Boltzmannbrain

What do you mean by the set of natural numbers? Each number in the second list is alone.

I thought your new proof for the new list was saying that every natural number (in their own individual sets) could not be listed.

I thought that the new list had all natural numbers in it. I thought it would be a complete list of all natural numbers where each is contained in its own set.

Okay, I will use those terms for now on. Thanks for this +1 Oh, then according to your proof, the second list is not complete either. I thought that we both thought it was?

I give up. I can not see any relevant reason why your proof works for the first list but not the second list. Why do the extra elements change anything about your proof?

I am thoroughly confused and tired. I have to think about this more.

I don't understand why it has to modified at all.

Then when I try to apply your proof by contradiction argument to the new list, it doesn't seem to work anymore. And the relevant properties of this simple list are quite parallel to the properties in the list in my OP. This seems to be a problem. Okay, but a set with all natural numbers, namely the set N, there is no longer that equivalence from the number of rows to an n existing in the set that equals the number of rows (that we called greatest element in the finite sets). In other words, each n is finite and cannot match the infinite number of rows, so the equivalence breaks. In addition to my OP, that related issue is also what is driving me crazy. Yeah, I understand that. But I am not sure why you are posting this.

Each n has to finite, not infinite. That's what I mean. Ah, of course! +1

I meant to ask if every natural number exists in the list. This is a similar situation as in my OP.

Yeah, these are fun.

Okay, that makes sense. +1 I have been thinking about your argument earlier regarding whether or not all the sets contain the set N. What if I just put the last natural number in each set, so it would look like this. 1 {1} 2 {2} 3 {3} . . . n {n} . (Every n in N is mapped to a row) Then I ask the same sort of the same question. Does every natural number exist in the list of each set? This would seem like I should run into the same kind of problem as the argument you made. But it would seem contradictory or at least counterintuitive that every n is not in each set. I am quite confused. Oh interesting! I will have to read up on this. +1 I had always intended it to be infinite. I just did not use the proper notation. I don't think you are seeing my issue for infinite n. If you look at the list I made in the OP, you will see that everything is very "equivalent". Number of objects in each set = greatest element in each set = nth row (nth set). Looks good and everything is fine, until we use every n (where every n in N is mapped to a row (as Genady pointed out are proper terms that I want to convey)). Now the equalities create a problem. They seem to cause an n to exist that does not end. Infinite sets do not have a greatest element, but it also seems logical that an n exists with no end somewhere in a set in the list. And this is a problem of course because the naturals have to be finite.

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

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?

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

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

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

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.

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?

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.

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

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.

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?

That makes sense. It is what I was thinking. Okay, that makes sense. I only meant that my intuition tells me recently that there is a smallest real. I think this all starting to sink in. Okay, this makes sense too. I see. Thanks, I forgot about that. I was only discussing it with you. I forgot/misunderstood the integral process. This seems interesting to me in that adding one real number is significant geometrically, but I don't know right how relevant it is to this discussion. Thank you very much for your help and incredible patience Sorry, I really appreciate your help, but I wasn't making the connection with what you were saying and my issue. Also, I did not have enough time to figure it out. I think I have finally understood this whole issue from my OP completely. Thank you for your help!

I read what you said about neighborhoods and infinity. Then I looked up neighborhoods because I have never heard of them before. But I still have no idea how it helps me understand anything about my issue, or even how it relates to the discussion. I am not sure if I am even ready to get into the topic of neighborhoods. It seems a little or a lot more advanced than what I have learnt so far.