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 is called -Erdős if for every function , there is a set of order type that is homogeneous for . In the notation of the partition calculus, is -Erdős if
![{\displaystyle f:[\kappa ]^{<\omega }\to \{0,1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25a4681fcbb9b0bc71e8b83a56d13c48a3bf6a3e)
- .
Under this definition, any cardinal larger than the least -Erdős cardinal is -Erdős.
The existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal , there is an -Erdős cardinal". In fact, for every indiscernible , satisfies "for every ordinal , there is an -Erdős cardinal in " (the Lévy collapse to make countable).
However, the existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for (using ordinal parameters), then the existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to . Thus, the existence of an -Erdős cardinal implies that the axiom of constructibility is false.
The least -Erdős cardinal is not weakly compact,[1]p. 39. nor is the least -Erdős cardinal.[1]p. 39
If is -Erdős, then it is -Erdős in every transitive model satisfying " is countable."
Dodd's Notion of Erdős Cardinals
For a limit ordinal , a cardinal is less often called -Erdős if for every closed unbounded and every function such that for all , there is a set of order-type that is homogeneous for .[2]p. 138.
![{\displaystyle f:[C]^{<\omega }\rightarrow \kappa }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94329901948a26da28d9d4ab4197276bfa4dcbb9)
![{\displaystyle x\in [C]^{<\omega }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a14d0b8810612a4653928f9748e7fc1d3a96acea)
An equivalent definition is that is -Erdős if for any , there is a set of order-type of order-indiscernibles for the structure such that:
![{\displaystyle (L_{\kappa }[A];\in ,A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0aba4afef7a2fed519da92f61cb2552b1c0d25)
- for every , , and
- for every , the set forms a set of order-indiscernibles for the structure .
![{\displaystyle (L_{\beta }[A];\in ,A)\prec (L_{\kappa }[A];\in ,A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43777de846a6582345e6ba2145cb01cb3f1527f9)
![{\displaystyle (L_{\kappa }[A];\in ,A,\xi )_{\xi <\gamma }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01855466d8b9e1eb6925f5ef77a29353ac96ef84)
The least cardinal to satisfy the partition relation is still -Erdős under this definition. Every -Erdős cardinal is an inaccessible limit of ineffable cardinals.[3]
See also
References
- Baumgartner, James E.; Galvin, Fred (1978). "Generalized Erdős cardinals and 0#". Annals of Mathematical Logic. 15 (3): 289–313. doi:10.1016/0003-4843(78)90012-8. ISSN 0003-4843. MR 0528659.
- Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
- Erdős, Paul; Hajnal, András (1958). "On the structure of set-mappings". Acta Mathematica Academiae Scientiarum Hungaricae. 9 (1–2): 111–131. doi:10.1007/BF02023868. ISSN 0001-5954. MR 0095124. S2CID 18976050.
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
Citations
- ^ a b F. Rowbottom, "Some strong axioms of infinity incompatible with the axiom of constructibility". Annals of Mathematical Logic vol. 3, no. 1 (1971).
- ^ A. J. Dodd (1982), The Core Model. Cambridge University Press. ISBN 978-0-521-28530-8
- ^ 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.