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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
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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
allows talk about proper classes.

--
Daryl McCullough
Ithaca, NY



Date Subject Author
11/2/10
Read Re: Mathematics as a language
Aatu Koskensilta
11/3/10
Read Re: Mathematics as a language
herb z
11/3/10
Read Re: Mathematics as a language
Herman Jurjus
11/3/10
Read Re: Mathematics as a language
Marshall
11/3/10
Read Re: Mathematics as a language
Herman Jurjus
11/4/10
Read Re: Mathematics as a language
herb z
11/4/10
Read Re: Mathematics as a language
Marshall
11/5/10
Read Re: Mathematics as a language
herb z
11/5/10
Read Re: Mathematics as a language
Herman Jurjus
11/6/10
Read Re: Mathematics as a language
herb z
11/6/10
Read Re: Mathematics as a language
James Dolan
11/6/10
Read Re: Mathematics as a language
Tim Little
11/6/10
Read Re: Mathematics as a language
Daryl McCullough
11/6/10
Read Re: Mathematics as a language
Marshall
11/6/10
Read Re: Mathematics as a language
Brian Chandler
11/6/10
Read Re: Mathematics as a language
Tim Little
11/7/10
Read Re: Mathematics as a language
lwalke3@lausd.net
11/8/10
Read Re: Mathematics as a language
Brian Chandler
11/7/10
Read Re: Mathematics as a language
herb z
11/7/10
Read Re: Mathematics as a language
Daryl McCullough
11/8/10
Read Re: Mathematics as a language
herb z
11/3/10
Read Re: Mathematics as a language
lwalke3@lausd.net
11/3/10
Read Re: Mathematics as a language
Marshall
11/4/10
Read Re: Mathematics as a language
herb z
11/4/10
Read Re: Mathematics as a language
herb z
11/4/10
Read Re: Mathematics as a language
herb z
11/3/10
Read Re: Mathematics as a language
Daryl McCullough
11/4/10
Read Re: Mathematics as a language
Bill Taylor
11/4/10
Read Re: Mathematics as a language
Daryl McCullough
11/5/10
Read Re: Mathematics as a language
herb z
11/4/10
Read Re: Mathematics as a language
herb z
11/4/10
Read Re: Mathematics as a language
Daryl McCullough
11/5/10
Read Re: Mathematics as a language
herb z
11/5/10
Read Re: Mathematics as a language
Daryl McCullough
11/4/10
Read Re: Mathematics as a language
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