In a previous question, I asked whether or not there was such as thing as negative transfinite numbers, such as negative omega, -ω.
I received an answer that suggested ω* was the solution I had been looking for (i.e. the order type of the negative integers).
I then came across an article about surreal numbers, which posited that -ω was born on day ω and was equal to:
$$ \{|-1,-2,-3, ...\} $$
So my question is, does
$$ -\omega = \{|-1,-2,-3, ...\} = \omega^* $$
?
And why or why not?
They are totally different kinds of object; there is no standard meaningful sense in which they could be equal. One is an ordered set (or an isomorphism class of ordered sets) and the other is a surreal number. Asking whether they are equal is much like asking whether the set of $2\times 2$ matrices with coefficients in $\mathbb{R}$ is equal to the real number $2$.
Your confusion may come from the fact that $\omega$ "is" both an order-type and a surreal number. This is because there is a canonical and standard embedding of the ordinals into the surreal numbers, and so it is customary to identify certain surreal numbers with ordinals. But there is no similar canonical way to consider arbitrary order types as surreal numbers. You could identify co-well ordered sets as surreal numbers by identifying them with the negatives of surreal numbers corresponding with the opposite well-order, and in doing so you would identify $\omega^*$ with $-\omega$. However, there is no standard convention to do so, and so it would not be customary to say $\omega^*$ is "equal" to $-\omega$. Moreover, this convention is incompatible with the convention to identify well-ordered sets with ordinals, since finite co-well ordered sets are also well-ordered and so would get identified with two different surreal numbers.