
Re: Mathematics as a language
Posted:
Nov 5, 2010 7:35 AM


herbzet says...
>Daryl McCullough wrote:
>> Well, the existence of a set that cannot be wellordered contradicts >> the existence of a wellordering of the universe of sets. Both are >> consistent, but they can't both exist. > >You mean if the universe of sets is wellordered, then every set is >wellordered?
Yes. To say that the universe is wellordered implies that every set is wellordered, since every set is a subset of the universe.
>What does "a well ordering of the universe of sets" mean, anyway?
A binary relation R(x,y) is a wellordering of a collection C if it is a total ordering and for every subcollection C' of C, if C' is nonempty, then C' has a minimal element, under the ordering R.
A wellordering of the universe is equivalent to the existence of a way to index elements of the universe with ordinals.
In ZFC, you can't actually talk about proper classes, so the claim "The universe is wellordered" can't be formulated, but it can in an extension such as MorseKelley set theory that allows talk about proper classes.
 Daryl McCullough Ithaca, NY

