Confusion with finite sets vs an infinite set of natural numbers

[Boltzmannbrain]

9 minutes ago, studiot said:

This rather hinges on what property of infinity is being invoked.

What sort of mathematical object do you think n is ?

An n is a natural number.  According to the paper you posted, we can define a natural number when it belongs to a hereditary set that is defined when x+1 is an element of F and when x is an element of F, which also has 1.  At least that is how I am understanding it. 

7 minutes ago, Genady said:

By "they" I mean each set. So, I need to rephrase my question:

If you define each set as "starting at 1 and increasing by 1", then it is infinite.

But if you define each set as "starting at 1, increasing by 1, and stopping when the row number is reached", then it is finite.

What is your definition?

The first one.  But I need to clear something up.  Your pronoun "it" must refer to the amount of sets in the list, right?  I say this because clearly each set is finite as we look down the list of sets.  Right? 

[Genady]

5 minutes ago, Boltzmannbrain said:

Your pronoun "it" must refer to the amount of sets in the list, right?

No. It refers to each individual set.

 

6 minutes ago, Boltzmannbrain said:

clearly each set is finite as we look down the list of sets.  Right? 

No. A set defined as "starting at 1 and increasing by 1" is infinite.

[Boltzmannbrain]

13 minutes ago, Genady said:

No. It refers to each individual set.

 

No. A set defined as "starting at 1 and increasing by 1" is infinite.

I wanted to talk about the list of sets that start at 1 and increase by 1.   The same list that is in my OP.

Edited May 1, 2023 by Boltzmannbrain

[studiot]

14 minutes ago, Boltzmannbrain said:

An n is a natural number.  According to the paper you posted, we can define a natural number when it belongs to a hereditary set that is defined when x+1 is an element of F and when x is an element of F, which also has 1.  At least that is how I am understanding it. 

Not quite.

n must be the value of a function that takes on natural numbers ( or the positive integers if you prefer) as its values.

That is the only way n can be a different natural numbers in different lines in your OP list.th

 

You are correct about (x+1) also called successor property of the natural numbers.
It is this property that invokes infinity and leads to the countable infinity property of the natural numbers.

Infinity has many properties, but the one we want in this case is that it cannot be reached.

The ancient Greeks called this the potential infinity because it is never actually realised.

3 minutes ago, Boltzmannbrain said:

I wanted to talk about the list of sets that start at 1 and increase by 1.   The same list that is in my OP.

This is imprecise and inaccurate.

Please try to be correct.

 

The list does not have a numerical value or increase by one.

Nor does a set have a numerical value.

 

I know what you mean (and your meaning is OK, but it isn't what your words are saying) Until  you  can state it correctly yourself I doubt if anyone can help you understand.

[Genady]

8 minutes ago, Boltzmannbrain said:

I wanted to talk about the list of sets that start at 1 and increase by 1.   The same list that is in my OP.

The sets shown in your OP don't fit the definition "start at 1 and increase by 1."

They fit the definition "start at 1, increase by 1, and stop at the row number."

[Boltzmannbrain]

19 minutes ago, studiot said:

Not quite.

n must be the value of a function that takes on natural numbers ( or the positive integers if you prefer) as its values.

That is the only way n can be a different natural numbers in different lines in your OP list.th

 

You are correct about (x+1) also called successor property of the natural numbers.
It is this property that invokes infinity and leads to the countable infinity property of the natural numbers.

Infinity has many properties, but the one we want in this case is that it cannot be reached.

The ancient Greeks called this the potential infinity because it is never actually realised.

 

Okay, I was thinking about this potential infinity vs infinity.  This might be where I am misleading myself.  Can we not talk about infinity in its totality?  For example, is it self-contradictory to say something like "all elements of N" or "every real number from 1 to 2"? 

I know I have read those kinds of terms get used, but maybe they cannot be used formally, or can they?  

 

Quote

 

This is imprecise and inaccurate.

Please try to be correct.

 

The list does not have a numerical value or increase by one.

Nor does a set have a numerical value.

 

I know what you mean (and your meaning is OK, but it isn't what your words are saying) Until  you  can state it correctly yourself I doubt if anyone can help you understand.

 

 

I don't know why I keep resorting to my own terms instead of using the proper terms.  I will try harder to use the proper terms in the future.  And if I don't know the correct term, I will say so.

 

2 hours ago, Genady said:

The sets shown in your OP don't fit the definition "start at 1 and increase by 1."

They fit the definition "start at 1, increase by 1, and stop at the row number."

Sorry, like I told @studiot I will try harder to use the correct terms in the future.

[Boltzmannbrain]

7 hours ago, Genady said:

The sets shown in your OP don't fit the definition "start at 1 and increase by 1."

They fit the definition "start at 1, increase by 1, and stop at the row number."

