weakly compact cardinal

Weakly compact cardinals are (large) infiniteMathworldPlanetmath cardinals which have a property related to the syntactic compactness theorem for first order logic. Specifically, for any infinite cardinal κ, consider the languagePlanetmathPlanetmath Lκ,κ.

This language is identical to first logic except that:

The weak compactness theorem for Lκ,κ states that if Δ is a set of sentencesMathworldPlanetmath of Lκ,κ such that |Δ|=κ and any θΔ with |θ|<κ is consistent then Δ is consistent.

A cardinal is weakly compact if the weak compactness theorem holds for Lκ,κ.

Title weakly compact cardinal
Canonical name WeaklyCompactCardinal
Date of creation 2013-03-22 12:50:53
Last modified on 2013-03-22 12:50:53
Owner Henry (455)
Last modified by Henry (455)
Numerical id 5
Author Henry (455)
Entry type Definition
Classification msc 03E10
Synonym weakly compact
Related topic CardinalNumber
Defines weakly compact cardinal
Defines weak compactness theorem