Vopěnka's principle

In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings.

Vopěnka's principle was first introduced by Petr Vopěnka and independently considered by Keisler, and was written up by Solovay, Reinhardt & Kanamori (1978). According to Pudlák (2013, p. 204), Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However before publishing his inconsistency proof he found a flaw in it.

Definition

Vopěnka's principle asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another. This cannot be stated as a single sentence of ZFC as it involves a quantification over classes. A cardinal κ is called a Vopěnka cardinal if it is inaccessible and Vopěnka's principle holds in the rank Vκ (allowing arbitrary SVκ as "classes"). [1]

Many equivalent formulations are possible. For example, Vopěnka's principle is equivalent to each of the following statements.

Strength

Even when restricted to predicates and proper classes definable in first order set theory, the principle implies existence of Σn correct extendible cardinals for every n.

If κ is an almost huge cardinal, then a strong form of Vopenka's principle holds in Vκ:

There is a κ-complete ultrafilter U such that for every {Ri: i < κ} where each Ri is a binary relation and RiVκ, there is S  U and a non-trivial elementary embedding j: RaRb for every a < b in S.

References

  1. 1 2 3 Kanamori, Akihiro (2003). The higher infinite : large cardinals in set theory from their beginnings (2nd ed.). Berlin [u.a.]: Springer. ISBN 9783540003847.
  2. 1 2 3 4 Rosicky, Jiří Adámek ; Jiří (1994). Locally presentable and accessible categories (Digital print. 2004. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 0521422612.
  3. Bagaria, Joan (23 December 2011). "C(n)-cardinals". Archive for Mathematical Logic 51 (3-4): 213–240. doi:10.1007/s00153-011-0261-8.

External links

Friedman, Harvey M. (2005), EMBEDDING AXIOMS  gives a number of equivalent definitions of Vopěnka's principle.


This article is issued from Wikipedia - version of the Tuesday, August 25, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.