Please help me with this apparent paradox: Does set of all naturals exist in {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...} ?

[Mates]

I was on another forum discussing an issue related to this one. I wanted to know if the set of natural numbers exists in the following set.

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

I was told that the set of all natural numbers is not in it. Okay fine, but then I asked if all natural numbers is in this set, and the poster said yes.

I would think that for every natural number there needs to be a set to go along with it. How can all natural numbers be there without being in one of the sets? Furthermore it is a bijection between each set and each natural number isn't it? If so, wouldn't that "break" the bijection?

 

[Dr.Peterson]

then I asked if all natural numbers is in this set, and the poster said yes.

You presumably mean to ask whether the set of all natural numbers is in this set, not whether every natural number is in the set.

Did you ask them to prove their claim? On what grounds do they say that?

I would think that for every natural number there needs to be a set to go along with it. How can all natural numbers be there without being in one of the sets? Furthermore it is a bijection between each set and each natural number isn't it? If so, wouldn't that "break" the bijection?

But your question, as stated, is not about a bijection. A number being in the set is not the same as the number corresponding to an element in the set.

 

[Mates]

You presumably mean to ask whether the set of all natural numbers is in this set, not whether every natural number is in the set.

No, that was my original question in the thread. The poster said that the set of natural numbers is not in the set of sets, but every natural number is ( as n goes to infinity).

But your question, as stated, is not about a bijection. A number being in the set is not the same as the number corresponding to an element in the set.

The bijection I am referring to is the correspondence between every natural number and every set.

 

[Dr.Peterson]

No, that was my original question in the thread.

Do you mean, that's what you asked in this thread, or in the other site? I just tried to correct your wording, which was unclear, by adding "set of".

The poster said that the set of natural numbers is not in the set of sets, but every natural number is ( as n goes to infinity).

Nonsense. Did he give a reason? Clearly 1 is not in the set, {1} is. And 2 is not in the set, but {1,2} is. Those are different things.

The bijection I am referring to is the correspondence between every natural number and every set.

I know that. But that's not involved in the question as stated. You asked about what elements are actually in the set {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}.

If you want help with your conversation with someone elsewhere, then you'll have to show us exactly what was said. We can't answer questions that have been paraphrased, or correct answers that have been paraphrased!

 

[Mates]

Do you mean, that's what you asked in this thread, or in the other site? I just tried to correct your wording, which was unclear, by adding "set of".

In the other thread I just wanted to know if there is set with all the natural numbers in it. The answer was no. Then I asked if all the natural numbers were in the set of sets, and the poster asnwered yes.

Nonsense. Did he give a reason? Clearly 1 is not in the set, {1} is. And 2 is not in the set, but {1,2} is. Those are different things.

I was told that all the natural numbers are contained in the set of sets. The poster said yes but not as elements.

I know that. But that's not involved in the question as stated. You asked about what elements are actually in the set {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}.

If you want help with your conversation with someone elsewhere, then you'll have to show us exactly what was said. We can't answer questions that have been paraphrased, or correct answers that have been paraphrased!

I actually only want to be clear on this regardless of what the poster said. My question to you is whether or not we can say that there are natural numbers contained in the set of sets.

If you really want to see our converstaion, I was talking to Whatkindoffred (my name is A_vat_in_the_brain) in Reddit here

https://www.reddit.com/r/math/comments/18xo0lq/_/kh6oa53

 

[Dr.Peterson]

In the other thread I just wanted to know if there is set with all the natural numbers in it. The answer was no. Then I asked if all the natural numbers were in the set of sets, and the poster asnwered yes.

No, you can't be accurately restating your question. Of course there is a set with all the natural numbers in it -- namely, the set of natural numbers, or the set of integers, or the set of reals. I presume you were asking about the specific set you mentioned. Am I wrong?

If he said that the set {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...} contains the natural numbers, the answer is certainly no. But surely you asked a different question than you claim. The precise wording matters!

The poster said yes but not as elements.

