Trying to understand an apparent inconsistency (sets and subsets of natural numbers)

[Mates]

I was thinking about sets and subsets of natural numbers, infinite and finite.

In the case of a finite set S containing only consecutive natural numbers, including 1, there will be an n that is also the number of elements in S. And conversely, the number of elements in S will be equal to some n, namely its greatest element. So if S has 5 elements, that means there must be the element 5 in the set. That is the set S = {1, 2, 3, 4, 5}.

Now here is where the consistency breaks down.

Since each n is finite (which it has to be by definition) in the set N, then using the logic above, there must only ever be a finite number of elements in N. But we know there has to be an infinite number of n in N.

I don't know if this is truly inconsistent, but it just seems inconsistent to me.

 

[JeffM]

If by the set N, we mean the set of all natural numbers, it is infinite, meaning without an upper bound.

First, If by n you mean a natural number and by S a set of n consecutive natural numbers, it simply is not true that n is necessarily in S. For example, if n = 3 and S = {6, 7, 8}, S has n consecutive elements, but n is not an element of S.

Second, I cannot see the logical inconsistency at all.

Let F be ANY finite set of natural numbers (whether consecutive or not). Then the number of elements in the set EXACTLY equals some natural number m. That is what we mean definitionally by saying F is finite. Let's further say that m > 2. Let's label the elements of F as [imath]\{e_1, \ ... \ e_m\}[/imath]. Now consider the number [imath]p = \displaystyle \prod_{k=1}^m e_k[/imath]. Now consider the set G containing all the elements of F and also p. This is a different set than F (because it contains p which is not equal to any element of F), and it has more elements than F. Therefore, no finite set contains all the natural numbers.

 

[Mates]

If by the set N, we mean the set of all natural numbers, it is infinite, meaning without an upper bound.

First, If by n you mean a natural number and by S a set of n consecutive natural numbers, it simply is not true that n is necessarily in S. For example, if n = 3 and S = {6, 7, 8}, S has n consecutive elements, but n is not an element of S.

I said that S has to include 1.

Let F be ANY finite set of natural numbers (whether consecutive or not). Then the number of elements in the set EXACTLY equals some natural number m. That is what we mean definitionally by saying F is finite. Let's further say that m > 2. Let's label the elements of F as [imath]\{e_1, \ ... \ e_m\}[/imath]. Now consider the number [imath]p = \displaystyle \prod_{k=1}^m e_k[/imath]. Now consider the set G containing all the elements of F and also p. This is a different set than F (because it contains p which is not equal to any element of F), and it has more elements than F. Therefore, no finite set contains all the natural numbers.

It's not that I see inconsistency for N to have infinite elements, it's more about there not being an infinite n, which I am aware goes against definition and other proofs.

The argument that I put in the OP appears to show that there should be an n infinitely big n if it is going to be consistent with the number of elements being infinite.

 

[NotJeffM]

Yes, you did say that the consecutive numbers must include 1. Sorry that I missed that.

So your argument is that if a set S consists of consecutive natural numbers starting from 1, then an element of S represents the cardinality of S. That can be shown to be true if S is a finite set. But it is obviously false if S is an infinite set because infinity is not a natural number. All you have done is to identify a difference between finite sets of natural numbers and infinite sets of natural numbers. There is absolutely no reason to expect an infinite set to have properties that are identical to those of a finite set.

 

[Mates]

Yes, you did say that the consecutive numbers must include 1. Sorry that I missed that.

Are you JeffM?

So your argument is that if a set S consists of consecutive natural numbers starting from 1, then an element of S represents the cardinality of S. That can be shown to be true if S is a finite set. But it is obviously false if S is an infinite set because infinity is not a natural number. All you have done is to identify a difference between finite sets of natural numbers and infinite sets of natural numbers. There is absolutely no reason to expect an infinite set to have properties that are identical to those of a finite set.

I understand what you are saying, but it doesn't resolve the inconsistency as we go from finite to infinite. It seems as though the n values cannot "match" an infinite number of elements.

Put in a different way, let's say we want to fill S to equal N. Starting from 1, we add a 1 to the greatest element to make the next element. In order to have infinite elements in S, doesn't there have to be an n of an infinite sum of 1's?

I hope I am not frustrating you or anyone else. I am just really interested in exploring this these issues that I have. I always try my best to understand every reply.

 

[NotJeffM]

Yes, I am JeffM. My credentials got scrambled on my Ipad so I had to pick a new name for using my Ipad.

I know I am repeating myself. Your argument assumes that infinite sets always act like finite sets. Sometimes they do and sometimes they do not, and your example identifies such a difference.

Here is another example. The finite set of all even natural numbers less than 21 has a cardinality that is less than the cardinality of the finite set of all natural numbers less than 21. But the cardinality of the set of all even natural numbers is the same as the cardinality of the set of all natural numbers. You simply cannot extrapolate the nature of the infinite from the nature of the finite.

