Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: Given a set , is there a disjoint set with an arbitrary cardinality?
Replies: 28   Last Post: Dec 4, 2012 5:50 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
jaakov

Posts: 11
Registered: 12/3/12
Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Posted: Dec 3, 2012 5:28 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 03.12.2012 17:51, Aatu Koskensilta wrote:
> jaakov<removeit_jaakov@deleteit_ro.ru> writes:
>

>> 1. k is not related to the cardinality of X.
>
> Yes, I misread your original question.
>

>> 2. Your lambda need not exist. To show that lambda exists, one has to
>> show that a set may not contain arbitrarily large ordinals. Is it a
>> known fact or simply false?

>
> For any set A, lambda = U{ alpha in A | alpha is an ordinal} is an
> upper bound on ordinals in A. Take the next k ordinals after lambda and
> you have your desired Y.
>

Thank you, that works. Let me just restate the proof.

Let lambda = sup { alpha in X | alpha is an ordinal}.

External theorem: lambda exists and is an ordinal.
Hope: its proof does not assume the axiom of regularity or set purity.

Let l be any ordinal greater than lambda.
Let Y = { l + a | a is an ordinal and a < k}.

We show:
1. card(Y) = k.
Consider f: k -> Y, a |-> l+a.
By definition of Y, f is onto. Since a1<a2 implies l+a1<l+a2,
f is one-to-one. Thus f is a bijection, so card(Y)=k.

2. Y is disjoint from X.
Assume some x in X and x in Y. By definition of Y there is
a<k such that x = l+a. Since x in X, l+a is an ordinal in X.
Thus l+a <= lambda. But lambda<l<=l+a. By transitivity lambda<lambda.
Contradiction!


Jaakov.


Date Subject Author
12/3/12
Read Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
forbisgaryg@gmail.com
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Aatu Koskensilta
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Carsten Schultz
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Aatu Koskensilta
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Aatu Koskensilta
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Carsten Schultz
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Aatu Koskensilta
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Butch Malahide
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/3/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Butch Malahide
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
forbisgaryg@gmail.com
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
William Elliot
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
forbisgaryg@gmail.com
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
William Elliot
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
William Elliot
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
William Elliot
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Shmuel (Seymour J.) Metz
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Spammer
12/4/12
Read Re: Given a set , is there a disjoint set with an arbitrary cardinality?
jaakov

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.