Karl Jones at 11:13, 25 April 2016

2016-04-25T11:13:46Z

Karl Jones at 16:31, 5 February 2016

2016-02-05T16:31:25Z

Karl Jones: Created page with "In the mathematical field of set theory, a '''large cardinal property''' is a certain kind of property of transfinite cardinal numbers. (TO DO: o..."

2015-09-06T15:50:59Z

Created page with "In the mathematical field of set theory, a '''large cardinal property''' is a certain kind of property of transfinite cardinal numbers. (TO DO: o..."

New page

In the [[Mathematics|mathematical field]] of [[set theory]], a '''large cardinal property''' is a certain kind of [[property]] of [[transfinite cardinal numbers]].

(TO DO: organize, cross-ref.)

== Description ==

Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα).

The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results.

In other words, they can be seen, in [[Dana Scott]]'s phrase, as quantifying the fact "that if you want more you have to assume more".

There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated.

Whether this is simply a linguistic convention, or something more, is a controversial point among distinct philosophical schools.

A large cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.

Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with ZFC.

These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via [[Gödel's]] [[second incompleteness theorem]]) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent).

== No generally agreed definition ==

There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those in the List of large cardinal properties are large cardinal properties.

== See also ==

* [[Cardinal number]]
* [[Mathematics]]
* [[Property]]
* [[Set theory]]
* [[Transfinite cardinal numbers]]

== External links ==

* [https://en.wikipedia.org/wiki/Large_cardinal Large cardinal] @ Wikipedia

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	In the [[Mathematics\|mathematical field]] of [[set theory]], a '''large cardinal property''' is a certain kind of [[property]] of [[transfinite cardinal numbers]].		In the [[Mathematics\|mathematical field]] of [[set theory]], a '''large cardinal property''' is a certain kind of [[property]] of [[transfinite cardinal numbers]].
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Large cardinal - Revision history

Karl Jones at 11:13, 25 April 2016

Karl Jones at 16:31, 5 February 2016

Karl Jones: Created page with "In the mathematical field of set theory, a '''large cardinal property''' is a certain kind of property of transfinite cardinal numbers. (TO DO: o..."