I wanted the rows to continue indefinitely.  I forgot to put more ellipsis under the n for indefinite rows in the OP.

With that said, I still do not see how my issue of the OP is resolved.

[Genady]

7 minutes ago, Boltzmannbrain said:

I wanted the rows to continue indefinitely.  I forgot to put more ellipsis under the n for indefinite rows in the OP.

With that said, I still do not see how my issue of the OP is resolved.

Let me describe how I understand your construction.

 

There is a list of numbered rows, one row for each natural number.

Each row has a set of natural numbers, which contains natural numbers between 1 and the row number (including).

 

Is this description correct?

 

Putting it more formally:

 

You construct a one-to-one (injective) map from set of natural numbers N to set of sets of natural numbers such that each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}.

 

Correct?

 

[studiot]

As I see your argument in plain English it is this:-

 

Taking you list of sets from post#1

Set the indexing line counter aside for the moment as it is not really needed.

Consider the set which contains every set on your list.

If such  a set exists, call it W .

The listing of W then appears as in your list of sets without the indexing.

IF you go on long enough why do you not arrive at the set {1,2,3,4...., (n-1), n (n++1)...}, why of course is N ?

Of course N is also the indexing set we have ignored up to now.

Note also that all the sets up to N are finite, but N itself is transfinite.

 

I mentioned Russel's Paradox which queries the existence of W.

This was one of the earliest expositions of many paradoxes that appeared around the 1890s to do with the size of sets.

Quote

Cantor's paradox - Wikipedia

In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.

This paradox is named for Georg Cantor, who is often credited with first identifying it in 1899 (or between 1895 and 1897). Like a number of "paradoxes" it is not actually contradictory but merely indicative of a mistaken intuition, in this case about the nature of infinity and the notion of a set. Put another way, it is paradoxical within the confines of naïve set theory and therefore demonstrates that a careless axiomatization of this theory is inconsistent.

Hints of these difficulties go right back to the Ancient Greeks and Zeno in particular,
although they did not have more modern set theory to place the questions in.

A proper course of study into the whys and wherefores of these matters takes more than a year so most folks don't attempt it but look for a quick fix explanation.

My offering to you is to consider the Greek approach, where they realised that there is more than one infinity.
They distinguished two types of infinity viz potential and actual infinity.

They believed that there are no instances of actual infinity, which we observe as for instance, the count of numbers between 1 and 2.

But their potential infinity does not exist' either for a different reason.

It does not exist because no finite process can ever get there.

In other words the process does not terminate or goes on forever.

Which is what I am suggesting is the reason why your list will never arrive at N.

 

Cantor's approach considering magnitudes has run into difficulties why has yet to be fully resolved.
There are at least three different mathematical/logical schemes to try to achieve this. 
After your year and more of study you would find that none are totally satisfactory as they all wrestle with the idea that some sets are just too big to be contained in other sets.

 

Hopefully you can now sleep happy at night.

 

 

[joigus]

20 hours ago, Boltzmannbrain said:

I don't agree.  The more I look into all of this the more strange and complicated it is.  

That should be your first clue. Usually, the more I look into anything, the less strange and complicated it seems.

And that's how it should be. Don't you think?

[Boltzmannbrain]

12 hours ago, Genady said:

Let me describe how I understand your construction.

 

There is a list of numbered rows, one row for each natural number.

Each row has a set of natural numbers, which contains natural numbers between 1 and the row number (including).

 

Is this description correct?

 

Yes

 

Quote

 

Putting it more formally:

 

You construct a one-to-one (injective) map from set of natural numbers N to set of sets of natural numbers such that each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}.

 

Correct?

 

 

Yes I believe that's correct, except you added one small difference from your other description.  You want the sets to go inside of a set.  I guess that's fine because I can't see it changing my original description. 

[Genady]

33 minutes ago, Boltzmannbrain said:

You want the sets to go inside of a set.

This is needed because mapping connects elements of two sets. In your case, each element of the domain set ("from") is a natural number, and each element of the codomain set ("to") is a set of natural numbers.

Very well. So, following the definition,

"each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}",

please define the issue that bothers you.

[Boltzmannbrain]

5 hours ago, studiot said:

As I see your argument in plain English it is this:-

 

Taking you list of sets from post#1

Set the indexing line counter aside for the moment as it is not really needed.

Consider the set which contains every set on your list.

If such  a set exists, call it W .

The listing of W then appears as in your list of sets without the indexing.

 

I understand up until here.

 

Quote

IF you go on long enough why do you not arrive at the set {1,2,3,4...., (n-1), n (n++1)...}, why of course is N ?

 

I do not understand what you are saying here. 

 

Quote

 

Of course N is also the indexing set we have ignored up to now.

Note also that all the sets up to N are finite, but N itself is transfinite.