That's an entirely different thing. At the least, it's misleading. It's nonsense to be an element, but not as an element!

My question to you is whether or not we can say that there are natural numbers contained in the set of sets.

If by "the set of sets" you mean {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}, the answer, once again, is NO. No natural number is an element of that set. Your phrase "contained in" is not well-defined.

The image doesn't help, as I can't see the original question and what "set 2" refers to.

Taking the link (which still doesn't show me everything), I see that you actually asked "does set 3 have all natural numbers in it (but not as elements)?", and he answered, "Yes, every natural number is in one of the sets that are in Set 3". So it is you that is using vague terms, and he restated your question in a way that makes sense.

 

[Mates]

"Yes, every natural number is in one of the sets that are in Set 3". So it is you that is using vague terms, and he restated your question in a way that makes sense.

Okay, I see.

My issue remains though. How can every natural number be in a set, but there is no set of all natural numbers int he set of sets?

 

[Dr.Peterson]

Okay, I see.

My issue remains though. How can every natural number be in a set, but there is no set of all natural numbers int he set of sets?

As I understand it, you are asking why the set {1, 2, 3, ...} is not an element of the set {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}.

That's because every element of the latter set is a finite set (the natural numbers less than or equal to some number n), and the set of all natural numbers is an infinite set. It simply isn't one of those finite sets.

There is no "infinitieth" set. Infinity is not a natural number.

 

[Mates]

As I understand it, you are asking why the set {1, 2, 3, ...} is not an element of the set {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}.

That's because every element of the latter set is a finite set (the natural numbers less than or equal to some number n), and the set of all natural numbers is an infinite set. It simply isn't one of those finite sets.

There is no "infinitieth" set. Infinity is not a natural number.

Then the term "all n" is a little ambiguous for me. "All n", as I understand it, means that we have exhausted every single n in the set of all natural numbers. If that is true, then there are no more natural numbers to consider. If there are no more natural numbers to consider, then there is no "next natural number" after n. In this sense, I would think that the set of all natural numbers should be in the set of sets.

 

[Dr.Peterson]

Then the term "all n" is a little ambiguous for me. "All n", as I understand it, means that we have exhausted every single n in the set of all natural numbers. If that is true, then there are no more natural numbers to consider. If there are no more natural numbers to consider, then there is no "next natural number" after n. In this sense, I would think that the set of all natural numbers should be in the set of sets.

You are just confused by the concept of infinity, which is normal for us finite people.

Infinite sets do not behave like finite sets. There is no last term, after which there is no next one. They can't be exhausted. "All" in this case doesn't mean we've reached an end, only that we are not excluding anything.

As I said, the elements of the set you are asking about are all finite sets. The set of all natural numbers is not one of them, because it is infinite. It has no last element, as all those finite sets do. And just as the set of natural numbers contains no last element, your set of finite sets has no last element; it is infinite, too.

You just have to learn to live with this.

By the way, the word "infinite" means "endless", "without an end". There is "no finish". If you even try to think about an end to it, you are contradicting yourself.

 

[Mates]

You are just confused by the concept of infinity, which is normal for us finite people.

Infinite sets do not behave like finite sets. There is no last term, after which there is no next one. They can't be exhausted. "All" in this case doesn't mean we've reached an end, only that we are not excluding anything.

As I said, the elements of the set you are asking about are all finite sets. The set of all natural numbers is not one of them, because it is infinite. It has no last element, as all those finite sets do. And just as the set of natural numbers contains no last element, your set of finite sets has no last element; it is infinite, too.

You just have to learn to live with this.

By the way, the word "infinite" means "endless", "without an end". There is "no finish". If you even try to think about an end to it, you are contradicting yourself.

I am very grateful for your help, but something still feels wrong. I have this ever-nagging feeling that "they" have something a little wrong with this part of math. The logic is not quite convincing like it is in every other math subject that I have ever come across.

That said, I realize that it is likely me that is still missing something that I need to be at ease with this.

