To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Erdős cardinal

From Wikipedia, the free encyclopedia

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, Lκ 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".

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.


This page was last edited on 1 September 2021, at 21:05
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.