Erdős cardinal

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In mathematics, an Erdős cardinal (named after Paul Erdős) is a certain kind of large cardinal number.

Formally, a cardinal number κ which is 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, is called an α-Erdős cardinal.

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 generic 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.

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

[edit] References

• 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 0444105352. 

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3540003843.