 

 

Yes, I believe that this is a part of the heart of the problem.  

 

Quote

 

I mentioned Russel's Paradox which queries the existence of W.

This was one of the earliest expositions of many paradoxes that appeared around the 1890s to do with the size of sets.

 

 

I understand Russel's paradox, but I not see how my issue is related.

 

Quote

 

Hints of these difficulties go right back to the Ancient Greeks and Zeno in particular,
although they did not have more modern set theory to place the questions in.

A proper course of study into the whys and wherefores of these matters takes more than a year so most folks don't attempt it but look for a quick fix explanation.

My offering to you is to consider the Greek approach, where they realised that there is more than one infinity.
They distinguished two types of infinity viz potential and actual infinity.

They believed that there are no instances of actual infinity, which we observe as for instance, the count of numbers between 1 and 2.

But their potential infinity does not exist' either for a different reason.

It does not exist because no finite process can ever get there.

In other words the process does not terminate or goes on forever.

Which is what I am suggesting is the reason why your list will never arrive at N.

 

 

Okay, I think I understand what you are saying, but can't N exhaust N?  If so, then my list of N rows, exhausts/creates a set with N elements, or vice versa.  The point being, that the 2 N's exhaust each other.  Or am I going off on a tangent?   

  

Quote

 

Cantor's approach considering magnitudes has run into difficulties why has yet to be fully resolved.
There are at least three different mathematical/logical schemes to try to achieve this. 
After your year and more of study you would find that none are totally satisfactory as they all wrestle with the idea that some sets are just too big to be contained in other sets.

 

 

Are you saying that my OP illustrates a problem in mainstream math that has yet to be resolved, or is my particular problem resolvable with mainstream math?

 

Quote

Hopefully you can now sleep happy at night.

 

That would be nice.

3 hours ago, joigus said:

That should be your first clue. Usually, the more I look into anything, the less strange and complicated it seems.

And that's how it should be. Don't you think?

Clue to what?

36 minutes ago, Genady said:

This is needed because mapping connects elements of two sets. In your case, each element of the domain set ("from") is a natural number, and each element of the codomain set ("to") is a set of natural numbers.

 

Oh thanks I did not know that.  +1

 

Quote

 

Very well. So, following the definition,

"each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}",

please define the issue that bothers you.

 

 

Actually, I do see a problem with your definition.  n limits the list of sets to always be finite.  We wanted an infinite list.  We wanted every n in N.

[Genady]

1 hour ago, Boltzmannbrain said:

n limits the list of sets to always be finite

No, it does not. What makes you think it does?

It says, "each n∈N", doesn't it?

Here is the definition again:

"each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}"

There is no limit on n.

[Boltzmannbrain]

29 minutes ago, Genady said:

No, it does not. What makes you think it does?

It says, "each n∈N", doesn't it?

Here is the definition again:

"each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}"

There is no limit on n.

Each n in N is finite.  Doesn't this imply that each set must be finite too?

Your even made a proof showing that each n is not sufficient to list every set.  But I don't agree with your proof anyways since it didn't work for the analogous list I made way back at the top of page 2 of this thread.

[Genady]

11 minutes ago, Boltzmannbrain said:

Each n in N is finite.  Doesn't this imply that each set must be finite too?

Yes, each set R(n) is finite.

You said something else in the previous post:

2 hours ago, Boltzmannbrain said:

the list of sets to always be finite

This is incorrect. The list of sets is not finite.

In other words, for each n,

the set R(n) is finite.

But

the set {R(n) | n∈N} is not finite.

Edited May 2, 2023 by Genady
typo

[Boltzmannbrain]

56 minutes ago, Genady said:

Yes, each set R(n) is finite.

You said something else in the previous post:

This is incorrect. The list of sets is not finite.

In other words, for each n,

the set R(n) is finite.

But

the set {R(n) | n∈N} is not finite.

 

Given your definition, each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}  do you agree that for each n∈N there is only a finite number of rows?

If not, here is what is in my head,

 

n = 1 implies 1 row

n = 2 implies 2 rows

n = 3 implies 3 rows

n = 4 implies 4 rows

n = 5 implies 5 rows

.

.

.

any finite n implies a finite number of rows

Every n is finite implies there can only ever be a finite number of rows.

This is my issue.

[Genady]

25 minutes ago, Boltzmannbrain said:

 

Given your definition, each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}  do you agree that for each n∈N there is only a finite number of rows?

If not, here is what is in my head,

 

n = 1 implies 1 row

n = 2 implies 2 rows

n = 3 implies 3 rows

n = 4 implies 4 rows

n = 5 implies 5 rows

.

.

.

any finite n implies a finite number of rows

Every n is finite implies there can only ever be a finite number of rows.

This is my issue.

