Confusion with finite sets vs an infinite set of natural numbers

[Boltzmannbrain]

For all n rows, this would be a list of all sets of natural numbers that increase by 1 starting from 1.

 

1 {1}

2 {1, 2}

3 {1, 2, 3}

4 {1, 2, 3, 4}

.

.

.

n

Every set listed here would have to be finite since every natural number is finite.

But if every possible set of increasing natural numbers (that increase by 1 starting from 1) is here, then how can the set of all natural numbers N be infinite?

Edited April 27, 2023 by Boltzmannbrain

[studiot]

  On 4/27/2023 at 2:03 AM, Boltzmannbrain said:

For all n rows, this would be a list of all sets of natural numbers that increase by 1 starting from 1.

 

1 {1}

2 {1, 2}

3 {1, 2, 3}

4 {1, 2, 3, 4}

.

.

.

n

Every set listed here would have to be finite since every natural number is finite.

But if every possible set of increasing natural numbers (that increase by 1 starting from 1) is here, then how can the set of all natural numbers N be infinite?

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Didn't you have a very long thread about this once before ?

I thought we had cleared up your misunderstanding but you are making the same mistake again.

There is no 'n' at the end of the list

 

An infinite list does not terminate,, by definition.

 

But trying to write an n at the end implies that there is a definite last number, n.

[Boltzmannbrain]

  On 4/27/2023 at 10:21 AM, studiot said:

Didn't you have a very long thread about this once before ?

I thought we had cleared up your misunderstanding but you are making the same mistake again.

There is no 'n' at the end of the list

 

An infinite list does not terminate,, by definition.

 

But trying to write an n at the end implies that there is a definite last number, n.

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Why would putting an n there imply a last number???  That's just common notation.  If you want to help me, just read what I put, and tell me exactly what I said that is false.

 

And trying to say this thread is the same as the other is just garbage.  They both are clearly different threads.

[studiot]

  On 4/27/2023 at 3:58 PM, Boltzmannbrain said:

Why would putting an n there imply a last number???  That's just common notation.  If you want to help me, just read what I put, and tell me exactly what I said that is false.

 

And trying to say this thread is the same as the other is just garbage.  They both are clearly different threads.

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And a nice pleasant discussion to you to.

 

Since the list has no end,  the convention if you are going to use 'n' , is to place it between two ellipsis thus

 

1

2

3

...

n

...

 

 

The elli[psis is used mathematically as a symbol to mean 'indefinite continuation' .

 

Normally the list would be displayed horizontally, but vertically is OK so I have followed your lead in presenting it this way.

 

 

[Boltzmannbrain]

  On 4/27/2023 at 5:16 PM, studiot said:

 

And a nice pleasant discussion to you to.

 

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I get defensive when posters try to make me look like an idiot.  What you said (unless it's true of course) is not a good way to start a thread if you want things to be pleasant.  Anyway let's move on.

 

  Quote

 

Since the list has no end,  the convention if you are going to use 'n' , is to place it between two ellipsis thus

 

1

2

3

...

n

...

 

 

The elli[psis is used mathematically as a symbol to mean 'indefinite continuation' .

 

Normally the list would be displayed horizontally, but vertically is OK so I have followed your lead in presenting it this way.

 

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Ok, you're right.  I should have put another ellipses.  +1 

[studiot]

  On 4/27/2023 at 2:03 AM, Boltzmannbrain said:

Every set listed here would have to be finite since every natural number is finite.

But if every possible set of increasing natural numbers (that increase by 1 starting from 1) is here, then how can the set of all natural numbers N be infinite?

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I did actually answer your question since your list was an incomplete representation of the stated infinite set N.

Of course N does not and can not appear on your list since N itself is not a natural number.

To put it another way the problem is confusing a set, N which is not finite, with its elements (natural numbers)  which are all finite.

[Boltzmannbrain]

  On 4/27/2023 at 7:01 PM, studiot said:

 

I did actually answer your question since your list was an incomplete representation of the stated infinite set N.

Of course N does not and can not appear on your list since N itself is not a natural number.

To put it another way the problem is confusing a set, N which is not finite, with its elements (natural numbers)  which are all finite.

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Then what did I say that was wrong (except for forgetting the ellipsis)? 

 

I made 2 premises and a conclusion (the conclusion is in the form of a question).  Please tell me which of the 3, or if all, are incorrect.  

If there is a contradiction here, I don't think it would be so much that the set of natural numbers has to finite (because it obviously can't be by its very own nature of never ending), but rather every n is not finite.  

[Genady]

  On 4/27/2023 at 2:03 AM, Boltzmannbrain said:

every possible set of increasing natural numbers (that increase by 1 starting from 1) is here

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This is wrong, i.e., not every possible set of increasing natural numbers (that increase by 1 starting from 1) is there.

The set {1, 2, 3, 4, ...} is a possible set of increasing natural numbers (that increase by 1 starting from 1), and it is not there.

[Boltzmannbrain]

  On 4/27/2023 at 8:47 PM, Genady said:

This is wrong, i.e., not every possible set of increasing natural numbers (that increase by 1 starting from 1) is there.