 

[Mates]

Yes, I am JeffM. My credentials got scrambled on my Ipad so I had to pick a new name for using my Ipad.

I know I am repeating myself. Your argument assumes that infinite sets always act like finite sets. Sometimes they do and sometimes they do not, and your example identifies such a difference.

Here is another example. The finite set of all even natural numbers less than 21 has a cardinality that is less than the cardinality of the finite set of all natural numbers less than 21. But the cardinality of the set of all even natural numbers is the same as the cardinality of the set of all natural numbers. You simply cannot extrapolate the nature of the infinite from the nature of the finite.

I am trying to use what you say here to resolve my issue, but I can not make the connection.

I will try to put this issue in a different form that I cannot even fathom how it can be resolved.

Let's consider the set N. In it are only finite natural numbers. This would seem to mean that a person could count to any of them in a finite amount of time. However, a person could not count all of the numbers in N in a finite amount of time. This seems to directly mean that there are natural numbers that are infinite.

There must be something crucial that I am misunderstanding.

 

[pka]

Let's consider the set N. In it are only finite natural numbers. This would seem to mean that a person could count to any of them in a finite amount of time. However, "a person could not count all of the numbers in N in a finite amount of time. This seems to directly mean that there are natural numbers that are infinite.

Lets go sentence by sentence.
1) You write "Let's consider the set N." By that I assume you mean [imath]{\bf N}=\{0,1,2,3,4,\cdots\}\text{ the set of }{\bf{\large~COUNTING}~ numbers}[/imath]
2) you write: "In it are only finite natural numbers" well of course: natural numbers are counting numbers!

3) The set [imath]{\bf N}[/imath] is an infinite set that is not bounded above. Do you know what it means to say in an ordered field a set is not bounded?
For [imath](\forall n\in{\bf N})(\exists j\in{\bf N})[n<j][/imath] Another way to say that: for any number in [imath]{\bf N}[/imath] there is a greater number in [imath]{\bf N}[/imath].

4) "a person could not count all of the numbers in N" Why would anyone who is working at this level write that?
Consider the set of powers of two: [imath]\mathscr{T}=\left\{2^n : n\in \bf N\right\}[/imath].
Now define [imath]f:{\bf N}\mapsto\mathscr{T}[/imath] by [imath]f(n)=2^n[/imath] It is relativity easy to show that [imath]f[/imath] is bijective.
In a few statements I proved that [imath]\mathscr{T}[/imath] is infinite. Moreover. I counted [imath]\mathscr{T}[/imath].

[imath][/imath][imath][/imath][imath][/imath]

 

[Mates]

2) you write: "In it are only finite natural numbers" well of course: natural numbers are counting numbers!

I meant that each number is finitely long. For example, none of them can go on forever like pi. However you still can count using infinite numbers like the set: {sq2, sq3, ....

3) The set [imath]{\bf N}[/imath] is an infinite set that is not bounded above. Do you know what it means to say in an ordered field a set is not bounded?
For [imath](\forall n\in{\bf N})(\exists j\in{\bf N})[n<j][/imath] Another way to say that: for any number in [imath]{\bf N}[/imath] there is a greater number in [imath]{\bf N}[/imath].

Yes, I understand this.

4) "a person could not count all of the numbers in N" Why would anyone who is working at this level write that?

I meant that a person (or Turing machine) would never be able to finish counting each number in N starting from 1. If it takes one second per number to count, the time it would take to finish would not end.

Consider the set of powers of two: [imath]\mathscr{T}=\left\{2^n : n\in \bf N\right\}[/imath].
Now define [imath]f:{\bf N}\mapsto\mathscr{T}[/imath] by [imath]f(n)=2^n[/imath] It is relativity easy to show that [imath]f[/imath] is bijective.
In a few statements I proved that [imath]\mathscr{T}[/imath] is infinite. Moreover. I counted [imath]\mathscr{T}[/imath].

[imath][/imath][imath][/imath][imath][/imath]

I do not understand how this resolves my issue.

 

[NotJeffM]

I shall try one last time. Not because I am annoyed but because I do not seem to be helping.

A finite set has a number of elements that correspond to a natural number. Whatever infinity may be, it is not a natural number as defined by the Peano axioms. Therefore, the number of elements in an infinite set is not a natural number. It seems to me that what you see as an inconsistency (namely that finite sets and infinite sets have certain differences) is exactly what should be expected. In short, your problem is not logical but rather psychological: infinity is weird.

 

[NotJeffM]

Let's consider the set N. In it are only finite natural numbers.

Correct.

This would seem to mean that a person could count to any of them in a finite amount of time.

Correct. Actually, what you are saying is you could count in finite time any finite set.

