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(Note: Most of what I know about transfinite ordinals comes from this Wikipedia page.)

According to Wikipedia, each ordinal number is defined by the set of those smaller than it. That is, 0={}, 1={0}, 2={0,1}, 3={0,1,2}. However, doesn't this mean that ℕ (the set of natural numbers) is an ordinal, since it is equal to {0,1,2,3,4,5,6,...}? Since Wikipedia says the smallest infinite ordinal is ω, does that mean ℕ = ω ? If it is, then why is the ω symbol used at all if it's just ℕ ? If not, then what did I do wrong?

Wikipedia also says that ω is the "order type" of ℕ. I don't know what that means; is it related to my question?

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    $\begingroup$ When when one writes $\Bbb N$, one might be thinking about the natural numbers as a set with addition. When one writes $\omega$, one might be thinking about the set of natural numbers as a model for the first infinite ordinal. Ordinals are well ordered, and the order we give $\omega$ is the usual order you know. You will also bump into $\aleph_0$, in which case we think about cardinals. Ordinals are special kind of cardinals. You can pick a book (instead of reading things off Wikipedia) and you'll find lots of details. The point I'm trying to make is that the underlying set might (...) $\endgroup$ – Pedro Tamaroff Mar 7 '15 at 3:18
  • $\begingroup$ (...) be the same, but the object itself carries more structure, and we want to stress this out with notation. $\endgroup$ – Pedro Tamaroff Mar 7 '15 at 3:20
  • $\begingroup$ It is possible to construct a set which has the same "order type" as $\Bbb N$, for example any set $\{A_i : i \in \Bbb N\}$ under the order $A_i < A_j \iff i < j$. The point being here, is that the "$A_i$" do not need to be numbers, they might refer to any kind of object (say, letters in an infinite alphabet, for example). Ordinal numbers carry an idea of "place" rather than "amount", in the same way that "second" is different than "two". $\endgroup$ – David Wheeler Mar 7 '15 at 3:38
  • $\begingroup$ Observe that $0\in\omega.$ Thus your equation $\mathbb N=\omega$ only works if you consider $0$ to be a natural number. It doesn't work if, like most people, you consider $1$ to be the first natural number. $\endgroup$ – bof Mar 7 '15 at 5:37
  • $\begingroup$ math.stackexchange.com/questions/150575/… $\endgroup$ – Asaf Karagila Mar 7 '15 at 8:28
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Yes and no.

There's one sense in which the answer is unambiguously yes: Within the most popular scheme for embedding all of ordinary mathematics inside axiomatic set theory, the set that represents the set of natural numbers is identical to the set that represents the first infinite ordinal -- and by the way also identical to the set that represents the cardinal number $\aleph_0$.

Set theorists usually work in a context where this formalization is taken for granted, so they tend to say that $\mathbb N$ is the ordinal $\omega$. That works well for them. (They rarely use the symbol $\mathbb N$ anyway, preferring $\omega$ even in contexts where other mathematicians would use $\mathbb N$).


The arguments for an answer of no are a bit softer and more philosophical. But a good argument can be made that in everyday (rigorous but not formalized) mathematics, neither the natural numbers nor the ordinal numbers "are really" sets.

The number $2$, for example, is a separate idea from a set: namely the property shared by all collections that "contain one more than one thing". Ordinary mathematics allows these ideas -- which are not themselves sets -- to be collected as a set, which we call $\mathbb N$. And the ordinal number $\omega$ is not a set, but instead the abstract idea of the "thing" that is common about all infinite well-ordered sets that have no infinite proper prefixes.

With this "everyday" view $\mathbb N$ and $\omega$ are not the same thing, and are not even the same kind of thing.

This view is by no means shared by all "everyday mathematicians", though. Many seem to accept at least in principle to consider the set-theoretic formalization to be "what is really going on" -- but they still usually write things as if $\mathbb N$ and $\omega$ are different things, considering this distinction a helpful hint to the reader about which way to think about the thing in question is most relevant in each case.


Note that it is somewhat rare in "everyday mathematics" -- that is, outside axiomatic set theory -- to speak about ordinals at all. So a quick first-approximation rule of thumb could be to write $\omega$ whenever you writing for set theorists and $\mathbb N$ whenever you're writing everyday mathematics; then you will usually not go wrong.

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The finite ordinal numbers, the finite cardinal natural numbers are all use the same symbols. Natural numbers are used for counting finite things, cardinal numbers are used for measuring how many objects something can hold, and ordinals numbers are used for specifying which part of a list something is located at. The reason you need two different number systems when you expand to the infinite is because you can extend a list past the natural numbers and the list will still contain the same amount of objects as the natural numbers.

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The first limit ordinal, and the first infinite ordinal, namely $\omega$, is the set of ordinals smaller than it. This is therefore the set of finite ordinals: $\omega=\{0,1,....\}$, which is also known as $\Bbb N$ by many serious mathematicians. However, some mathematicians—number theorists in particular—find it more convenient to exclude zero from the natural numbers. This reduced set, which was considered as an abstraction in its own right long before the natural numbers became identified with the von Neumann ordinals, was first rigorously formalized by Peano and denoted $\bf N$. Modern treatments of Peano's axioms often start with zero and use the symbol $\Bbb N$. For the purposes of number theory (and most of the rest of mathematics) it doesn't matter what the natural numbers are in terms of sets. What does matter is whether zero is included or not. It might be convenient if we used $\Bbb N$ and $\bf N$ respectively for the natural numbers with and without zero, but unfortunately the notation has not settled.

It is unlikely that people doing research involving ordinal numbers will start using $\Bbb N$ for $\omega$, because it is a well-established convention to use Greek letters for ordinals.

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