Boltzmannbrain

I think you are saying that we can't glean anything from the finite sets of the forementioned type (start at 1 and increase by 1), let's call type T, to the infinite sets of type T. Is that accurate?

A set that starts at 1, increases by 1 and has infinite elements has the properties of N. Isn't that sufficient to construct N?

I was referring to a set equal to N. I meant a jump in comparing the finite vs infinite sets that start at 1 and increase by 1.

This may be the best way that I can state my confusion. What I am trying to show is that the finite sets of natural numbers, that start at 1 and increase by 1, seem quite logical when comparing both sides (the number of elements versus the input n). For example 5 elements in the set implies an input of n = 5. Infinite elements implies an input of n = ??? How do we make this asymmetrical jump to an infinite number of elements with finite inputs??? I am not sure what you are saying that I don't agree with. My issue is really quite simple. Take a set of natural numbers that start at 1 and increase by 1. When this kind of set is finite, we know there must be a finite n that also equals the number of elements that it has. When this kind of set has an infinite number of elements, we know there must be an infinite n that also equals the number of elements that it has. That is where my logic is leading me (for better or worse). It leads me to believe that there must be an infinitely large n in the set of all natural numbers. It is perfectly symmetrical and perfectly proportional. Why should it even be wrong?

I just wanted to show how the symmetry breaks when going from R (finite) to N (infinite) Why can't I do this?

@Genady just went over this with me. What keeps me confused is how the symmetry below gets broken. Just going from some set R to the set of natural numbers N does not make sense to me. Let's continue to use each n∈N is mapped to set R(n)= {x∈N | 1≤x≤n} 1 element in R --> n = 1 2 elements in R --> n = 2 3 elements in R --> n =3 4 elements in R --> n =4 . . . infinite elements in N --> n = a finite number Or put more generally, finite --> finite finite --> finite finite --> finite . . . infinite --> finite How this symmetry between the left side and the right side gets broken is my main issue.

Okay, thanks for your patience. I forgot to take into account the set symbols. +1 My same issue still lingers if you care to continue. You say, "R(n) is finite. {R(n) | n∈N} is not". Is every R(n) finite in LIST?

Why does R(n) change to infinite when n is an element of N instead of just an n? This is not a facetious question. I think we have come to the absolute heart of my issue. I see what you are saying. There are certainly many reasons to explain away my issue. But there are still reasons that maintain my issue from being resolved.

I know. I was trying to make a point. I agree. But you wrote, "the set {R(n) | n∈N} is not finite" a few posts ago.

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?

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.

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.

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.

I understand up until here. I do not understand what you are saying here. Yes, I believe that this is a part of the heart of the problem. I understand Russel's paradox, but I not see how my issue is related. 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? 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? That would be nice. Clue to what? Oh thanks I did not know that. +1 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.

Yes 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.

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.

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

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.

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. 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?

By "they, I meant each set is finite. If what you mean by "they" is the amount of sets, then I agree, the amount of sets are infinite. This is the heart of my confusion. Somehow, it seems that the amount of sets can be infinite while each particular set is not (that is if we want every element in the set to be a finite natural number).

I mean the list in my OP. The sets are natural numbers that start at 1 and increase by 1. So I can only suppose they would all be finite (which is the crux of the issue).

I thought I just put in a very thorough review of our conversation from scratch. But we can move on if you want, and I will attempt to explain my confusion in a very brief and direct way. The confusion stems from my OP. Instead of asking whether or not the list has every set of natural numbers, what happens if I just define the list to have every set (starting from 1 and increasing by 1)? Can I do this? If I can, then what about your proof? I tried to explain how your proof does not seem to work for what I think is a directly analogous example.

My mistake, I did not look at your second proof close enough. My mind was going in a completely different direction than yours, apparently. So, I only assumed a different type of analogous proof. I hope I come across a lot more direct this time. Let's rewind a little. Your first proof shows that S = {x| x∈N & x≥1} cannot be listed since each row can be listed as L = {x| x∈N & x≥1 & x≤l} and L ≠ S. Or in other words, for any row in my original list, there is always at least one more row. Moreover, your proof also shows that the greatest element in each of the sets in my original list is its limiting factor that disallows S, from your proof, to exist in my list. These seem to be the principles of your proof that are relevant to explaining how my list does not have every set, and thus resolving my issue in the OP. The original list was: 1 {1} 2 {1, 2} 3 {1, 2, 3} 4 {1, 2, 3, 4} . . . Your proof made perfect sense to me, and I was happy ... but then I thought of an analogous situation. Since the relevant part of your proof really only concerns the nth set (lth to be exact) and the greatest element in each set in my list, I thought of a list that only shows the greatest element in each set. 1 {1} 2 {2} 3 {3} 4 {4} . . . So I hope you can see where I am going with this. If we use this same relevant principles from your original proof, even though it still seems totally sound, you say that every natural number is in the second list. I don't agree. The more I look into all of this the more strange and complicated it is.

I thought the proof (second proof) proved that every natural number cannot be listed alone in its own set.

I understand partial sums, and I think I understand the rest of what you are saying here. But I do not understand how this addresses the post you quoted. The list I made on page 2 is suppose to have every natural number as every row maps to every n in N. At least that's what I wanted. My main question is whether or not every natural number would be listed alone in its own set. Why would I think that the set of natural numbers would be there? I don't understand what you are getting at.