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Why all the properties of a sequence or a series or a sequence of functions or a series of functions remain unchanged irrespective of which of $\mathbb N$ & $\omega$ we are using as an index set? Is it because $\mathbb N$ is equivalent to $\omega$?

Sugata Adhya
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    Actuallly $\Bbb N$ is equal to $\Bbb N\cup{0}$, in my book. It may be worth while to state explicitly that you are assuming the convention that $0\notin\Bbb N$ as you obviously are. – Marc van Leeuwen Dec 16 '12 at 10:31
  • @MarcvanLeeuwen That's always annoyed me. Every teacher I've ever had defined $0\notin\mathbb N$ and so I'm used to that, but we already have perfectly good notation for both sets: $\mathbb N$ for natural numbers and $\omega$ for whole numbers. I'm borrowing $\omega$ from set theory here (as the first limit ordinal), but it is conveniently equal to $\mathbb N\cup{0}$, and it looks like a W to boot. Also available alternative notation is $\mathbb Z^{\geq 0}=\omega$ and $\mathbb Z^+=\mathbb N$. – Mario Carneiro Dec 16 '12 at 10:48
  • @MarioCarneiro: The notation for the set of whole numbers is $\mathbb Z$. The set of natural numbers is $\omega=\mathbb N={0,1,2,3,\ldots}$. – hmakholm left over Monica Dec 16 '12 at 11:48
  • Also related http://math.stackexchange.com/questions/150575/what-is-the-difference-between-omega-and-mathbbn – Asaf Karagila Dec 16 '12 at 11:54
  • @HenningMakholm Clearly we're speaking past each other. For me, natural numbers = counting numbers = positive integers = $\mathbb N={1,2,\dots}$; whole numbers = nonnegative integers = $\omega={0,1,2,\dots}$; integers = $\mathbb Z={\dots,-1,0,1,\dots}$. I've changed my answer to make this convention explicit, and I recommend someone change the OP to make this clear as well (right now it says $\mathbb N$ vs. $\omega$, which is evidently notationally ambiguous). – Mario Carneiro Dec 16 '12 at 12:29
  • @Mario: My experience is the exact opposite: the natural numbers are the non-negative integers, denoted by $\Bbb N$ or, for the set-theoretically more sophisticated, $\omega$, and the positive integers are denoted by $\Bbb Z^+$. I was out of grad school before it really registered that some people use $\Bbb N$ to mean $\Bbb Z^+$, and I still find it unnatural (and, to be honest, mildly annoying, though I can’t say exactly why). – Brian M. Scott Dec 16 '12 at 18:40
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    @BrianM.Scott I would gladly switch my notation if I knew that everyone was on the other side, but this is just one of those bad notation peculiarities with no consensus. See http://en.wikipedia.org/wiki/Natural_number for an overview of the history and arguments for each side and a few unambiguous alternative notations. One clear one which I use sometimes is $\mathbb N-{0}$ vs. $\mathbb N\cup{0}$, since it doesn't need any new symbols and works with both systems. – Mario Carneiro Dec 16 '12 at 20:59

2 Answers2

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There is a trivial 1-1 correspondence from $\mathbb N=\{1,2,\dots\}$ to $\omega=\{0,1,2,\dots\}$ given by $f(n)=n-1$. This mapping is a bijection, an order isomorphism, an isometry, and preserves limits, in the sense that $f(n)\to\infty$ if $n\to\infty$ (although it would be hard not to satisfy this). Thus practically every property we care about is preserved when we switch from one index set to the other, almost to the point that we don't need to pay attention to which one we are working with.

Mario Carneiro
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It is because $\omega$ and $\mathbb N$ are just different names for the same set. Their members are the same, and so by the Axiom of Extensionality they are the same set.

hmakholm left over Monica
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