I have to prove that if $\alpha$ is ordinal and $\beta \in \alpha$ then $\beta$ is oridinal. To do this I wanted to prove that for every $x\in\beta$, $x\in\alpha$.
This seems like it should be true, but given the definition of ordinal numbers I have in my textbook, it doesn't seem so. It states the following
"A set $x$ is an ordinal number if $(a, \in)$ is transitive and well-ordered". Firstly, I'm pretty sure that a well-order implies transitivity, but that's not the problem here.
I can construct a set like $\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$ which is well-ordered with the $\in$ relation, but $\{\emptyset\}\in\{\emptyset,\{\emptyset\}\}$ and $\{\emptyset,\{\emptyset\}\}\in\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$ and $\{\emptyset\}\notin\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$.
Is this definition wrong, or does this property really not hold for ordinals? Or maybe I'm just missing something.
Your definition is correct and here is the proof you are looking for: Let $\alpha$ be an ordinal and let $\beta \in \alpha$. We show that $\beta$ has the two properties you mentioned:
Well-ordering: Let $X$ be a non-empty subset of $\beta$. From the transitivity of $\alpha$, it follows that $X \subseteq \alpha$ and since $\alpha$ is well-ordered, $X$ must contain a least element. Now, we show that the order $\in$ is total in $\beta$. Assume that $x,y \in \beta$. By the transitivity of $\alpha$, we have $x,y \in \alpha$ and therefore $x=y$, $x\in y$ or $y \in x$. Therefore, $\beta$ is a well-order.
Transitivity: Assume $x \in y \in \beta$. Then $x \in y \in \alpha$ and hence $x \in \alpha$. Since $\alpha$ is a well-order we must have $x=\beta$, $\beta \in x$ or $x \in \beta$ but the first two possibilities lead to a contradiction since $\beta$ is a well-order. Hence $x \in \beta$.
$\textbf{Edit:}$ Here is an inductive definition with which you can produce some ordinal numbers $\alpha_0,\alpha_1,...$:
$$\alpha_0=\emptyset$$ $$\alpha_{n+1}=\alpha_n \cup \{\alpha_n\}$$
You can verify that all these $\alpha_i$ are indeed ordinals. However, note that the above definition does not generate all ordinals.
Some examples:
$$\alpha_0=\emptyset$$
$$\alpha_1=\{\emptyset\}$$
$$\alpha_2=\{\emptyset,\{\emptyset\}\}$$
$$\alpha_3=\{\emptyset,\{\emptyset\}, \{\emptyset,\{\emptyset\}\} \}$$