What's the definition of $\omega$?

Question

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$?

Are the following equivalent definition of $\omega$:

1. $\omega$ is the initial ordinal of $\aleph_0$.

2. $\omega$ is the least/first infinite ordinal.

3. $\omega$ is the set of all finite ordinals.

4. $\omega$ is the first non-zero limit ordinal

If yes, are there any more equivalent definitions, not on this list?

Answer 1

I'd like to summarise what I have learnt from this question:

Point (1) is circular since $\aleph_0$ is defined to be the cardinality of $\omega$.

Let's assume that we define $\omega$ to be the first ordinal of infinite cardinality. Then it must contain all finite ordinals since the ordinals are a linear order with respect to $\subseteq$. From this it is immediately clear that (2) and (3) are equivalent. It is similarly easy to see that (4) is equivalent to (3).

Answer 2

$\omega$ is defined to be the set of all finite ordinals.

This is provably equivalent to the assertion "the least infinite ordinal" or "the least limit ordinal" (note that $0$ is not a limit ordinal. It is $0$).

It can be stated as the smallest inductive set. Or the set of all ordinals whose rank is finite.

Indeed it is also defined to be $\aleph_0$.

Answer 3

I prefer to say: $\omega$ is the order type of the natural numbers with its usual order. All those others are theorems or common identifications.

Answer 4

Here is a definition that works even without the axiom of infinity, in which case $\omega$ can be a proper class. Namely, $\omega$ is the class of finite ordinals. An ordinal $\alpha$ is finite if $\alpha=0=\emptyset$ or $\alpha$ is a successor ordinal that has only $0$ and other succesor ordinals as predecessors.