However, a person could not count all of the numbers in N in a finite amount of time.

Correct.

This seems to directly mean that there are natural numbers that are infinite.

Not at all. You keep relying on the incorrect idea that ANY set of sequential natural numbers, starting with one, contains the number that describes the set’s own cardinality. That proposition can be proved true for finite sets of that type. Have you tried to prove it for the infinite set of that type? How did you define natural number in that proof?

There must be something crucial that I am misunderstanding.

Yes. You are relying on a “theorem” that is false. You could of course try to define a natural number that has the properties of countable infinity, but then you would not mean by natural number what anyone else means by natural number. You would then need to prove what is the arithmetic of natural numbers based on the new definition. You should note that an important part of the standard definition involves the successor function. That is out for your new number.

 

[Mates]

Correct.

Correct. Actually, what you are saying is you could count in finite time any finite set.

Correct.

Not at all. You keep relying on the incorrect idea that ANY set of sequential natural numbers, starting with one, contains the number that describes the set’s own cardinality. That proposition can be proved true for finite sets of that type. Have you tried to prove it for the infinite set of that type? How did you define natural number in that proof?

Yes. You are relying on a “theorem” that is false. You could of course try to define a natural number that has the properties of countable infinity, but then you would not mean by natural number what anyone else means by natural number. You would then need to prove what is the arithmetic of natural numbers based on the new definition. You should note that an important part of the standard definition involves the successor function. That is out for your new number.

Thank you for this reply. I made this diagram to make things as clear as possible. I have two questions.

Is there a difference between infinity and "goes to infinity"?

Even though there is no greatest element in this infinite set S, elements are going to infinity nevertheless. So the question is, how can the "size" (I do not know the correct term there) of n never end but also be finite?

 

[NotJeffM]

When push comes to shove, I am a finitist, meaning that, for me, discussions of infinity and real numbers are an intellectual game. We can have no experiential intuition about such things. They are creatures of the human imagination like mandrake roots, unicorns, honest government, etc. So, if you want to think about such things, you need to be prepared for things that are not intuitive.

Natural numbers are defined as having successors, meaning another natural number that is exactly 1 greater. Natural numbers also have a least member, zero or one depending on which definition is picked, which is not itself a successor. Infinity does not have a successor in that sense, and is not a successor to any natural number.

Technically, “going to infinity” is not even defined. What is actually meant is the idea that something is true for all natural numbers greater than some specified natural number. (PKA may be able to give you a more technically exact definition.) But it is self-evidently true that there is no natural number greater than all natural numbers. Suppose there was a greatest natural number g Then 1 + g is a natural number greater than the greatest natural number g. Nonsense. So the ”number” of elements in the set of ALL natural numbers cannot be a natural number.

What you are doing is imagining a last natural number, which would indeed number the set of natural numbers. But there is no last natural number.

 

[Mates]

When push comes to shove, I am a finitist, meaning that, for me, discussions of infinity and real numbers are an intellectual game. We can have no experiential intuition about such things. They are creatures of the human imagination like mandrake roots, unicorns, honest government, etc. So, if you want to think about such things, you need to be prepared for things that are not intuitive.

I am trying to be as open as I can. After years of obsessing about infinity, I have not heard much about it that surprises me anymore. Having said that, now I would like to actually understand better by knowing how I am deviating from "mainstream" mathematics.

Natural numbers are defined as having successors, meaning another natural number that is exactly 1 greater. Natural numbers also have a least member, zero or one depending on which definition is picked, which is not itself a successor. Infinity does not have a successor in that sense, and is not a successor to any natural number.

Technically, “going to infinity” is not even defined. What is actually meant is the idea that something is true for all natural numbers greater than some specified natural number. (PKA may be able to give you a more technically exact definition.) But it is self-evidently true that there is no natural number greater than all natural numbers. Suppose there was a greatest natural number g Then 1 + g is a natural number greater than the greatest natural number g. Nonsense. So the ”number” of elements in the set of ALL natural numbers cannot be a natural number.

What you are doing is imagining a last natural number, which would indeed number the set of natural numbers. But there is no last natural number.

I am just putting together a few principles of math, as I understand them, and giving you one of their conclusions. I am not trying to find a last number or anything new like that. An infinitely long natural number - or whatever you want to call it - seems to be where my mistaken principle/s are taking me. I just want to know exactly where I am going wrong.

 

[NotJeffM]

I am trying to be as open as I can. After years of obsessing about infinity, I have not heard much about it that surprises me anymore. Having said that, now I would like to actually understand better by knowing how I am deviating from "mainstream" mathematics.




I am just putting together a few principles of math, as I understand them, and giving you one of their conclusions. I am not trying to find a last number or anything new like that. An infinitely long natural number - or whatever you want to call it - seems to be where my mistaken principle/s are taking me. I just want to know exactly where I am going wrong.