The set {1, 2, 3, 4, ...} is a possible set of increasing natural numbers (that increase by 1 starting from 1), and it is not there.

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Okay, I am taking this to mean that the set of all natural numbers is not there.  This is good.  But I don't understand why it isn't there.  

Maybe you say this because I forgot the extra vertical ellipsis after the n?  

[Genady]

  On 4/27/2023 at 8:56 PM, Boltzmannbrain said:

Okay, I am taking this to mean that the set of all natural numbers is not there.  This is good.  But I don't understand why it isn't there.  

Maybe you say this because I forgot the extra vertical ellipsis after the n?  

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I don't know what you mean in the second statement, but here is a proof that this set is not there.

Let's define the set:

S = {x| x∈N & x≥1}

Let's assume that the set S is in the list. Then there is a line, l, in the list with this set on it.

The set on line l is:

L = {x| x∈N & x≥1 & x≤l}

But 

L ≠ S

which contradicts the assumption.

Thus, S in not in the list.

[Boltzmannbrain]

  On 4/27/2023 at 9:11 PM, Genady said:

I don't know what you mean in the second statement, but here is a proof that this set is not there.

Let's define the set:

S = {x| x∈N & x≥1}

Let's assume that the set S is in the list. Then there is a line, l, in the list with this set on it.

The set on line l is:

L = {x| x∈N & x≥1 & x≤l}

But 

L ≠ S

which contradicts the assumption.

Thus, S in not in the list.

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I am lost.  It looks like S is the set of natural numbers, or S = N.  Then the part that confuses me is that the set L has x < or = 1 in it.  Where is that coming from?

 

Note: I forgot to put another downward ellipsis under the n in the OP.

Edited April 28, 2023 by Boltzmannbrain

[Genady]

  On 4/27/2023 at 11:59 PM, Boltzmannbrain said:

I am lost.  It looks like S is the set of natural numbers, or S = N.  Then the part that confuses me is that the set L has x < or = 1 in it.  Where is that coming from?

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It is not 1, it is l (the letter):

L = {x| x∈N & x≥1 & x≤l}

[Boltzmannbrain]

  On 4/28/2023 at 12:04 AM, Genady said:

It is not 1, it is l (the letter):

L = {x| x∈N & x≥1 & x≤l}

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Right.  But I am still confused.  Why does this set L have to be a line in S?  And why doesn't S = L?

[Genady]

  On 4/28/2023 at 12:26 AM, Boltzmannbrain said:

Right.  But I am still confused.  Why does this set L have to be a line in S?  And why doesn't S = L?

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L is not a line in S. L is a set on a line number l in your list.

If this is clear, I'll go to the second question.

[Boltzmannbrain]

  On 4/28/2023 at 12:30 AM, Genady said:

L is not a line in S. L is a set on a line number l in your list.

If this is clear, I'll go to the second question.

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Okay, so l is a sequence of digits like 1, 2, 3, ...  and L is the set of those digits?

[Genady]

  On 4/28/2023 at 12:36 AM, Boltzmannbrain said:

Okay, so l is a sequence of digits like 1, 2, 3, ...  and L is the set of those digits?

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No, l is not a sequence. It is a number. It is a number of some line in your list. We don't know which line it is, thus we call it l.

[Boltzmannbrain]

  On 4/28/2023 at 12:38 AM, Genady said:

No, l is not a sequence. It is a number. It is a number of some line in your list. We don't know which line it is, thus we call it l.

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Okay, I understand.  And now why does L have to be in S?  And why doesn't S = L?

[Genady]

  On 4/28/2023 at 12:43 AM, Boltzmannbrain said:

Okay, I understand.  And now why does L have to be in S?  And why doesn't S = L?

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L does not have to be in S, and it not in S. L is a set on a line in your list.

Each line in your list has a set. L is one of these sets, the one on the line number l.

[Boltzmannbrain]

  On 4/28/2023 at 12:46 AM, Genady said:

L does not have to be in S, and it not in S. L is a set on a line in your list.

Each line in your list has a set. L is one of these sets, the one on the line number l.

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Just so I am clear, I will give an example.

Let's say the line l = 5.  

This would be shown in the list as

5 {1, 2, 3, 4, 5}

Then there is a set L "on it" (I put this in quotes because I assuming what it means here).

L can be something like {2, 3, 4} (and maybe in other words L is a subset of the numbers on line 5?)

Is this the idea so far?

[Genady]

  On 4/28/2023 at 1:00 AM, Boltzmannbrain said:

 

Just so I am clear, I will give an example.

Let's say the line l = 5.  

This would be shown in the list as

5 {1, 2, 3, 4, 5}

Then there is a set L "on it" (I put this in quotes because I assuming what it means here).

L can be something like {2, 3, 4} (and maybe in other words L is a subset of the numbers on line 5?)

Is this the idea so far?

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The last two lines above are wrong.

The set on the line 5 in your list is {1, 2, 3, 4, 5}. This is the set L for l=5, i.e.,

L = {1, 2, 3, 4, 5}.

