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Mike Kelly science forum Guru Wannabe
Joined: 30 Mar 2006
Posts: 119

Posted: Fri Jul 21, 2006 9:13 am Post subject:
Re: The List of All Lists



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 j1 )
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{(i2)}.
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. 

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Patricia Shanahan science forum Guru Wannabe
Joined: 13 May 2005
Posts: 214

Posted: Fri Jul 21, 2006 10:10 am Post subject:
Re: The List of All Lists



Russell Easterly wrote:
....
Quote:  I proved in another thread that any nonempty 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
nonempty proper subset of the natural numbers that is transitive and
wellordered?
If so, I missed it and would welcome a reference to the article.
Patricia 

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Patricia Shanahan science forum Guru Wannabe
Joined: 13 May 2005
Posts: 214

Posted: Fri Jul 21, 2006 10:30 am Post subject:
Re: The List of All Lists



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 j1 )
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{(i2)}.
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 findthefallacy
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 

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Patricia Shanahan science forum Guru Wannabe
Joined: 13 May 2005
Posts: 214

Posted: Fri Jul 21, 2006 10:33 am Post subject:
Re: The List of All Lists



Patricia Shanahan wrote:
Quote:  Russell Easterly wrote:
...
I proved in another thread that any nonempty 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
nonempty proper subset of the natural numbers that is transitive and
wellordered?

That should be "infinite nonempty proper subset".
There are, of course, plenty of finite nonempty proper subsets of the
natural numbers, which are transitive, wellordered, and have a largest
element.
Quote: 
If so, I missed it and would welcome a reference to the article.
Patricia 


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Rupert science forum Guru
Joined: 18 May 2005
Posts: 372

Posted: Fri Jul 21, 2006 10:38 am Post subject:
Re: The List of All Lists



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 j1 )
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{(i2)}.
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 nonempty
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 nonempty 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. 

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