FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math
The List of All Lists
Post new topic   Reply to topic Page 2 of 2 [20 Posts] View previous topic :: View next topic
Goto page:  Previous  1, 2
Author Message
Mike Kelly
science forum Guru Wannabe


Joined: 30 Mar 2006
Posts: 119

PostPosted: Fri Jul 21, 2006 9:13 am    Post subject: Re: The List of All Lists Reply with quote

Russell Easterly wrote:
Quote:
"Patricia Shanahan" <pats@acm.org> wrote in message
news:y5Zvg.2660$157.889@newsread3.news.pas.earthlink.net...
Russell Easterly wrote:
I will prove there exists a largest "effective" natural number.
This is related to the Well Ordering the Reals thread,
but I think you will find this proof unique.

The list of all lists contains a largest "effective" natural number.
Assume I am given a list of unique lists of natural numbers, A.
By unique, I mean each list in A differs from every other list in A.

Given two lists, A_i and A_j, we know these
lists are unique and differ at some finite position, k.
Define the effective number, k=F(i,j), as the
first position where list A_i differs from list A_j.

I can now find the largest effective number for a given j.
For each j define maxj=max( F(i,j) for i=1 to j-1 )
as the largest effective number for all i less than j.

Define maxn as the largest effective number.
For each j, if maxj > maxn, let maxn = maxj.

Even if every list in A is infinite, the largest effective
number, maxn, will be finite. We can distinguish any two
sets by only looking at the first maxn elements in
the two lists.

Suppose A_1 is N, and A_i, for every i>1, is N-{(i-2)}.

A_1 = {0, 1, 2, 3, ... }
A_2 = {1, 2, 3, 4, ... }
A_3 = {0, 2, 3, 4, ... }
A_4 = {0, 1, 3, 4, ... }
etc.

What is maxn?

The last one.
You haven't defined which one that is, but there must be one
if every list differs from every other list.

Why?

--
mike.
Back to top
Patricia Shanahan
science forum Guru Wannabe


Joined: 13 May 2005
Posts: 214

PostPosted: Fri Jul 21, 2006 10:10 am    Post subject: Re: The List of All Lists Reply with quote

Russell Easterly wrote:
....
Quote:
I proved in another thread that any non-empty proper
subset of the set of all natural numbers that is transitive
and well ordered must have a largest element.
This means any inductive set that doesn't contain
omega is finite.
....


Did you ever prove, rather than merely assume, that there is any
non-empty proper subset of the natural numbers that is transitive and
well-ordered?

If so, I missed it and would welcome a reference to the article.

Patricia
Back to top
Patricia Shanahan
science forum Guru Wannabe


Joined: 13 May 2005
Posts: 214

PostPosted: Fri Jul 21, 2006 10:30 am    Post subject: Re: The List of All Lists Reply with quote

Russell Easterly wrote:
Quote:
"Patricia Shanahan" <pats@acm.org> wrote in message
news:y5Zvg.2660$157.889@newsread3.news.pas.earthlink.net...
Russell Easterly wrote:
I will prove there exists a largest "effective" natural number.
This is related to the Well Ordering the Reals thread,
but I think you will find this proof unique.

The list of all lists contains a largest "effective" natural number.
Assume I am given a list of unique lists of natural numbers, A.
By unique, I mean each list in A differs from every other list in A.

Given two lists, A_i and A_j, we know these
lists are unique and differ at some finite position, k.
Define the effective number, k=F(i,j), as the
first position where list A_i differs from list A_j.

I can now find the largest effective number for a given j.
For each j define maxj=max( F(i,j) for i=1 to j-1 )
as the largest effective number for all i less than j.

Define maxn as the largest effective number.
For each j, if maxj > maxn, let maxn = maxj.

Even if every list in A is infinite, the largest effective
number, maxn, will be finite. We can distinguish any two
sets by only looking at the first maxn elements in
the two lists.
Suppose A_1 is N, and A_i, for every i>1, is N-{(i-2)}.

A_1 = {0, 1, 2, 3, ... }
A_2 = {1, 2, 3, 4, ... }
A_3 = {0, 2, 3, 4, ... }
A_4 = {0, 1, 3, 4, ... }
etc.

What is maxn?

The last one.
You haven't defined which one that is, but there must be one
if every list differs from every other list.

Prove it. I didn't define which is the last one because there isn't one.

By the way, how are you getting on with MoeBlee's find-the-fallacy
challenge? I think the fallacy is a bit too subtle for you to have any
chance of finding it, but you may surprise me.

Patricia
Back to top
Patricia Shanahan
science forum Guru Wannabe


Joined: 13 May 2005
Posts: 214

PostPosted: Fri Jul 21, 2006 10:33 am    Post subject: Re: The List of All Lists Reply with quote

Patricia Shanahan wrote:
Quote:
Russell Easterly wrote:
...
I proved in another thread that any non-empty proper
subset of the set of all natural numbers that is transitive
and well ordered must have a largest element.
This means any inductive set that doesn't contain
omega is finite.
...

