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I am currently reading a book in set theory which defines $V$ the universe of sets thus:

(1) $V_0=\emptyset$

(2) $V_{\alpha +1} = \mathcal{P} (V_{\alpha})$

(3) If $\beta$ is a limit ordinal, then $V_\alpha = \bigcup \{V_\beta |\beta < \alpha\}$

Shouldn't that say "if $\alpha$ is a limit ordinal?"

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    Yes, it should. – 2017-01-13
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    That's an unusually interesting title for a math book. My favorite math books are "Calculus," "Topics in Algebra," "Topology," and "Algebraic Topology." This is why no one refers to math books by name. – 2017-01-14
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    If you insist on including the title, I believe that Hugh has been a Professor for over 20 years now. – 2017-01-14
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    Upon retrospect, it was inappropriate to call out a respected figure in the mathematical community over a minor (possibly typographical) error. I've edited my question to remove reference to the title. Thank you for the correction. – 2017-01-14

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Yes, it should be $\alpha$. This is clear because $\beta$ is a dummy variable. (I'm sure there's a more set theoretic term for it, but it eludes me).

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    You might be looking for "indexing variable" – 2017-01-13
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    @StellaBiderman [Bound variable](https://en.wikipedia.org/wiki/Free_variables_and_bound_variables)? – 2017-01-13
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    @bof no, bounded vs free variables have to do with quantification of propositions. $P(x)$ is free, but $\forall x P(x)$ is bound. It doesn't really apply here. – 2017-01-13
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    @StellaBiderman The symbol $$\bigcup_{x\in S}$$ is not on this list of [variable-binding operators](https://en.wikipedia.org/wiki/Free_variables_and_bound_variables#Variable-binding_operators) but it is analogous to $$\sum_{x\in S}.$$ At any rate, the use of the term "bound variable" is not limited to symbolic logic. – 2017-01-14
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    Interesting. I've only seen the term applies to $\forall$ and $\exists$ – 2017-01-14
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    @StellaBiderman I've seen "bound variable" used for any context where there is a symbol which binds a variable to a specific context. So in my book quantifiers count, but so do indexed operations, and even the set-builder notation "$\{x:$". – 2017-01-14
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    @NoahSchweber Good to know. This is probably a side effect of studying logic: I'm used to using the term in Model Theory when talking about quantifier elimination. – 2017-01-14
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    @StellaBiderman I mean, I also learned mine studying logic :). I think the point is that in first-order model theory, quantifiers are the *only* thing that exist to bind variables. But there are plenty of points in logic where we want to consider variables bound by more general operators. – 2017-01-14
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    Riiiiiight! Thanks :) – 2017-01-14