relation κ → ( α ) < ω {\displaystyle \kappa \rightarrow (\alpha )^{<\omega }} {\displaystyle \kappa \rightarrow (\alpha )^{<\omega }} is still α {\displaystyle \alpha } {\displaystyle \alpha }-Erdős under this definition. Every ω {\displaystyle \omega } {\displaystyle \omega }-Erdős cardinal is an inaccessible limit of ineffable cardinals.[3]

Strength relative to other large cardinals

For any ordinal α, α-Erdős cardinal is stronger than α-iterable cardinal, but weaker than α+1-iterable cardinal. virtually rank-into-rank cardinal is weaker than ω-Erdős cardinal. If α < ω1, then α-Erdős cardinal is weaker than Silver cardinal. ω1-Erdős cardinal is stronger than zero sharp.

See also

References

Citations

  1. F. Rowbottom, "Some strong axioms of infinity incompatible with the axiom of constructibility". Annals of Mathematical Logic vol. 3, no. 1 (1971).
  2. A. J. Dodd (1982), The Core Model. Cambridge University Press. ISBN 978-0-521-28530-8
  3. Wilson, Trevor M. (2019). "Weakly Remarkable Cardinals, Erdős Cardinals, and the Generic Vopěnka Principle". The Journal of Symbolic Logic. 84 (4): 1711–1721. arXiv:1807.02207. doi:10.1017/jsl.2018.76.