Did you ever prove, rather than merely assume, that there is any
non-empty proper subset of the natural numbers that is transitive and
well-ordered?

That should be "infinite non-empty proper subset".

There are, of course, plenty of finite non-empty proper subsets of the
natural numbers, which are transitive, well-ordered, and have a largest
element.

Quote:

If so, I missed it and would welcome a reference to the article.

Patricia
Back to top
Rupert
science forum Guru


Joined: 18 May 2005
Posts: 372

PostPosted: Fri Jul 21, 2006 10:38 am    Post subject: Re: The List of All Lists Reply with quote

Russell Easterly wrote:
Quote:
"Rupert" <rupertmccallum@yahoo.com> wrote in message
news:1153460812.700396.327670@b28g2000cwb.googlegroups.com...

Russell Easterly wrote:
"Rupert" <rupertmccallum@yahoo.com> wrote in message
news:1153459847.203837.56200@75g2000cwc.googlegroups.com...

Russell Easterly wrote:
"Patricia Shanahan" <pats@acm.org> wrote in message
news:y5Zvg.2660$157.889@newsread3.news.pas.earthlink.net...
Russell Easterly wrote:
I will prove there exists a largest "effective" natural number.
This is related to the Well Ordering the Reals thread,
but I think you will find this proof unique.

The list of all lists contains a largest "effective" natural
number.
Assume I am given a list of unique lists of natural numbers, A.
By unique, I mean each list in A differs from every other list in
A.

Given two lists, A_i and A_j, we know these
lists are unique and differ at some finite position, k.
Define the effective number, k=F(i,j), as the
first position where list A_i differs from list A_j.

I can now find the largest effective number for a given j.
For each j define maxj=max( F(i,j) for i=1 to j-1 )
as the largest effective number for all i less than j.

Define maxn as the largest effective number.
For each j, if maxj > maxn, let maxn = maxj.

Even if every list in A is infinite, the largest effective
number, maxn, will be finite. We can distinguish any two
sets by only looking at the first maxn elements in
the two lists.

Suppose A_1 is N, and A_i, for every i>1, is N-{(i-2)}.

A_1 = {0, 1, 2, 3, ... }
A_2 = {1, 2, 3, 4, ... }
A_3 = {0, 2, 3, 4, ... }
A_4 = {0, 1, 3, 4, ... }
etc.

What is maxn?

The last one.

There is no last one. She defined an A_i for every positive integer i.

Prove it.

She defined an A_i for every positive integer i.

There is no last positive integer because for every positive integer i,
i+1 is also a positive integer, showing that i is not the last positive
integer.

You can start by proving every natural number exists and is finite.

Tell me what axioms I am allowed to use, and define "finite".

Incidentally, I don't see why the burden of proof lies with me. She
challenged you to give the value of maxn. It's your job to substantiate
your claim that there is a last one.

Then you can prove there isn't a finite number of finite numbers.


Sure, I'll do that. Just tell me which axioms I'm allowed to use. Do
you want me to do it in ZF?

Sure, let's use ZF.
You start by defining the set of all natural numbers as the
intersection of all transfinite inductive sets.

I want to talk about this intersection.
I am assuming transfinite, inductive sets are transitive,
well ordered and that every inductive set contains
every natural number.

Some inductive sets also include non-empty
limit ordinals like omega.

Obviously, omega will be in the intersection
of two inductive sets that both contain omega.
Since the set of natural numbers doesn't contain
omega, there must be at least one inductive set
that doesn't contain omega.

I proved in another thread that any non-empty proper
subset of the set of all natural numbers that is transitive
and well ordered must have a largest element.

My apologies. This is actually correct.

Quote:
This means any inductive set that doesn't contain
omega is finite.


However, this does not follow. The set of natural numbers is an
inductive set which does not contain omega, and it is not finite.
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 2 of 2 [20 Posts] Goto page:  Previous  1, 2
View previous topic :: View next topic
The time now is Tue Dec 12, 2017 12:24 pm | All times are GMT
Forum index » Science and Technology » Math
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts Best fit orthogonal basis for list of vectors chengiz@my-deja.com num-analysis 4 Wed Jul 19, 2006 6:16 pm
No new posts help with lists operations omk Math 0 Sat Jul 08, 2006 12:17 pm
No new posts where is the newsgroup/mailing-list for optimization? gino Math 2 Thu Jul 06, 2006 4:57 pm
No new posts list of wheels kkrish Mechanics 0 Fri Jun 23, 2006 3:54 am
No new posts how to calculate/list all fit values john_woo@canada.com Math 4 Tue Jun 13, 2006 6:32 pm

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.0448s ][ Queries: 16 (0.0142s) ][ GZIP on - Debug on ]