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Just a quick question:

If a = [A] and a belongs to N (set of all natural numbers) doesn't that mean that A is a subset of N?

The reason I'm asking this is because I'm trying to prove the theorem that the set of all natural numbers is closed under addition.

The definition of addition that I have is the following: Let a, b, c be natural numbers. We write a + b = c if and inly if there exists sets A and B such that a = [A], b = [B], A intersection B is nullset and c = [A union B]

I was thinking if a and b are natural numbers then A and B are subsets of N, and so is their union, which makes c also a natural number.

Thank you

user1563544
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  • $a=[A]$ or $a\in[A]$? – Asaf Karagila Mar 21 '14 at 00:29
  • a = [A] where a belongs to N – user1563544 Mar 21 '14 at 00:29
  • Okay, so I'm very unclear as to what is $[A]$ and what is $a$. Also, what exactly do you mean by the set of natural numbers? Is this is the context of set theory, where we take $\omega$? Or is this in a different context, where numbers are not treated as sets (not in particular anyway)? Or is it that $\Bbb N$ is in fact a subset of $\Bbb C$ when constructed via some canonical method from $\omega$? – Asaf Karagila Mar 21 '14 at 00:31
  • a is a equivalence set on A. Well, N = {1, 2, 3, ...} :) – user1563544 Mar 21 '14 at 00:33
  • What is an equivalence set? Do you mean an equivalence relation, the set of ordered pairs, or do you mean a partition of $A$? Or is this a particular equivalence class of $A$? Or is this ... what? – Asaf Karagila Mar 21 '14 at 00:35
  • You are not answering my questions. Is $1={\varnothing}$ and $2={1}$ and $3={1,2}$ and so on? If not, what does that mean for a set to be a natural number at all? – Asaf Karagila Mar 21 '14 at 00:36
  • Asaf, sorry I answered you wrong. I updated the question. – user1563544 Mar 21 '14 at 00:41
  • Now it's slightly better; but you still haven't answered my questions. What are natural numbers for you? And what do you mean by $[A]$? Do you mean $|A|$? The cardinality of $A$? – Asaf Karagila Mar 21 '14 at 00:43
  • These are the natural numbers N = {1, 2, 3...}. 1={∅} and 2={1} and 3={1,2} etc... I'm not sure to be honest about the latter, I read it from the book. – user1563544 Mar 21 '14 at 00:45
  • Perhaps it is better to be sure about your foundations, before setting to prove something which is very foundational (i.e. depends on your interpretation of certain notions into the mathematical universe). – Asaf Karagila Mar 21 '14 at 00:51
  • Wait, I think 1 = {a}, 2 = {a, b}, 3 = {a, b, c} etc – user1563544 Mar 21 '14 at 00:51
  • Yeah, I give up. You should first sit and understand the definition of a natural number before tackling the two questions that you have asked about them. As I said, and as comments to your previous question said, this is a foundational issue which depends on how you define the natural numbers (and to answer "Well, 1,2,3,..." is not a valid answer to this question. At all). – Asaf Karagila Mar 21 '14 at 00:52

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