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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:

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.
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 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
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" 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
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 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
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" 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.
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: 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.

OK

 Quote: 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.

OK, although why you call this the "effective" number I don't follow.

 Quote: 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.

OK.

 Quote: Define maxn as the largest effective number.

What if there isn't a largest effective number?

 Quote: For each j, if maxj > maxn, let maxn = maxj.

What if there is no largest maxj?

 Quote: Even if every list in A is infinite, the largest effective number, maxn, will be finite.

How can something that doesn't exist be finite?

 Quote: We can distinguish any two sets by only looking at the first maxn elements in the two lists.

What about two lists that differ only after maxn+1 elements?

--
mike.
Han de Bruijn
science forum Guru

Joined: 18 May 2005
Posts: 1285

Posted: Fri Jul 21, 2006 8:56 am    Post subject: Re: The List of All Lists

Virgil wrote:

 Quote: Note that in ZF and NBG there is no set of all sets.

Oh no! They found out that there is a _class_ of all sets. How smart!

Han de Bruijn
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Fri Jul 21, 2006 8:08 am    Post subject: Re: The List of All Lists

In article <qI6dnTZq4ZPZ7F3ZnZ2dnUVZ_oydnZ2d@comcast.com>,
"Russell Easterly" <logiclab@comcast.net> wrote:

 Quote: "Rupert" wrote in message news:1153460812.700396.327670@b28g2000cwb.googlegroups.com... 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.

What do YOU mean by an inductive set? The usual definition of containing
the empty set and being closed under successor (if x is a member so is
(x union {x}) ) does not necessitate transistivity nor well-ordering by
membership, nor by inclusion.

For example if one starts with {0,{0,2}, {0,2,4}, ...} and forms the
closure under successorship, the result will not be either transitive or
well ordered.
 Quote: 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.

Do you mean that omega is to be a subset of or a member of the two
sets?

 Quote: Since the set of natural numbers doesn't contain omega, there must be at least one inductive set that doesn't contain omega.

It is called the set of natural numbers, or the set of finite ordinals.

 Quote: Either the set of all natural numbers contains omega or the set of all natural numbers is finite.

Russell manages to mess up in every post.

The set of all naturals numbers IS omega as an ordinal, at least n the
von Neumann sense.
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Fri Jul 21, 2006 7:49 am    Post subject: Re: The List of All Lists

In article <RdmdnW68ZMYk-l3ZnZ2dnUVZ_uudnZ2d@comcast.com>,
"Russell Easterly" <logiclab@comcast.net> wrote:

 Quote: "Rupert" wrote in message news:1153459847.203837.56200@75g2000cwc.googlegroups.com... Russell Easterly wrote: "Patricia Shanahan" 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. You can start by proving every natural number exists and is finite. Then you can prove there isn't a finite number of finite numbers.

In ZF and NBG this has already been proved. Multiply proved.

And even in ZF-I it cannot be disproved.

If Russell claims it false, he must provide the axiom system in which it
is false.

 Quote: There exist two non-equal lists that don't differ at any finite position?

There is Patricia Shanahan's list of lists one list whose set of first
differences from other lists is larger than any finite cardinality .
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Fri Jul 21, 2006 7:42 am    Post subject: Re: The List of All Lists

In article <hoednf-FmJua-13ZnZ2dnUVZ_vCdnZ2d@comcast.com>,
"Russell Easterly" <logiclab@comcast.net> wrote:

 Quote: "Patricia Shanahan" 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.

Not in ZF.

Not in NBG.

So what set theory is Russell looking at now?
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Fri Jul 21, 2006 7:41 am    Post subject: Re: The List of All Lists

In article <laednSNOfMXQy13ZnZ2dnUVZ_ridnZ2d@comcast.com>,
"Russell Easterly" <logiclab@comcast.net> wrote:

 Quote: 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.

In what system are you assuming that a list of all lists exists?

And what is your definition of a list"

The standard definition is that a list is a function whose domain is a
countable ordinal no larger than N.
Note that in ZF and NBG there is no set of all sets.

 Quote: 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.