 

[Steven G]

You need to think about this for awhile until you agree with what Dr Peterson has been saying. It really is true that every member of this set is itself a set of finite elements. The natural numbers is not one of them (it's infinite) and can't be in the set you are considering.

Whatever you do, just don't accept it. Think about it until you are satisfied with the conclusion.

 

[Mates]

You need to think about this for awhile until you agree with what Dr Peterson has been saying. It really is true that every member of this set is itself a set of finite elements. The natural numbers is not one of them (it's infinite) and can't be in the set you are considering.

Whatever you do, just don't accept it. Think about it until you are satisfied with the conclusion.

One thing that I am frustrated with is why n is limited to some element in the nth set. If there is no n being left out, why do we have to stop at some n, which leaves out an infinite number of more n's. This seems like a direct contradiction to me.

 

[Dr.Peterson]

I am very grateful for your help, but something still feels wrong. I have this ever-nagging feeling that "they" have something a little wrong with this part of math. The logic is not quite convincing like it is in every other math subject that I have ever come across.

That said, I realize that it is likely me that is still missing something that I need to be at ease with this.

One thing you need to understand is that in math, we (or "they") make a definition, and follow where it leads; often what results is not an understanding of the real world, but a created "world" whose foundation is our definitions and axioms. Our real world is not flat, so planar geometry results in conclusions contrary to our experiences on a round earth (and maybe curved space). Our real world is finite, so a geometry that implies you can go forever in any direction is contrary to our experiences. In math, if something doesn't fit your experience, you just have to let yourself "live" in that mathematical world, knowing that it does not describe things you can actually experience.

One thing that I am frustrated with is why n is limited to some element in the nth set. If there is no n being left out, why do we have to stop at some n, which leaves out an infinite number of more n's. This seems like a direct contradiction to me.

We don't "stop at some n"! Each element of this set is a finite set that "stops at some n" because that's how you (or someone else?) defined the set. But the next element in the list stops at the next n, and so on. No n is being left out in the overall set.

Again, you need to focus on the definition, and not expect it to be something it is not. The definition says what is in the set, and that is a collection of finite sets, each of which consists of all natural numbers up to some natural number n.

If you think there's a contradiction, tell us exactly what that contradiction is! If it's just a feeling, set that feeling aside and follow the definition.

 

[Mates]

One thing you need to understand is that in math, we (or "they") make a definition, and follow where it leads; often what results is not an understanding of the real world, but a created "world" whose foundation is our definitions and axioms. Our real world is not flat, so planar geometry results in conclusions contrary to our experiences on a round earth (and maybe curved space). Our real world is finite, so a geometry that implies you can go forever in any direction is contrary to our experiences. In math, if something doesn't fit your experience, you just have to let yourself "live" in that mathematical world, knowing that it does not describe things you can actually experience.


We don't "stop at some n"! Each element of this set is a finite set that "stops at some n" because that's how you (or someone else?) defined the set. But the next element in the list stops at the next n, and so on. No n is being left out in the overall set.

Again, you need to focus on the definition, and not expect it to be something it is not. The definition says what is in the set, and that is a collection of finite sets, each of which consists of all natural numbers up to some natural number n.

If you think there's a contradiction, tell us exactly what that contradiction is! If it's just a feeling, set that feeling aside and follow the definition.

Yes, I only want to stay within the mathematical definitions, and leave the real world completely out of it.

Here is the part that I think is a contradiction. Every n that exists in the set of sets must be in some set. If all n are truly in that set of sets, then there has to be a set that contains all of these n's in the set of sets. The set of all natural numbers would have to be there.

I don't understand how we can get around this issue.

 

[fresh_42]

A set of sets is always problematic. An element [imath] n\in \mathbb{N} [/imath] is not an element of [imath]S:= \{\{1\},\{1,2\},\{1,2,3\},\ldots\} [/imath] since [imath] S [/imath] contains only sets, and [imath] n [/imath] isn't a set. [imath] \{n\} [/imath] isn't an element of [imath] S [/imath] either, except when [imath] n=1. [/imath] All elements of [imath] S [/imath] are finite sets, so [imath] \{N\} \not\in S [/imath] because it is infinte.

