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Topic: Mathematics as a language
Replies: 35   Last Post: Nov 8, 2010 1:53 AM

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 Daryl McCullough Posts: 2,871 Registered: 1/29/05
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 well-ordered contradicts
>> the existence of a well-ordering of the universe of sets. Both are
>> consistent, but they can't both exist.

>
>You mean if the universe of sets is well-ordered, then every set is
>well-ordered?

Yes. To say that the universe is well-ordered implies that every set
is well-ordered, 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 well-ordering of a collection C if
it is a total ordering and for every subcollection C' of C, if C'
is non-empty, then C' has a minimal element, under the ordering R.

A well-ordering 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 well-ordered" can't be formulated, but
it can in an extension such as Morse-Kelley set theory that

--
Daryl McCullough
Ithaca, NY

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