What is a long number?

And what principles are you using?

 

[Mates]

What is a long number?

I put "infinitely long". It would look like a natural number except with no end, something like pi without the decimal 314......
But this is definitely not what I want to focus on, unless you think it will help me.

And what principles are you using?

These are probably more like definitions, but these are what I was referring to:

1. The total elements in the set of natural numbers N are infinite
2. Each element is finite
3. Every increasing element from the set N added to some empty set S is a greater number by at least 1

My conclusion:

1. An infinite number of elements added to S creates a number that increases by 1 an infinite number of times.

 

[NotJeffM]

I put "infinitely long". It would look like a natural number except with no end, something like pi without the decimal 314......
But this is definitely not what I want to focus on, unless you think it will help me.

It will definitely not help you. What you are imagining is a representation of a number rather than a number. Assuming that representation in meaningful, can you prove that the represented number is a natural number? What attributes does the number represented have and are they the same as the attributes of a natural number?

These are probably more like definitions, but these are what I was referring to:

1. The total elements in the set of natural numbers N are infinite

Let’s start by being a bit more exact.

The total number of elements in the set of all natural numbers (N) is infinite.

Or to avoid getting confused by different senses of the word “number,”

The cardinality of the set of all natural numbers (N) is infinity.

2. Each element is finite

A natural number is not infinite.

Alternatively, infinity is not a natural number.

3. Every increasing element from the set N added to some empty set S is a greater number by at least 1

This is gobbledygook.

The elements in N are specific whole numbers like three. Is three an increasing number or a decreasing number? Can you give an example of an increasing natural number so I can try to translate what you are saying into something meaningful.

”some empty set” implies there are multiple empty sets. What is the feature that distinguishes one empty set from another?

Let’s not worry about that too much. We will assume that there is at least one set that is empty (S) and work with S.

Assuming three is one of the increasing numbers in N, are you really saying that by transferring it from N to S, three becomes some number greater than three such as four or five.

Perhaps you are trying to say something like

If every element in set U is an element in set V and at least one element in set V is not an element in set U, then the cardinality of set V is greater than the cardinality of set U.

My conclusion:

1. An infinite number of elements added to S creates a number that increases by 1 an infinite number of times.

Is what you mean [imath]\displaystyle \sum_{k=1}^{\infty} 1 = \infty[/imath]

Is that your inconsistency?

What is it inconsistent with?

 

[Mates]

Assuming three is one of the increasing numbers in N, are you really saying that by transferring it from N to S, three becomes some number greater than three such as four or five.

No, the 3 is still in S. So, 3+1 (for n =1) is added to the elements 1, 2, 3 that already exist in S. Then 4+n goes into S etc.

Perhaps you are trying to say something like

If every element in set U is an element in set V and at least one element in set V is not an element in set U, then the cardinality of set V is greater than the cardinality of set U.

No, not really. I want to show it a specific way. I want N to fill S with increasing natural numbers.

Is what you mean [imath]\displaystyle \sum_{k=1}^{\infty} 1 = \infty[/imath]

Is that your inconsistency?

What is it inconsistent with?

Yes, that is it. Infinity would have to be in the set S.

 

[NotJeffM]

No, the 3 is still in S. So, 3+1 (for n =1) is added to the elements 1, 2, 3 that already exist in S. Then 4+n goes into S etc.

You do understand that a set is defined by its elements so talking about a set having different elements from itself makes absolutely no sense. If n = 1, doesn’t S have just the single element of one.

No, not really. I want to show it a specific way. I want N to fill S with increasing natural numbers.

S cannot have different elements. That is why I talked about different sets U and V. You could posit a sequence of sets such as S_1, S_2, where each set contains as elements the natural numbers, starting with one through the index of S. Notice that every one of those sets contains a finite number of elements.

Yes, that is it. Infinity would have to be in the set S.

You keep saying that. Prove it.

 

[Mates]

You do understand that a set is defined by its elements so talking about a set having different elements from itself makes absolutely no sense. If n = 1, doesn’t S have just the single element of one.

S cannot have different elements. That is why I talked about different sets U and V. You could posit a sequence of sets such as S_1, S_2, where each set contains as elements the natural numbers, starting with one through the index of S.

Ok, I didn't know that.

Notice that every one of those sets contains a finite number of elements.

Thanks, that makes sense. I am close to dropping this issue.

You keep saying that. Prove it.

Here is just one last idea that your summation equation made me think of. This is also my attempted proof.


Assumptions:

- Every natural number in increasing order is greater than the previous number by 1.

- There is an infinite number of natural numbers.

Conclusion 1:

- 1 is being added to a previous natural number an infinite number of times.

Final Conclusion:

- There must be an infinite natural number


That is all I got. If you find something wrong with that (and I understand you), I will finally be at peace with all this.