All you can say is, that [imath] n [/imath] is an element of infinitely many sets that constitute the elements of [imath] S [/imath]. As I said, sets of sets are very special.

 

[blamocur]

If all n are truly in that set of sets, then there has to be a set that contains all of these n's in the set of sets.

This is not correct. Which definitions/axioms/theorems make you think that?

 

[Dr.Peterson]

Here is the part that I think is a contradiction. Every n that exists in the set of sets must be in some set. If all n are truly in that set of sets, then there has to be a set that contains all of these n's in the set of sets. The set of all natural numbers would have to be there.

I've been talking around an ambiguity in your language, trying to clarify what you are referring to. Now I need to be very specific.

When you say "the set of sets" I have presumed (hoped?) that you meant, as I said here, one particular set of sets:

I presume you were asking about the specific set you mentioned.

If by "the set of sets" you mean {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}, the answer, once again, is NO. No natural number is an element of that set.

You never said that assumption was wrong. Since you initially asked about this particular set of sets, our answers have been about that. We've taken you at your word.

But here I think you are not talking about that particular set, but about the set of all sets. Or else you are confusing the two, and thinking that they are the same. I say that because it is utter nonsense to say "Every n that exists in [this particular] set of sets must be in some set," but it would be true (with a caveat) to say that "Every n that exists in the set of all sets must be in some set." (In either case, I'm taking your "in" to mean "an element of one of the sets constituting the big set", and not "an element of the big set"; the set's elements are sets, not numbers!)

So, please answer me directly: Are you thinking that {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...} is the set of all sets?? Or are you not really talking about this particular set of sets in the first place, but something else?

You talked from the start about the "bijection between each set and each natural number", though, as I said, that is irrelevant to the question as you asked it. Yes, there is a bijection between the set of natural numbers and {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}; but when we write {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...}, we are talking about that set exactly as written, not about any set in one-to-one correspondence with it. For example, we don't say that the set {A, B, C} is the same set as {1, 2, 3}. I'm wondering if you are making that mistake, and thinking of {{1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4} ...} as "the set containing any set with cardinality 1, and any set with cardinality 2, and so on -- which would be the set of all (non-empty, finite) sets.

But even this would not be the set of all sets, because it would still omit infinite sets!

Before we can solve your problem, we have to agree on exactly what you mean. That's part of the reason I emphasized definitions. You defined a particular set, but you don't seem to be talking about that set at all. Definitions and notation are taken very seriously in math; they are intended to mean exactly what they say, and no more. If you say one thing but mean another, then communication is impossible.

 

[Mates]

A set of sets is always problematic. An element [imath] n\in \mathbb{N} [/imath] is not an element of [imath]S:= \{\{1\},\{1,2\},\{1,2,3\},\ldots\} [/imath] since [imath] S [/imath] contains only sets, and [imath] n [/imath] isn't a set. [imath] \{n\} [/imath] isn't an element of [imath] S [/imath] either, except when [imath] n=1. [/imath] All elements of [imath] S [/imath] are finite sets, so [imath] \{N\} \not\in S [/imath] because it is infinte.

All you can say is, that [imath] n [/imath] is an element of infinitely many sets that constitute the elements of [imath] S [/imath]. As I said, sets of sets are very special.

Yes, what you say here here is what I meant from the start and how I understand it. I will try to use better wording though.

 

[Mates]

This is not correct. Which definitions/axioms/theorems make you think that?

I will try to explain why I think there is a contradiction using the correct concepts.

Every n (natural number) that exists in the set of sets (I am refering to the set in the OP; let's call it S) must be in some subset. If all n are elements of the subsets in S, then there can only be one of these subsets. It would be identical to the set of all natural numbers.