[pzkpfw]

Boltzmannbrain, now knowing n in your OP is not meant to itself be infinity ("Note: I forgot to put another downward ellipsis under the n in the OP."), then its line would be:

n {1, 2, 3, 4, ..., n-1, n }

n is not infinity, the set does not have infinite numbers in it. If n = 1000, then there are 1000 numbers in the set.

Happy so far?

Now add 1 to n. And again, and again. Each time you add another number to the set.

1000 { 1, 2, 3, ..., 999, 1000 }

1001 { 1, 2, 3, ..., 1000, 1001 }

1002 { 1, 2, 3, ..., 1001, 1002 }

When/why do you stop? Is there a last n?

Edited April 28, 2023 by pzkpfw

[Genady]

If we will continue the discussion of my proof that the set S is not in the list, the following clarification of the list might help:

l=1 L={1}

l=2 L={1, 2}

l=3 L={1, 2, 3}

l=4 L={1, 2, 3, 4}

...

Or, equivalently:

l=1, L = {x| x∈N & x≥1 & x≤1}

l=2, L = {x| x∈N & x≥1 & x≤2}

l=3, L = {x| x∈N & x≥1 & x≤3}

l=4, L = {x| x∈N & x≥1 & x≤4}

...

[Boltzmannbrain]

  On 4/28/2023 at 2:33 AM, Genady said:

If we will continue the discussion of my proof that the set S is not in the list, the following clarification of the list might help:

l=1 L={1}

l=2 L={1, 2}

l=3 L={1, 2, 3}

l=4 L={1, 2, 3, 4}

...

Or, equivalently:

l=1, L = {x| x∈N & x≥1 & x≤1}

l=2, L = {x| x∈N & x≥1 & x≤2}

l=3, L = {x| x∈N & x≥1 & x≤3}

l=4, L = {x| x∈N & x≥1 & x≤4}

...

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Yeah, I think I understand now. 

When we put something like 1, 2, 3,  ... n ... as n goes to infinity, is that the same as saying "for all n element of N"? 

How would have I wrote it if I wanted every n in the set of N to be assigned a row, or is this not possible?   +4

  On 4/28/2023 at 2:02 AM, pzkpfw said:

Boltzmannbrain, now knowing n in your OP is not meant to itself be infinity ("Note: I forgot to put another downward ellipsis under the n in the OP."), then its line would be:

n {1, 2, 3, 4, ..., n-1, n }

n is not infinity, the set does not have infinite numbers in it. If n = 1000, then there are 1000 numbers in the set.

Happy so far?

Now add 1 to n. And again, and again. Each time you add another number to the set.

1000 { 1, 2, 3, ..., 999, 1000 }

1001 { 1, 2, 3, ..., 1000, 1001 }

1002 { 1, 2, 3, ..., 1001, 1002 }

When/why do you stop? Is there a last n?

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I wanted the list to have all n of the set of all natural numbers N.

Edited April 28, 2023 by Boltzmannbrain

[Genady]

  On 4/28/2023 at 3:21 AM, Boltzmannbrain said:

When we put something like 1, 2, 3,  ... n ... as n goes to infinity, is that the same as saying "for all n element of N"?

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I wouldn't say in this case, "as n goes to infinity", because n doesn't "go" at all here, but rather is one arbitrary member of the set / sequence.

Also, the expression, "for all n element of N", doesn't make sense to me in this case.

  On 4/28/2023 at 3:21 AM, Boltzmannbrain said:

How would have I wrote it if I wanted every n in the set of N to be assigned a row

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One could write that there is mapping from the set N to set of rows such that every n in N is mapped to a row {1,2,3, ..., n}. See, e.g., Mapping | mathematics | Britannica.

[studiot]

  On 4/27/2023 at 6:51 PM, Boltzmannbrain said:

I get defensive when posters try to make me look like an idiot.  What you said (unless it's true of course) is not a good way to start a thread if you want things to be pleasant.  Anyway let's move on.

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I am sorry you felt like that since nothing could be further from the truth.

You are most definitely not an idiot since greater minds that yours or mine have been baffled by this question.

I have noticed since that I mistakenly assumed your set S to be a set of number when in fact I see now that you stated clearly a set of sets.

 

This brings us to the crux of greater minds since this is exactly the situation brought about by Russel's Paradox.

That is when you try to apply Cantor - ZFC naive set theory to infinite sets that cannot be members of themselves.

 

This is why Russell and Whitehead introduced the theory of types or classes, which is basically a reclassification of sets introducing a hierarchy of set types.

This also paved the way for 'orders of logic'  so ZFC is first order, infinite sets of sets is second order which is needed to correctly analyse  infinite first order sets of numbers.

In general you need a higher order of logic than the one you want to analyse and there are an unknown, perhaps indefinite or infinite, count of orders.

This situation lead, in turn, to Godel's famous theorems about the subject.

 

The best plain explanation of all this I have come across is put forwards in Hofstadter's award winning book

Godel Escher Back