Not necessarily. There may not be one.

Consider the following:

A_0 = N = { 0 ,1, 2, ...},
and for i > 0, A_i is the list of elements of N in normal order except
omitting i-1.

Then for j = 0 there is no largest effective number, k(F(0,i) = i-1 for
all i > 0, so has no max.

So Russell is wrong again, as usual.
Rupert
science forum Guru

Joined: 18 May 2005
Posts: 372

Posted: Fri Jul 21, 2006 6:24 am    Post subject: Re: The List of All Lists

Russell Easterly wrote:
 Quote: "Rupert" wrote in message news:1153460812.700396.327670@b28g2000cwb.googlegroups.com... Russell Easterly wrote: "Rupert" wrote in message news:1153459847.203837.56200@75g2000cwc.googlegroups.com... Russell Easterly wrote: "Patricia Shanahan" 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.

All inductive sets are transfinite, and the definition of "transfinite"
usually comes a bit later in the game, so I'd prefer to leave
"transfinite" out.

They're not. The intersection of all of them is.

 Quote: well ordered

Well-ordered by what relation?

 Quote: and that every inductive set contains every natural number.

Yep, fine.

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

Yes.

 Quote: 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.

Yes.

 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

Well-ordered by what? The membership relation?

 Quote: must have a largest element.

This is false. Point me to the proof and I'll find the error for you.

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

No inductive set is finite, and there exist inductive sets not
containing omega.

 Quote: I proved that any subset of N that contains all but one element of N is not transitive.

Yes, that can be proved.

 Quote: A_0 = {1,2,3,...} A_1 = {0,1,2,...} A_2 = {0,1,3,...} ... Either the set of all natural numbers contains omega or the set of all natural numbers is finite.

No.

 Quote: Russell - 2 many 2 count
Russell Easterly
science forum Guru Wannabe

Joined: 27 Jun 2005
Posts: 199

Posted: Fri Jul 21, 2006 6:16 am    Post subject: Re: The List of All Lists

"Rupert" <rupertmccallum@yahoo.com> wrote in message
 Quote: Russell Easterly wrote: "Rupert" wrote in message news:1153459847.203837.56200@75g2000cwc.googlegroups.com... Russell Easterly wrote: "Patricia Shanahan" 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 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.
This means any inductive set that doesn't contain
omega is finite.

I proved that any subset of N that contains all but
one element of N is not transitive.

A_0 = {1,2,3,...}
A_1 = {0,1,2,...}
A_2 = {0,1,3,...}
....

Either the set of all natural numbers contains omega
or the set of all natural numbers is finite.

Russell
- 2 many 2 count
Rupert
science forum Guru

Joined: 18 May 2005
Posts: 372

Posted: Fri Jul 21, 2006 5:46 am    Post subject: Re: The List of All Lists

Russell Easterly wrote:
 Quote: "Rupert" wrote in message news:1153459847.203837.56200@75g2000cwc.googlegroups.com... Russell Easterly wrote: "Patricia Shanahan" 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.

 Quote: 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.

 Quote: 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?

 Quote: You haven't defined which one that is, but there must be one if every list differs from every other list. False. There exist two non-equal lists that don't differ at any finite position?

No.

You claimed: if every list differs from every other list, there must be
a last list. If we take the contrapositive, we get: if there is no last
list, there exist two lists on the list which are the same. I claim
there is no last list, but I also claim that your conditional is false.
So I don't claim that there exist two lists on the list which are the
same; rather, I claim the opposite.

 Quote: Russell - 2 many 2 count
Russell Easterly
science forum Guru Wannabe

Joined: 27 Jun 2005
Posts: 199

Posted: Fri Jul 21, 2006 5:35 am    Post subject: Re: The List of All Lists

"Rupert" <rupertmccallum@yahoo.com> wrote in message
 Quote: Russell Easterly wrote: "Patricia Shanahan" 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.
You can start by proving every natural number exists and is finite.
Then you can prove there isn't a finite number of finite numbers.

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

There exist two non-equal lists that don't differ at any finite position?

Russell
- 2 many 2 count

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