No, for each n there is one and only one row, the row number n. This row has nothing to do with other rows.

For each n there is one set, R(n). The list is a set of these sets. Let's call it LIST.

This set, LIST is defined so that for each n∈N the set R(n)∈LIST, and for each element Q∈LIST there exists n∈N such that Q=R(n).

There is no "implies" anywhere in the definitions.

[Boltzmannbrain]

42 minutes ago, Genady said:

No, for each n there is one and only one row, the row number n. This row has nothing to do with other rows.

For each n there is one set, R(n). The list is a set of these sets. Let's call it LIST.

This set, LIST is defined so that for each n∈N the set R(n)∈LIST, and for each element Q∈LIST there exists n∈N such that Q=R(n).

There is no "implies" anywhere in the definitions.

I used "implies" in the way that we would say, If n =5, then there are 5 rows/sets, or equivalently, n = 5 implies 5 rows/sets.  I don't think I am doing anything wrong here.

Furthermore, n is the input, right?  That means that the number of sets is the output.  There is no infinite number that n can be that allows an output of infinite sets. 

Doesn't the proportionality of

n = 1 = 1 set

n = 2 = 2 sets

n = 3 = 3 sets

n = 4 = 4 sets

.

.

.

n = infinity = infinite sets

Why not this?  But I suppose this has been thought of already.

But looking at this list, doesn't it seem wrong that the left side is always finite but the right side is not always finite?

 

I see your argument too.  We want infinite sets such that every n in N is mapped to a set.  But then my problem gets switched to there being an n that is infinite, which is not allowed either.

 

Edited May 2, 2023 by Boltzmannbrain

[Genady]

7 minutes ago, Boltzmannbrain said:

If n =5, then there are 5 rows/sets ...

However, if n=5 there are no 5 rows/sets, but one.

If you want to discuss a different mapping, then define it first.

[Boltzmannbrain]

1 hour ago, Genady said:

However, if n=5 there are no 5 rows/sets, but one.

If you want to discuss a different mapping, then define it first.

Oh oops, sorry.  But this does not change anything about the point I am trying to make.  Let's still use your definition, each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n}

 

n = 1 ---> 1 element

n = 2 ---> 2 elements

n = 3 ---> 3 elements

n = 4 ---> 4 elements

.

.

.

n is always finite ---> finite elements

n = infinity ---> infinite elements (of course this last example is not permitted, but it is the only thing that makes sense to me at the moment.)

Shouldn't the n value stay proportional to the number of elements? 

What n in N gives a set with no end?

Edited May 2, 2023 by Boltzmannbrain

[Genady]

5 minutes ago, Boltzmannbrain said:

n = infinity ---> infinite elements (of course this last example is not permitted, but it is the only thing that makes sense to me at the moment.)

This last example not only is not permitted, but it does not have any meaning in the set of natural numbers. Infinity is not an element of this set. This example does not make sense.

 

7 minutes ago, Boltzmannbrain said:

Shouldn't the n value stay proportional to the number of elements?

Yes, each R(n) has n elements.

 

8 minutes ago, Boltzmannbrain said:

What n in N gives a set with no end?

None. Each R(n) is finite. 

[pzkpfw]

30 minutes ago, Boltzmannbrain said:

...

n is always finite ---> finite elements

n = infinity ---> infinite elements (of course this last example is not permitted, but it is the only thing that makes sense to me at the moment.)

Shouldn't the n value stay proportional to the number of elements? 

What n in N gives a set with no end?

 

Just a few posts ago you wrote (my bold) "I forgot to put more ellipsis under the n for indefinite rows in the OP."

This is what I tried to show with:

1 { 1 }

2 { 1, 2 }

... { 1, 2, ... }

There is no single finite n in N that gives a set with no end. The list ( { 1, 2, ... } ) is infinite, and the row number is also infinite.

Try thinking of the list ( { 1, 2, ... } ) as X on a graph and the list of lists as the Y. It's unbounded on both axis.

[Boltzmannbrain]

26 minutes ago, Genady said:

This last example not only is not permitted, but it does not have any meaning in the set of natural numbers. Infinity is not an element of this set. This example does not make sense.

 

I know.  I was trying to make a point.

 

Quote

Yes, each R(n) has n elements.

 

I agree.

 

Quote

None. Each R(n) is finite.

 

But you wrote, "the set {R(n) | n∈N} is not finite" a few posts ago.

[Genady]

1 minute ago, Boltzmannbrain said:

But you wrote, "the set {R(n) | n∈N} is not finite" a few posts ago.

Correct. R(n) is finite. {R(n) | n∈N} is not.

R(n) = {1, 2, 3, and all other numbers up to n}

LIST = {R(n) | n∈N} = {R(1), R(2), R(3), and all other R(n)'s}