Erdős cardinal

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).

The Erdős cardinal κ(α) is defined to be the least cardinal such that for every function  f : κ^{< ω} → {0, 1}, there is a set of order type α that is homogeneous for  f  (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal κ(α) is the smallest cardinal such that

κ(α) → (α)^{< ω}

Existence of zero sharp implies that the constructible universe L satisfies "for every countable ordinal α, there is an α-Erdős cardinal". In fact, for every indiscernible κ, L_κ satisfies "for every ordinal α, there is an α-Erdős cardinal in Coll(ω, α) (the Levy collapse to make α countable)".

However, existence of an ω₁-Erdős cardinal implies existence of zero sharp. If  f  is the satisfaction relation for L (using ordinal parameters), then existence of zero sharp is equivalent to there being an ω₁-Erdős ordinal with respect to  f . And this in turn, the zero sharp implies the falsity of axiom of constructibility, of Kurt Gödel.

If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable".

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 528659 
• 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: 111–131. doi:10.1007/BF02023868. ISSN 0001-5954. MR 0095124 

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.